dieudonné module theory, part ii: the classical theory...dieudonné module theory, part ii: the...
TRANSCRIPT
Dieudonneacute module theory part IIthe classical theory
Alan Koch
Agnes Scott College
May 24 2016
Alan Koch (Agnes Scott College) 1 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 2 41
Pick a prime p gt 2
Recall it is unreasonable to think that every finite abelian Hopf algebraH of p-power rank over an Fp-algebra R can be classified withDieudonneacute modules
Last time we focused on Hopf algebras with a nice coalgebrastructure
This time we will make assumptions on both R and H
The assumptions on R are very restrictive and the ones on H aremore palatable
Alan Koch (Agnes Scott College) 3 41
The situation du jour (et demain)
Throughout this talk let R = k be a perfect field of characterisic p
Any finite k -Hopf algebra H can be written as
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
whereHr r and Hlowast
r r are both reduced k -algebrasHr ℓ is a reduced and Hlowast
r ℓ is a local k -algebraHℓr is a local and Hlowast
r ℓ is a reduced k -algebraHℓℓ and Hlowast
ℓℓ are both local k -algebras
If p ∤ dimk H then H = Hr r
Alan Koch (Agnes Scott College) 4 41
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
Reduced k -Hopf algebras are ldquoclassified they correspond to finitegroups upon which Gal(kk) acts continuously
Then Hr r and Hr ℓ are classified and by duality so is Hℓr
Thus the most mysterious Hopf algebras are the ones of the form Hℓℓ
We will call these ldquolocal-localrdquo and assume that all Hopf algebras in thistalk are finite abelian and local-local over the perfect field k
If H is local-local then a result of Waterhouse is that H is a ldquotruncatedpolynomial algebrardquo ie
H sim= k [t1 tr ](tpn1
1 tpnr
r )
so the algebra structure isnrsquot terrible
Alan Koch (Agnes Scott College) 5 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 2 41
Pick a prime p gt 2
Recall it is unreasonable to think that every finite abelian Hopf algebraH of p-power rank over an Fp-algebra R can be classified withDieudonneacute modules
Last time we focused on Hopf algebras with a nice coalgebrastructure
This time we will make assumptions on both R and H
The assumptions on R are very restrictive and the ones on H aremore palatable
Alan Koch (Agnes Scott College) 3 41
The situation du jour (et demain)
Throughout this talk let R = k be a perfect field of characterisic p
Any finite k -Hopf algebra H can be written as
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
whereHr r and Hlowast
r r are both reduced k -algebrasHr ℓ is a reduced and Hlowast
r ℓ is a local k -algebraHℓr is a local and Hlowast
r ℓ is a reduced k -algebraHℓℓ and Hlowast
ℓℓ are both local k -algebras
If p ∤ dimk H then H = Hr r
Alan Koch (Agnes Scott College) 4 41
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
Reduced k -Hopf algebras are ldquoclassified they correspond to finitegroups upon which Gal(kk) acts continuously
Then Hr r and Hr ℓ are classified and by duality so is Hℓr
Thus the most mysterious Hopf algebras are the ones of the form Hℓℓ
We will call these ldquolocal-localrdquo and assume that all Hopf algebras in thistalk are finite abelian and local-local over the perfect field k
If H is local-local then a result of Waterhouse is that H is a ldquotruncatedpolynomial algebrardquo ie
H sim= k [t1 tr ](tpn1
1 tpnr
r )
so the algebra structure isnrsquot terrible
Alan Koch (Agnes Scott College) 5 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Pick a prime p gt 2
Recall it is unreasonable to think that every finite abelian Hopf algebraH of p-power rank over an Fp-algebra R can be classified withDieudonneacute modules
