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TRANSCRIPT
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The graded module category of a generalized Weyl algebraFinal Defense
Robert WonAdvised by: Daniel Rogalski
May 2, 2016 1 / 39
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Overview
1 Graded rings and things
2 Noncommutative is not commutative
3 Noncommutative projective schemes
4 Z-graded ringsThe first Weyl algebraGeneralized Weyl algebras
5 Future direction
May 2, 2016 2 / 39
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Graded k-algebras
• Throughout, k an algebraically closed field, chark = 0.• A k-algebra is a ring A with 1 which is a k-vector space such that
λ(ab) = (λa)b = a(λb) for λ ∈ k and a, b ∈ A.
Examples• The polynomial ring, k[x]• The ring of n× n matrices, Mn(k)• The complex numbers, C = R · 1 + R · i
Graded rings and things May 2, 2016 3 / 39
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Graded k-algebras
• Γ an abelian (semi)group• A Γ-graded k-algebra A has k-space decomposition
A =⊕γ∈Γ
Aγ
such that AγAδ ⊆ Aγ+δ.• If a ∈ Aγ , deg a = γ and a is homogeneous of degree γ.
Graded rings and things May 2, 2016 4 / 39
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Graded k-algebras
Examples• Any ring A with trivial grading• k[x, y] graded by N• k[x, x−1] graded by Z• If you like physics: superalgebras graded by Z/2Z• Or combinatorics: symmetric and alternating polynomials
graded by Z/2Z
RemarkUsually, “graded ring” means N-graded.
Graded rings and things May 2, 2016 5 / 39
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Graded modules and morphisms
• A graded right A-module M:
M =⊕i∈Z
Mi
such that Mi · Aj ⊆Mi+j.• Finitely generated graded right A-modules with graded module
homomorphisms form a category gr-A.
ExampleN-graded ring k[x, y]
• Homogeneous ideals (x), (x, x + y), (x2 − xy) are gradedsubmodules
• Nonhomogeneous ideals (x + 3), (x2 + y) are not
Graded rings and things May 2, 2016 6 / 39
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Graded modules and morphisms
• Idea: Study A by studying its module category, mod-A(analogous to studying a group by its representations).
• If mod-A ≡ mod-B, we say A and B are Morita equivalent.
Remarks• For commutative rings, Morita equivalent⇔ isomorphic• Mn(A) is Morita equivalent to A• Morita invariant properties of A: simple, semisimple,
noetherian, artinian, hereditary, etc.
Graded rings and things May 2, 2016 7 / 39
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Proj of a graded k-algebra
• A a commutative N-graded k-algebra• The scheme Proj A• As a set: homogeneous primes ideals of A not containing A≥0• Topological space: the Zariski topology with closed sets
V(a) = {p ∈ Proj A | a ⊆ p}
for homogeneous ideals a of A• Structure sheaf: localize at homogeneous prime ideals
Noncommutative is not commutative May 2, 2016 8 / 39
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Proj of a graded k-algebra
Interplay beteween algebra and geometry
algebra geometrygraded k-algebra A space Proj A
homogeneous prime ideals pointsI ⊆ J V(I) ⊇ V(J)
factor rings subschemeshomogeneous elements of A functions on Proj A
Noncommutative is not commutative May 2, 2016 9 / 39
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Noncommutative is not commutative
Noncommutative rings are ubiquitous.
Examples• Ring A, nonabelian group G, the group algebra A[G]• Mn(A), n× n matrices• Differential operators on k[t], generated by t· and d/dt is
isomorphic to the first Weyl algebra
A1 = k〈x, y〉/(xy− yx− 1).
If you like physics, position and momentum operators don’t commutein quantum mechanics.
Noncommutative is not commutative May 2, 2016 10 / 39
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Noncommutative is not commutative
• Localization is different.• Given A commutative and S ⊂ A multiplicatively closed,
a1s−11 a2s−12 = a1a2s
−11 s−12
• If A noncommutative, can only form AS−1 if S is an Ore set.
DefinitionS is an Ore set if for any a ∈ A, s ∈ S
sA ∩ aS 6= ∅.
