algebraic cycles, branes and de rham-wittstien101/talks/nagoya1.pdf · 2010-11-17 · algebraic...
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Algebraic Cycles, Branes and De Rham-Witt
Jan Stienstra
Nagoya, 23 November 2010
talk at workshopWitt vectors, Foliations and absolute De Rham cohomology
Algebraic Cycles: web of intersecting subvarieties of a variety.
Branes: boundary conditions for open strings in a manifold.
1
X smooth, projective algebraic variety over alg. closed field k.
Zd(X) := free abelian group on the set of irreducible
subvarieties of codimension d in X.
Bd(X) := subgroup of Zd(X) generated by cycles div(f),
the divisor of zeros and poles of a rational function f
on a codimension d− 1 subvariety of X.
codimension d algebraic cycle on X is an element of Zd(X).
rational equivalence:
α, β ∈ Zd(X) : α ∼ β ⇔ α− β ∈ Bd(X)
d-th Chow group of X
CHd(X) := Zd(X) /Bd(X)
2
Chow ring of X
CH•(X) =⊕
d≥0
CHd(X)
Product on CH•(X) given by intersecting algebraic cycles.
Do not use set theoretic unions and intersections!
Multiplicities are subtle!
3
K0(X) := Grothendieck group of the category of
locally free OX-modules of finite type.
K0(X) is a λ-ring, with λi-operation coming from the
ith exterior power operation on locally free sheaves.
Gr•K0(X) := graded ring associated with the
γ-filtration on K0(X).
K0(coh(X)) := Grothendieck group of the category of
coherent sheaves on X.
Gr•K0(coh(X)) := graded group associated with filtration on
K0(coh(X)) by codimension of support
of coherent sheaf on X.
4
For smooth X the natural map is an isomorphism:
K0(X)≃−→ K0(coh(X))
Grothendieck-Riemann-Roch:
For X smooth projective over a field there are natural homomor-
phisms of graded rings, which are isomorphisms modulo torsion
Gr•K0(X)≃−→ Gr•K0(coh(X))
≃←− CH•(X)
Grothendieck’s Chern character gives an isomorphism forevery d
GrdK0(X)⊗Q≃←− ξ ∈ K0(X)⊗Q | ψn(ξ) = nd ξ , ∀n
here ψn is the n-th Adams operation.
Remark: Quillen has defined higher algebraic K-groups Ki(X)(i ≥ 0) of the category of locally free sheaves of finite type onX. These also carry an action by Adams operations. The cor-
responding eigenspaces are the motivic cohomology groups
of X. Bloch has defined the higher Chow groups of X.
Upon tensoring with Q the higher Chow groups coincide withthe motivic cohomology groups.
5
Quillen’s higher algebraic K-theory of exact categories.
For commutative ring R with unit Kd(R) is the dth algebraic K-group of the category of finitely generated projectiveR-modules.
For scheme X let Kd,X denote the Zariski sheaf associated with
the pre-sheaf
affine open U = Spec(R) 7→ Kd(R)
Theorem: CHd(X) ≃ Hd(X,Kd,X)
For d = 1 this is in fact the classical formula
Pic(X) ≃ H1(X,O∗X)
For d = 2 this result is due to Bloch.For general d it was conjectured by Gersten and proven by
Quillen.
6
Question (Spencer Bloch, 1970’s):
Determine, for smooth projective X, the structure of the functor
augmented Artinian local k-algebras −→ Abelian groups
A 7→ Hm(X,Kn,X×A/X)
Here Kn,X×A/X is the Zariski sheaf associated with the pre-sheaf
affine open U = Spec(R) 7→ ker(Kn(R⊗k A)→ Kn(R))
Classical theory tells that for m = n = 1 this functor is pro-
representable by the formal group of the Picard variety of X.
Artin and Mazur have considered the case n = 1 and arbi-trary m. They gave conditions for when this functor is pro-
representable by a formal group.
e.g. n = 1 , m = 2 gives the formal Brauer group of X.
7
In 1976 Bloch proposed the case m = n = 2 to me as a researchproblem for my PhD thesis.
One PhD thesis and three post-doc jobs later I formulated theanswer as follows (published in Crelle 355 (1985)):
Theorem. Let X be a smooth projective variety over a perfect
field k of characteristic p > 0.Then there is a functorial homomorphism for every m and n:(Hm(X,WΩ•X)⊗R WΩn−1−•
A
)−→ Hm(X,Ks
n,X×A/X)
of which the kernel and the cokernel are both essentially 0.
Main ingredients in the proof of the theorem are
• detailed information about the De Rham-Witt complex givenin the works of Bloch, Deligne, Illusie and Raynaud
• an alternative construction of the De Rham-Witt complexbased on the K-theory of categories of finitely generated
projective modules equiped with an endomorphism.