Last time we focused on Hopf algebras with a nice coalgebrastructure
This time we will make assumptions on both R and H
The assumptions on R are very restrictive and the ones on H aremore palatable
Alan Koch (Agnes Scott College) 3 41
The situation du jour (et demain)
Throughout this talk let R = k be a perfect field of characterisic p
Any finite k -Hopf algebra H can be written as
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
whereHr r and Hlowast
r r are both reduced k -algebrasHr ℓ is a reduced and Hlowast
r ℓ is a local k -algebraHℓr is a local and Hlowast
r ℓ is a reduced k -algebraHℓℓ and Hlowast
ℓℓ are both local k -algebras
If p ∤ dimk H then H = Hr r
Alan Koch (Agnes Scott College) 4 41
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
Reduced k -Hopf algebras are ldquoclassified they correspond to finitegroups upon which Gal(kk) acts continuously
Then Hr r and Hr ℓ are classified and by duality so is Hℓr
Thus the most mysterious Hopf algebras are the ones of the form Hℓℓ
We will call these ldquolocal-localrdquo and assume that all Hopf algebras in thistalk are finite abelian and local-local over the perfect field k
If H is local-local then a result of Waterhouse is that H is a ldquotruncatedpolynomial algebrardquo ie
H sim= k [t1 tr ](tpn1
1 tpnr
r )
so the algebra structure isnrsquot terrible
Alan Koch (Agnes Scott College) 5 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
The situation du jour (et demain)
Throughout this talk let R = k be a perfect field of characterisic p
Any finite k -Hopf algebra H can be written as
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
whereHr r and Hlowast
r r are both reduced k -algebrasHr ℓ is a reduced and Hlowast
r ℓ is a local k -algebraHℓr is a local and Hlowast
r ℓ is a reduced k -algebraHℓℓ and Hlowast
ℓℓ are both local k -algebras
If p ∤ dimk H then H = Hr r
Alan Koch (Agnes Scott College) 4 41
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
Reduced k -Hopf algebras are ldquoclassified they correspond to finitegroups upon which Gal(kk) acts continuously
Then Hr r and Hr ℓ are classified and by duality so is Hℓr
Thus the most mysterious Hopf algebras are the ones of the form Hℓℓ
We will call these ldquolocal-localrdquo and assume that all Hopf algebras in thistalk are finite abelian and local-local over the perfect field k
If H is local-local then a result of Waterhouse is that H is a ldquotruncatedpolynomial algebrardquo ie
H sim= k [t1 tr ](tpn1
1 tpnr
r )
so the algebra structure isnrsquot terrible
Alan Koch (Agnes Scott College) 5 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
H sim= Hr r otimes Hr ℓ otimes Hℓr otimes Hℓℓ
Reduced k -Hopf algebras are ldquoclassified they correspond to finitegroups upon which Gal(kk) acts continuously
Then Hr r and Hr ℓ are classified and by duality so is Hℓr
Thus the most mysterious Hopf algebras are the ones of the form Hℓℓ
We will call these ldquolocal-localrdquo and assume that all Hopf algebras in thistalk are finite abelian and local-local over the perfect field k
If H is local-local then a result of Waterhouse is that H is a ldquotruncatedpolynomial algebrardquo ie
H sim= k [t1 tr ](tpn1
1 tpnr
r )
so the algebra structure isnrsquot terrible
Alan Koch (Agnes Scott College) 5 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Suggested readings
M Demazure and P Gabriel Groupes Algeacutebriques - Tome 1Good news partially translated into English as Introduction toAlgebraic Geometry and Algebraic GroupsBad news not the part you need (chapter 5)A Grothendieck Groupes de Barsotti-Tate et Cristaux deDieudonneacuteExplicit connection between Dieudonneacute modules and HopfalgebrasT Oda The first de Rham cohomology group and Dieudonneacutemodules The section on Dieudonneacute modules gave me my firstexplicit examplesR Pink Finite group schemeshttpspeoplemathethzch pinkftpFGSCompleteNotespdfGood intro to Dieudonneacute modules with proofs (group schemepoint of view)