Noncommutative is not commutative May 2, 2016 11 / 39
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Noncommutative is not commutativeEven worse, not enough (prime) ideals.
The Weyl algebra
k〈x, y〉/(xy− yx− 1)
is a noncommutative analogue of k[x, y] but is simple.
The quantum polynomial ringThe N-graded ring
k〈x, y〉/(xy− qyx)
is a “noncommutative P1” but for qn 6= 1 has only threehomogeneous ideals (namely (x), (y), and (x, y)).
Noncommutative is not commutative May 2, 2016 12 / 39
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Sheaves to the rescue
• The Beatles (paraphrased):
“All you need is sheaves.”
• Idea: You can reconstruct the space from the sheaves.
Theorem (Rosenberg, Gabriel, Gabber, Brandenburg)Let X,Y be quasi-separated schemes. If qcoh(X) ≡ qcoh(Y)then X and Y are isomorphic.
Noncommutative is not commutative May 2, 2016 13 / 39
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Sheaves to the rescue
• All you need is modules
TheoremLet X = Proj A for a commutative, f.g. k-algebra A generated indegree 1.
(1) Every coherent sheaf on X is isomorphic to M̃ for some f.g.graded A-module M.
(2) M̃ ∼= Ñ as sheaves if and only if there is an isomorphismM≥n ∼= N≥n.
• Let gr-A be the category of f.g. A-modules. Take the quotientcategory
qgr-A = gr-A/fdim-A
• The above says qgr-A ≡ coh(Proj A).
Noncommutative is not commutative May 2, 2016 14 / 39
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Noncommutative projective schemes
A (not necessarily commutative) connected graded k-algebra A is
A = k⊕ A1 ⊕ A2 ⊕ · · ·
such that dimk Ai
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Noncommutative curves
• “The Hartshorne approach”• A a k-algebra, V ⊆ A a k-subspace generating A spanned by{1, a1, . . . , am}.
• V0 = k, Vn spanned by monomials of length ≤ n in the ai.• The Gelfand-Kirillov dimension of A
GKdim A = lim supn→∞
logn(dimk Vn)
• GKdimk[x1, . . . , xm] = m.• So noncommutative projective curves should have GKdim 2.
Noncommutative projective schemes May 2, 2016 16 / 39
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Noncommutative curves
Theorem (Artin-Stafford, 1995)Let A be a f.g. connected graded domain generated in degree 1with GKdim(A) = 2. Then there exists a projective curve Xsuch that
qgr-A ≡ coh(X)
Or “noncommutative projective curves are commutative”.
Noncommutative projective schemes May 2, 2016 17 / 39
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Noncommutative surfaces
• The “right” definition of a noncommutative polynomial ring?
DefinitionA a f.g. connected graded k-algebra is Artin-Schelter regular if
(1) gl.dimA = d
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Noncommutative surfaces
Theorem (Artin-Tate-Van den Bergh, 1990)Let A be an AS regular ring of dimension 3 generated in degree 1.Either(a) qgr-A ≡ coh(X) for X = P2 or X = P1 × P1, or(b) A � B and qgr-B ≡ coh(E) for an elliptic curve E.
• Or “noncommutative P2s are either commutative or contain acommutative curve”.
• Other noncommutative surfaces (noetherian connected gradeddomains of GKdim 3)?
• Noncommutative P3 (AS-regular of dimension 4)?