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Theorem. Let X be a smooth projective variety over a perfectfield k of characteristic p > 0.Then there is a functorial homomorphism for every m and n:(Hm(X,WΩ•X)⊗R WΩn−1−•
A
)−→ Hm(X,Ks
n,X×A/X)
of which the kernel and the cokernel are both essentially 0.
• Ksn,X×A/X is the symbol part of Kn,X×A/X, i.e.
the image under multiplication in K-theory of
(1 +OX ⊗k mA)⊗Z O∗X×A ⊗Z . . .⊗Z O
∗X×A ;
mA is the maximal ideal of A.
• R = W (k)[F, V, d] is the Raynaud ring;W (k) the p-typical Witt vectors of k;F is Frobenius, V Verschiebung,
d the derivation in the De Rham-Witt complex of X:
WOXd→ WΩ1
Xd→ . . .
d→WΩi
Xd→ . . .
• Hm(X,WΩ•X) :=
Hm(X,WOX)d→ Hm(X,WΩ1
X)d→ . . .
d→ Hm(X,WΩi
X)d→ . . .
viewed as an R-module.
• WΩ•A is the formal De Rham-Witt complex of A.
• We call a functor G : augm.Art.loc.k-algs → Ab.grps
essentially zero if for every A there is a surjection A′ ։ Ainducing the zero map G(A′)→ G(A).
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The alternative construction of the De Rham-Witt complex works
for any commutative unital ring R, without assumptions
about the characteristic or about regularity.
End(R) denotes the exact category whose
• objects: pairs (M,α) consisting of a finitely generated pro-jective R-module M and an R-linear endomorphism α ofM
• morphisms: (M,α)→ (M ′, α′) is an R-linear mapf : M →M ′ s.t. fα = α′f .
• short exact sequences: underlying sequence of R-modulesis exact
Ki(End(R)), for i = 0, 1, 2, . . ., denote Quillen’s K-groups of the
exact category End(R).
10
K∗(End(R)) =⊕
i≥0
Ki(End(R))
is a graded commutative ring with product induced by tensorproduct
(M,α)⊗ (M ′, α′) = (M ⊗M ′, α⊗ α′)
There is a Frobenius operator Fn for every n ≥ 1 induced
by(M,α) 7→ (M,αn)
There is a Verschiebung operator Vn for every n ≥ 1 induced
by
(M,α) 7→ (M⊕n,
0 0 . . . 0 α
1 0 . . . 0 00 1 . . . 0 0... 0 . . . 0
...0 0 . . . 1 0
)
There is also a map
d : Ki(End(R)) → Ki+1(End(R)) ,
constructed as follows.
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Consider the polynomial ring in one variable Z[u] and the bi-exact functor
End(R) × End(Z[u]) −→ End(R)
(M,α) , (N, β) 7→ (M ⊗Z[u] N , 1⊗ β)
here M is considered as a Z[u]-module: um = αm, ∀m ∈M .
This induces maps
Ki(End(R)) ⊗Z Kj(End(Z[u])) −→ Ki+j(End(R)) .
The map
d : Ki(End(R)) → Ki+1(End(R)) ,
comes from a particular element in K1(End(Z[u])).
Remark. The above construction of the map d is a special caseof a general construction of functorial operations onK∗(End(R)).
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Relations:
V1 = F1 = 1
FnFm = Fnm , VnVm = Vnm (∀n,m)
FnVn = n1 (∀n)
VpFp = p1 if p prime and pR = 0
FnVm = VmFn if (n,m) = 1
Vn d = n d Vn , d Fn = nFn d (∀n)
Fn d Vn = d if n odd , F2 d V2 = d if 2R = 0
2 d2 = 0; d2 = 0 if 2R = 0
Fn(a b) = (Fn a) (Fn b) (∀n)
Vn(aFnb) = (Vna) b (∀n)
d(a b) = (da) b + (−1)ia (db)
for all a ∈ Ki(End(R)) and b ∈ Kj(End(R))
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Exact functors
End(R) −→ P (R) , (M,α) 7→ M
P (R) −→ End(R) , M 7→ (M, 0)
where P (R) = category of fin. gen. projective R-modules.
These induce direct sum decomposition:
Ki(End(R)) = Ki(R) ⊕ Ki(End(R))
with
Ki(End(R)) := ker(Ki(End(R))→ Ki(R))
Theorem(Almkvist, Grayson)
Let W(R) denote the ring of big Witt vectors of R.Then the map
K0(End(R)) −→ (1 + tR[[t]])× = W(R)
[M,α]− [M, 0] 7→ det(1− tα)−1
is an injective homomorphism of rings, which commutes withthe Frobenius and Verschiebung operators.