Alan Koch (Agnes Scott College) 6 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
The gist
What is a k -Hopf algebra
It is a k -algebra and a k -coalgebra with compatible structures
A Dieudonneacute module is a single module (over a ring TBD) whichencodes both structures
In Mondayrsquos talk the coalgebra structure was assumed and thealgebra structure was encoded by the action of F on Dlowast(H)
Alan Koch (Agnes Scott College) 7 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 8 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Witt polynomials
For each n isin Zge0 define Φn isin Z[Z0 Zn] by
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Note Φn depends on the choice of prime p which we assume to beldquoourrdquo p
Define polynomials SnPn isin Z[X0 XnY0 Yn] implicitly by
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Fact Yes the coefficients are all integers
Alan Koch (Agnes Scott College) 9 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Φn(Z0 Zn) = Z pn
0 + pZ pnminus1
1 + middot middot middot+ pnZn
Φn(S0 Sn) = Φn(X0 Xn) + Φn(Y0 Yn)
Φn(P0 Pn) = Φn(X0 Xn)Φn(Y0 Yn)
Example (low hanging fruit)
Φ0(Z0) = Z0
S0(X0Y0) = X0 + Y0
P0(X0Y0) = X0Y0
Exercise 1 Prove
S1((X0X1) (Y0Y1)) = X1 + Y1 minus1p
pminus1983142
i=1
983072pi
983073X i
0Y pminusi0
P1((X0X1) (Y0Y1)) = X p0 Y1 + X p
1 Y0 + pX1Y1
Alan Koch (Agnes Scott College) 10 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Let W (Z) = (w0w1w2 ) wi isin Z for all i Define addition andmultiplication on W (Z) by
(w0w1 ) +W (x0 x1 ) = (S0(w0 x0)S1((w0w1) (x0 x1)) )
(w0w1 ) middotW (x0 x1 ) = (P0(w0 x0)P1((w0 x1) (w0 x1)) )
These operations make W (Z) into a commutative ring
Actually W (minus) is a Z-ring scheme meaning that for all Z-algebras AW (A) is a ring
In particular set W = W (k) and call this the ring of Witt vectors withcoefficients in k (or ldquoWitt vectorsrdquo for short)
Alan Koch (Agnes Scott College) 11 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Properties of Witt vectors Show each of the following
Exercise 2 W is a ring of characteristic zero
Exercise 3 W is an integral domain
Exercise 4 (1 0 0 ) is the multiplicative identity of W
Exercise 5 W (Fp) sim= Zp
Exercise 6 W (Fpn) is the unramified extension of Zp of degree n
Exercise 7 The element (0 1 0 0 ) isin W acts as mult by p
Exercise 8 pnW is an ideal of W
Exercise 9 Let Wn = (w0w1 wnminus1) Then Wn is a ring withoperations induced from W
Exercise 10 WpnW sim= Wn In particular W0sim= k
Alan Koch (Agnes Scott College) 12 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Of particular importance will be two operators F V on W given by
F (w0w1w2 ) = (wp0 w
p1 w
p2 )
V (w0w1w2 ) = (0w0w1 )
F is called the Frobenius and V is called the VerschiebungShow each of the following
Exercise 11 FV = VF = p (multiplication by p)
Exercise 12 F V both act freely on W only F acts transitively
Exercise 13 If w isin pnW then Fw Vw isin pnW Thus F and V makesense on Wn as well
Exercise 14 Wn is annihilated by V n+1
Exercise 15 Any w = (w0w1w2 ) isin W decomposes as
(w0w1w2 ) =infin983142
i=0
pi(wpminusi
i 0 0 )
Alan Koch (Agnes Scott College) 13 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 14 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
The Dieudonneacute ring
Now let wσ be the Frobenius on w isin W (invertible since k is perfect)
Let E = W [F V ] be the ring of polynomials with