Noncommutative projective schemes May 2, 2016 19 / 39
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The first Weyl algebra
• Throughout, “graded” really meant N-graded• Recall the first Weyl algebra:
A1 = k〈x, y〉/(xy− yx− 1)
• Simple noetherian domain of GK dim 2• A1 is not N-graded
Z-graded rings May 2, 2016 20 / 39
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The first Weyl algebra
A1 = k〈x, y〉/(xy− yx− 1)
• Ring of differential operators on k[t]• x←→ t·• y←→ d/dt
• A is Z-graded by deg x = 1, deg y = −1• (A1)0 ∼= k[xy] 6= k• Exists an outer automorphism ω, reversing the grading
ω(x) = y ω(y) = −x
Z-graded rings May 2, 2016 21 / 39
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Sierra (2009)
• Sue Sierra, Rings graded equivalent to the Weyl algebra• Classified all rings graded Morita equivalent to A1• Examined the graded module category gr-A1:
−3 −2 −1 0 1 2 3
• For each λ ∈ k \ Z, one simple module Mλ• For each n ∈ Z, two simple modules, X〈n〉 and Y〈n〉
X Y
• For each n, exists a nonsplit extension of X〈n〉 by Y〈n〉 and anonsplit extension of Y〈n〉 by X〈n〉
Z-graded rings May 2, 2016 22 / 39
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Sierra (2009)
• Computed Pic(gr-A1), the Picard group (group ofautoequivalences) of gr-A1
• Shift functor, S :
• Autoequivalence ω:
Z-graded rings May 2, 2016 23 / 39
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Sierra (2009)
Theorem (Sierra)There exist ιn, autoequivalences of gr-A1, permuting X〈n〉 and Y〈n〉and fixing all other simple modules.
Also ιiιj = ιjιi and ι2n ∼= Idgr-A
Theorem (Sierra)
Pic(gr-A1) ∼= (Z/2Z)(Z) o D∞ ∼= Zfin o D∞.
Z-graded rings May 2, 2016 24 / 39
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Smith (2011)
• Paul Smith, A quotient stack related to the Weyl algebra
• Proves that Gr-A1 ≡ Qcoh(χ)• χ is a quotient stack “whose coarse moduli space is the affine line
Spec k[z], and whose stacky structure consists of stacky pointsBZ2 supported at each integer point”
• Gr-A1 ≡ Gr(C,Zfin) ≡ Qcoh(χ)
Z-graded rings May 2, 2016 25 / 39
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Smith (2011)
• Zfin the group of finite subsets of Z, operation XOR• Constructs a Zfin-graded ring
C :=⊕
J∈Zfin
hom(A1, ιJA1) ∼=k[xn | n ∈ Z]
(x2n + n = x2m + m | m,n ∈ Z)
∼= k[z][√
z− n | n ∈ Z]
where deg xn = {n}• C is commutative, integrally closed, non-noetherian PID
Theorem (Smith)There is an equivalence of categories
Gr-A1 ≡ Gr(C,Zfin).
Z-graded rings May 2, 2016 26 / 39
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Z-graded rings
Theorem (Artin-Stafford, 1995)Let A be a f.g. connected N-graded domain generated in degree1 with GKdim(A) = 2. Then there exists a projective curve Xsuch that
qgr-A ≡ coh(X).
Theorem (Smith, 2011)A1 is a f.g. Z-graded domain with GKdim(A1) = 2. There existsa commutative ring C and quotient stack χ such that
gr-A1 ≡ gr(C,Zfin) ≡ coh(χ).
Z-graded rings May 2, 2016 27 / 39
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Generalized Weyl algebras (GWAs)
• Introduced by V. Bavula• Generalized Weyl algebras and their representations (1993)• D a ring; σ ∈ Aut(D); a ∈ Z(D)• The generalized Weyl algebra D(σ, a) with base ring D
D(σ, a) =D〈x, y〉(
xy = a yx = σ(a)dx = xσ(d), d ∈ D dy = yσ−1(d), d ∈ D
)
Theorem (Bell-Rogalski, 2015)Every simple Z-graded domain of GKdim 2 is graded Moritaequivalent to a GWA.
Z-graded rings May 2, 2016 28 / 39
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Generalized Weyl algebras
• For us, D = k[z]; σ(z) = z− 1; a = f (z)
A(f ) ∼=k〈x, y, z〉(
xy = f (z) yx = f (z− 1)xz = (z + 1)x yz = (z− 1)y
)• Two roots α, β of f (z) are congruent if α− β ∈ Z
Example (The first Weyl algebra)Take f (z) = z
D(σ, a) =k[z]〈x, y〉(
xy = z yx = z− 1zx = x(z− 1) zy = y(z + 1)
) ∼= k〈x, y〉(xy− yx− 1)
= A1.