Moreover the Teichmuller lifting is given by
R → K0(End(R)) → W(R)
x 7→ [R, x]− [R, 0] 7→ (1− xt)−1
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The inverse map from the image of K0(End(R)) in W(R) backto K0(End(R)) is given by
(1−
n∑
j=1
rjtj
)−1
7→
R⊕n,
0 0 . . . 0 rn1 0 . . . 0 rn−1
0 1 . . . 0 rn−2... 0 . . . 0
...0 0 . . . 1 r1
− [R⊕n, 0]
We will describe a decreasing filtration by homogeneous ideals
FilnK∗(End(R)) , n = 1, 2, 3, . . .
such that the Frobenius maps Fm, Verschiebung maps Vm andthe derivation d extend to the completion
K∗(End(R))∧ := lim←n
K∗(End(R))/FilnK∗(End(R))
and such thatK0(End(R))∧ = W(R) .
15
For a commutative unital ring A let Nil(A) denote the full exactsubcategory of End(A) with objects those (M,α) for which α is
nilpotent.
The fundamental theorem of K-theory gives an isomor-phism for every i ≥ 0:
NKi+1(A) ≃ Ki(Nil(A))
where
NKi+1(A) := coker(Ki+1(A)→ Ki+1(A[u]))
Ki(Nil(A)) := ker(Ki(Nil(A))→ Ki(A))
for the maps induced by the inclusion A → A[u] and the forget-ful functor Nil(A)→ P (A), respectively.
Let s0 , s1 : Ki+1(A[u]) → Ki+1(A) denote the homorphisms
induced by the substitutions u 7→ 0 and u 7→ 1, respectively.Then s1 − s0 gives a well defined homorphism
s1 − s0 : NKi+1(A) −→ Ki+1(A) .
By composing the latter homomorphism with the isomorphism
from the fundamental theorem we obtain, for every i ≥ 0, ahomomorphism
∫ 1
0
: Ki(Nil(A)) −→ Ki+1(A) .
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The bi-exact functor
End(R) × Nil(A) −→ Nil(R⊗Z A)
(M,α) , (N, β) 7→ (M ⊗Z N , α⊗ β)
induces
Ki(End(R)) ⊗Z Kj(Nil(A)) −→ Ki+j(Nil(R⊗Z A))
a , b 7→ ab
and by composition with∫ 1
0 :
Ki(End(R)) ⊗Z Kj(Nil(A)) −→ Ki+j+1(R⊗Z A)
a , b 7→
∫ 1
0
ab
Remark: The above pairing (a, b) 7→∫ 1
0 ab underlies the map
(Hm(X,WΩ•X)⊗R WΩn−1−•
A
)−→ Hm(X,Ks
n,X×A/X)
in my theorem.
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Nil(A) carries an increasing filtration by full exact subcategories:
FilnNil(A) := (M,α) | αn = 0 .
This yields an increasing filtration on the K-groups:
FilnKj(Nil(A)) := image(Kj(FilnNil(A))→ Kj(Nil(A)))
Now define
FilnKi(End(R)) :=
a ∈ Ki(End(R))
∣∣∣∣∣∀A , ∀j , ∀b ∈ Kj(FilnNil(A))∫ 1
0 ab = 0 in Ki+j+1(R⊗Z A)
Ki(End(R))∧ := lim←n
Ki(End(R))/FilnKi(End(R))
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We define the generalized De Rham-Witt complex of R asthe completion of the smallest subring of K∗(End(R))∧ which
contains W(R) and is closed under the operations d, Fn, Vn.
The generalized formal De Rham-Witt complex of a com-mutative unital ring A is similarly defined from K∗(Nil(A)) and
the multiplicative group
W(A) := (1 + uAnil[u])× ⊂ NK1(A) = K0(Nil(A))
of polynomials with constant term 1 and all other coefficientsnilpotent in A. No need for completion.
If R and A are rings of prime characteristic p one can split off theDe Rham-Witt complex WΩ•R of R and the formal De Rham-
Witt complex WΩ•A of A from the above generalized forms bymeans of the idempotent operator (µ is the Mobius function):
∑
n, p∤n
µ(n)
nVnFn =
∏
ℓ prime 6=p
(1−
1
ℓVℓFℓ
)
19
In physics branes are boundary conditions for open strings ina target manifoldM.
One speaks of D-branes (D for Dirichlet) if the boundary con-
ditions specify the position of the endpoints of the string.
One speaks of D-branes of B-type if the boundary condi-tions require the string to end on a holomorphic submanifold inthe target manifoldM.