FV = VF = p Fw = wσF wV = Vwσw isin W
We call E the Dieudonneacute ring
Note that E is commutative if and only if k = Fp
Alan Koch (Agnes Scott College) 15 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
One more construction
Let us view Wn as a group scheme (as opposed to the ring Wn(k))
Let W mn be the mth Frobenius kernel of the group scheme Wn ie for
any k -algebra A
W mn (A) = (a0 a1 anminus1) ai isin A apm
i = 0 for all 0 le i le n minus 1
Some properties of W mn
Clearly F Wn rarr Wn restricts to F W mn rarr W m
n Also V Wn rarr Wn restricts to V W m
n rarr W mn
For each k -algebra A W (k) acts on W (A) through the algebrastructure map k rarr AEvery local-local Hopf algebra represents a subgroup scheme ofsome (W m
n )r
Alan Koch (Agnes Scott College) 16 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
There are maps ι W mn rarr W m+1
n (inclusion) and ν W mn rarr W m
n+1(induced by V ) such that
W mn
ιminusminusminusminusrarr W m+1n983117983117983175ν ν
983117983117983175
W mn+1
ιminusminusminusminusrarr W m+1n+1
Then W mn is a direct system (where the partial ordering
(nm) le (nprimemprime) is given by the conditions n le nprime m le mprime)
Let983153W = limminusrarr
mnW m
n
(Not standard notation)
Alan Koch (Agnes Scott College) 17 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Let H be a local-local Hopf algebra Define
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
The actions of F and V on 983153W induces actions on Dlowast(H) as does theaction of W (k)
Thus Dlowast(H) is an E-module
Furthermore Dlowast(minus) is a covariant functor despite the fact this isreferred to as ldquocontravariant Dieudonneacute module theoryrdquo
Alan Koch (Agnes Scott College) 18 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Properties of DlowastDlowast(H) is killed by some power of F and V Dlowast(H) is finite length as a W -modulelengthW Dlowast(H) = logp dimk HDlowast(H1 otimes H2) sim= Dlowast(H1)times Dlowast(H2)Note Spec(H1 otimes H2) sim= Spec(H1)times Spec(H2)
Theorem (Main result in Dieudonneacute module theory)Dlowast induces a categorical equivalence
983110983114
983112
p- power ranklocal-local
k-Hopf algebras
983111983115
983113 minusrarr
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113
We call E-modules satisfying the above Dieudonneacute modules
Alan Koch (Agnes Scott College) 19 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
This is not everyonersquos definition
Dieudonneacute modules can also be used to describep-power rank Hopf algebras (local separable)p-divisible groupsgraded Hopf algebrasprimitively generated Hopf algebrasa slew of other things
Alan Koch (Agnes Scott College) 20 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Another legal disclaimer
983110983114
983112
E-modulesof finite length over W (k)
killed by a power of F and V
983111983115
983113 = ldquoDieudonneacute modulesrdquo
Some literature will describe a Dieudonneacute module as a triple (MF V )where
M is a finite length W (k)-moduleF M rarr M is a nilpotent σ-semilinear mapV M rarr M is a nilpotent σminus1-semilinear map
This is just a different way to describe the same thing
Alan Koch (Agnes Scott College) 21 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
A Hopf algebra interpretation
Dlowast(H) = Homk -gr983054983153W Spec(H)
983055
Each W mn is an affine group scheme represented by
Hmn = k [t0 tnminus1](tpm
1 tpm
2 tpm
nminus1)
with comultiplications induced from Witt vector addition
∆(ti) = Si((t0 otimes 1 ti otimes 1) (1 otimes t0 1 otimes ti))
In theory we could possibly write something like
Dlowast(H) = Homk -Hopf(H limlarrminusmn
Hmn)
However I have never found this to be helpful