Z-graded rings May 2, 2016 29 / 39
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Generalized Weyl algebras
Properties of A(f ):
• Noetherian domain• Krull dimension 1• Simple if and only if no congruent roots
• gl.dim A(f ) =
1, f has neither multiple nor congruent roots2, f has congruent roots but no multiple roots∞, f has a multiple root
• We can give A(f ) a Z grading by deg x = 1, deg y = −1, deg z = 0
Z-graded rings May 2, 2016 30 / 39
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Questions
Focus on quadratic f (z) = z(z + α). For these GWAs A(f ):• What does gr-A(f ) look like?• What is Pic(gr-A(f ))?• Can we construct a commutative Γ-graded ring C such that
gr(C,Γ) ≡ gr-A(f )?
Z-graded rings May 2, 2016 31 / 39
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Results
If α ∈ k \ Z (distinct roots):
−2
α− 2
−1
α− 1
0
α
1
α+1
2
If α = 0 (double root):
−3 −2 −1 0 1 2 3
If α ∈ N+ (congruent roots):
Z-graded rings May 2, 2016 32 / 39
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Results
Theorem(s) (W)In all cases, there exist numerically trivial autoequivalences ιnpermuting “X〈n〉” and “Y〈n〉” and fixing all other simple modules.
Corollary (W)For quadratic f , Pic(gr-A(f )) ∼= Zfin o D∞.
Z-graded rings May 2, 2016 33 / 39
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Results
Let α ∈ k \ Z. Define a Zfin × Zfin graded ring C:
C =⊕
(J,J′)∈Zfin×Zfin
homA(A, ι(J,J′)A
)∼=
k[an, bn | n ∈ Z](a2n + n = a2m + m, a2n = b2n + α | m,n ∈ Z)
with deg an = ({n}, ∅) and deg bn = (∅, {n})
Theorem (W)There is an equivalence of categories gr(C,Zfin × Zfin) ≡ gr-A.
Z-graded rings May 2, 2016 34 / 39
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Results
Let α = 0. Define a Zfin-graded ring B:
B =⊕
J∈Zfin
homA(A, ιJA) ∼=k[z][bn | n ∈ Z]
(b2n = (z + n)2 | n ∈ Z).
Theorem (W)B is a reduced, non-noetherian, non-domain of Kdim 1 withuncountably many prime ideals.
Theorem (W)There is an equivalence of categories gr(B,Zfin) ≡ gr-A.
Z-graded rings May 2, 2016 35 / 39
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Results
• In congruent root (α ∈ N+), we have finite-dimensional modules.• Consider the quotient category
qgr-A = gr-A/fdim-A.
Theorem (W)There is an equivalence of categories gr(B,Zfin) ≡ qgr-A.
Z-graded rings May 2, 2016 36 / 39
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The upshot
In all cases,• There exist numerically trivial autoequivalences permuting X and
Y and fixing all other simples.• Pic(gr-A(f )) ∼= Zfin o D∞.• There exists a Zfin-graded commutative ring R such that
qgr-A(f ) ≡ gr(R,Zfin).
Z-graded rings May 2, 2016 37 / 39
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Questions• Zfin-grading on B gives an action of SpeckZfin on Spec B
χ =
[Spec B
SpeckZfin
]• Other Z-graded domains of GK dimension 2? Other GWAs?
• U(sl(2)), Down-up algebras, quantum Weyl algebra, SimpleZ-graded domains
• Construction of B: {Fγ | γ ∈ Γ} ⊆ Pic(gr-A)
B =⊕γ∈Γ
Homqgr-A(A,FγA)
• Opposite view: O a quasicoherent sheaf on χ:⊕n∈Z
HomQcoh(χ)(O,SnO)
Future direction May 2, 2016 38 / 39
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Thank you!
May 2, 2016 39 / 39
Graded rings and thingsNoncommutative is not commutativeNoncommutative projective schemesZ-graded ringsThe first Weyl algebraGeneralized Weyl algebras
Future direction