Chien-Hao Liu and Shing-Tung Yau are working on a projectto find structures in algebraic geometry to model D-branes of B-
type.
see e.g. their paperD-branes and Azumaya noncommutative geometry: From Polchin-ski to Grothendieck
Two key ideas:
• Polchinski: stacks of branes (≈ branes with multiplicities)
should be described by matrix valued functions onM.
• Grothendieck: instead of substructures ofM use structuresthat map intoM.
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Liu and Yau implement these ideas in what they call anAzumaya scheme with fundamental module.
The visible part of such a structure consists of a scheme X and
a locally free OX-module of finite type E .
The (as yet) invisible part of such a structure can only be probedthrough the set of morphisms from it to other schemes Y(acting as target spaces likeM).
Liu and Yau define these sets of morphisms as
Mor((X, E), Y ) :=
coherent OX×Y -modules E on
X × Y with p1∗E = E
They remark that it may suffice to “probe” the structure of theAzumaya scheme with fundamental module by taking Y from a
fixed (well chosen) collection of “basic spaces”.
They also say (parenthetically)
“Furthermore, from .......... , one expects that one finally has toconsider everything in the derived(-category) sense.”
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Comments and Questions.
• The assignment
Y 7→ Mor((X, E), Y )
is a covariant functor from the category of schemes to thecategory of sets.
One should compare this with standard constructions inmoduli problems where one uses a contravariant functor:
F : schemes→ sets
and shows that there is a scheme Z such that
F (Y ) = Mor(Y, Z) for all schemes Y .
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• What does it mean:has to consider everything in the derived(-category) sense?
Are X and Y fixed and can only E vary?
For fixed X and Y the E ’s form an exact category:
Brane(X; Y ) :=
coherent OX×Y -modules E on X × Y s.t.
p1∗E is locally free of finite type on X
Should one then take the corresponding derived category?
Or can one take its K-theory K∗(Brane(X; Y )) instead?
Should one then vary Y and consider for fixed X the co-variant functor
Y 7→ K∗(Brane(X; Y ))
on the category of schemes?
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Liu-Yau in affine context:
X = Spec(R)
Y = Spec(A)
E = finitely generated projective R-module M
ThenMor((X, E), Y ) = Homrings(A,EndR(M)) .
For fixed (R,M) the question becomes:
Study the contravariant functor:
commutative unital rings −→ sets
A 7→ Homrings(A,EndR(M))
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For fixed R and A we have the exact category
End(R;A) :=
pairs (M,ϕ)
∣∣∣∣M fin. gen. proj. R-mod.ϕ : A→ EndR(M) ring hom.
Example:
End(R; Z[u]) = End(R)
So there is a link with De Rham-Witt when Azumaya schemes
are “probed” by morphisms to the affine line.
A ring homomorphism f : A′ → A induces an exact functor
End(R;A) −→ End(R;A′) , (M,ϕ) 7→ (M,ϕf)
This makes the construction contravariantly functorial in A.
25
Recall the question Spencer Bloch asked in 1970’s:
Determine, for smooth projective X, the structure of the functor
augmented Artinian local k-algebras −→ Abelian groups
A 7→ Hm(X,Kn,X×A/X)
and notice behind the facade of K-theory, sheaves and coho-
mology on the affine level the exact category
fin. gen. proj. R ⊗k A-modules
Being an augmented Artinian local algebra over the field k thering A is a finite dimensional k-vector space.
Therefore every fin. gen. proj. R ⊗k A-module M is also
a fin. gen. proj. R-module with A-module structure given bya ring homomorphism ϕ : A→ EndR(M)
A ring homomorphism f : A′ → A induces an exact functor
f.g.pr. R ⊗k A′-mod.
−⊗R⊗kA′(R⊗kA)
−→ f.g.pr. R⊗k A-mod.
This makes the construction covariantly functorial in A.
26
As Liu and Yau remarked, it may suffice to “probe” the struc-ture of the Azumaya scheme with fundamental module by taking
Y from a fixed (well chosen) collection of “basic spaces”.
Do toric varieties constitute a good collection of “basic spaces”for that purpose?
Since toric varieties are locally modeled on commutative semi-groups this would mean in the affine context :
For a commutative unital ring R and and a commutative semi-
group S look at the exact category of representations of S infinitely generated projective R-modules:
Rep(R;S) :=
pairs (M,ϕ)
∣∣∣∣M fin. gen. proj. R-mod.ϕ : S → EndR(M) s.-grp hom.
Restricting to representations by nilpotent endomorphisms shouldprobably be the same as working with Artinian local algebras.
Question:
Working with Artinian local algebras is a kind of analysis.Are there other sensible ways to bring in (non-Archimedian)
analysis?
c.f. Payne’s work on analytification and tropicalization usinghomomorphisms from commutative semi-groups into the semi-
ring (R≥0 , max , ·).
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