Alan Koch (Agnes Scott College) 22 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
No wait come back
Suppose we are given a Dieudonneacute module What is thecorresponding Hopf algebra
Suppose V N+1M = 0 Let H = k [Tm m isin M] subject to therelations
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
T(w0w1 )m = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
mm1m2 isin M (w0w1 ) isin W
Define ∆ H rarr H otimes H by
∆(Tm) = SN((TV Nmotimes1TV Nminus1motimes1 Tmotimes1) (1otimesTV Nm 1otimesTm))
Then H is a local-local k -Hopf algebra
Alan Koch (Agnes Scott College) 23 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
The gist part 2
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
The action of F describes the (interesting) algebra structure
The action of V describes the coalgebra structure
Exercise 16 Show that the N above need not be minimal
Exercise 17 Let M be a Dieudonneacute module Show that T0 = 0
Alan Koch (Agnes Scott College) 24 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 25 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
TFm = (Tm)p
Tm1+m2 = SN((TV Nm1 TVm1 Tm1) (TV Nm2
TVm2 Tm2)
Twm = PN((wpminusN
0 wpminusN
N ) (TV Nm TVmTm)
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
ExampleThe simplest possible (nontrivial) E-module is a k -vector space M ofdimension 1 with F and V acting triviallyIn other words M = EE(F V ) In particular N = 1 Let M havek -basis x and let H be the Hopf algebra with Dlowast(H) = M
Exercise 18 Prove H is generated as a k -algebra by t where t = Tx
Sincetp =
983054Tx
983055p= TFx = T0 = 0
we have H = k [t ](tp) Also by the formulas above t is primitive
Alan Koch (Agnes Scott College) 26 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Primitively generated flashback
Suppose H is primitively generated (and local-local) and letM = Dlowast(H)
Since
∆(Tm) = SN((TV Nm otimes 1 Tm otimes 1) (1 otimes TV Nm 1 otimes Tm))
it follows that we may take N = 0
Thus VM = 0 hence M can be viewed as a module over k [F ] with
TFm = (Tm)p
Tm1+m2 = Tm1 + Tm2
Twm = w0Tm
In this case itrsquos the same module as yesterday
Alan Koch (Agnes Scott College) 27 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
ExampleLet M = E(F mV n) = Dlowast(H)
Exercise 19 What is pM
Exercise 20 Exhibit a k -basis for M
Exercise 21 Prove that
H = k [t1 tn](tpm
1 tpm
n )
Exercise 22 Write out the comultiplication (in terms of Witt vectoraddition)
Exercise 23 Show that Spec(H) = W mn
Alan Koch (Agnes Scott College) 28 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Let H = k [t ](tp5) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp3i otimes tp3(pminusi)1
What is Dlowast(H)Let M = Dlowast(H) and suppose there is an x isin M such that t = Tx Then
0 = tp5=
983054Tx
983055p5= TF 5x
so F 5x = 0 from which it follows that F 5M = 0 but F 4M ∕= 0 Also
∆(t) = S1983054((Tx)
p3 otimes 1Tx otimes 1) (1 otimes (Tx)p3 1 otimes Tx)
983055
= S1983054TF 3x otimes 1Tx otimes 1) (1 otimes TF 3x 1 otimes Tx)
Then N = 1 (ie V 2M = 0) and Vm = F 3m Thus
M sim= EE(F 5F 3 minus V V 2) = EE(F 5F 3 minus V )
This can be confirmed by working backwardsAlan Koch (Agnes Scott College) 29 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Let H = k [t1 t2](tp2
1 tp2
2 ) with
∆(t1) = t1 otimes 1 + 1 otimes t1
∆(t1) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
What is Dlowast(H)Let M = Dlowast(H) and suppose x y isin M satisfy t1 = Tx t2 = Ty Clearly F 2x = F 2y = 0 and
∆(t1) = S0(Tx otimes 1 1 otimes Tx)
= S1((0 otimes 1Tx otimes 1) (1 otimes 0 1 otimes Tx))
∆(t2) = S1983054((Tx)
p otimes 1Ty otimes 1) (1 otimes (Tx)p 1 otimes Ty )
983055
= S1983054(TFx otimes 1Ty otimes 1) (1 otimes TFx 1 otimes Ty )
983055
Thus Vx = 0 and Vy = Fx So
M = Ex + Ey F 2M = V 2M = 0Vx = 0Fx = Vy
Alan Koch (Agnes Scott College) 30 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Exercises
Write out the Hopf algebra for each of the following Be as explicit asyou can
Exercise 24 M = E(F 2F minus V )
Exercise 25 M = EE(F 2V 2)
Exercise 26 M = EE(F 2 pV 2)
Exercise 27M = Ex + Ey F 4x = 0F 3y = 0Vx = F 2y V 2x = 0Vy = 0
Alan Koch (Agnes Scott College) 31 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
More exercises
Find the Dieudonneacute module for each of the following
Exercise 28 H = k [t ](tp4) t primitive
Exercise 29 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tpi1 otimes tp(pminusi)
1
Exercise 30 H = k [t1 t2](tp3
1 tp2
2 ) t1 primitive and
∆(t2) = t2 otimes 1 + 1 otimes t2 +pminus1983142
i=1
1i(p minus i)
tp2i1 otimes tp2(pminusi)
1
Alan Koch (Agnes Scott College) 32 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 33 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
How does duality work
Very nicely with the proper definitions
For an E-module M we define its dual Mlowast to be
Mlowast = HomW (MW [pminus1]W )
This is an E-module where F and V act on φ M rarr W [pminus1]W asfollows
(Fφ)(m) = (φ(Vm))σ
(Vφ)(m) = (φ(Fm))σminus1
Note The σ σminus1 make Fφ Vφ into maps which are W -linear
Alan Koch (Agnes Scott College) 34 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Mlowast = HomW (MW [pminus1]W )
PropertiesIf M is a Dieudonneacute module so is Mlowast(EE(F nV m))lowast = EE(F mV n)ldquoDuality interchanges the roles of F and V rdquoDlowast(Hlowast) = (Dlowast(H))lowast
Alan Koch (Agnes Scott College) 35 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
An example
Let H = Fp[t ](tp3) with
∆(t) = t otimes 1 + 1 otimes t +pminus1983142
i=1
1i(p minus i)
tp2i otimes tp2(pminusi)
The choice of k = Fp is only for ease of notation
Then M = EE(F 3F 2 minus V ) = kx oplus kFx oplus kF 2x as W -modules
We wish to compute HomW (MW [pminus1]W ) = HomZp(MQpZp)
Alan Koch (Agnes Scott College) 36 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
HomZp(EE(F 3F 2 minus V )QpZp)
Note that any φ kx oplus kFx oplus kF 2x rarr QpZp is determined by theimages of x Fx and F 2x
Note also that V (Fx) = V (F 2x) = 0 which will be useful later
Since px = FVx = F (F 2x) = F 3x = 0
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
where φ isin Mlowast is given by
φ(x) = wφ(Fx) = y
φ(F 2x) = z
Alan Koch (Agnes Scott College) 37 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
φ(x) = w φ(Fx) = y φ(F 2x) = z
Mlowast sim= (w y z) isin (QpZp)3 pw py pz isin Zp
Now F acts on φ by
(Fφ)(x) = (φ(Vx))σ = (φ(F 2x))σ = zσ = z (Fφ)(y) = (Fφ(z)) = 0
and V acts on φ by
(Vφ)(x) = (φ(Fx))σminus1
= yσminus1= y
(Vφ)(Fx) = (φ(F 2x))σminus1
= zσminus1= z
(Vφ(z)) = 0
Thus F (w y z) = (z 0 0) V (w y z) = (y z 0)Clearly for all u = (w y z) we have V 3u = 0 and V 2u = FuIt follows that Mlowast = EE(V 3V 2 minus F )
Alan Koch (Agnes Scott College) 38 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
H = Fp[t ](tp3) Mlowast = EE(V 3V 2 minus F )
Exercise 31 Using Mlowast give the algebra structure for Hlowast (Hint Itrequires two generators)
Exercise 32 Using Mlowast give the coalgebra structure for Hlowast
Exercise 33 What changes if k ∕= Fp
Alan Koch (Agnes Scott College) 39 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
Outline
1 Overview
2 Witt Vectors
3 Dieudonneacute Modules
4 Some Examples
5 Duality
6 Want More
Alan Koch (Agnes Scott College) 40 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41
More is coming tomorrow
Thank you
Alan Koch (Agnes Scott College) 41 41