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Z-graded noncommutative projective geometryAlgebra Seminar
Robert WonUniversity of California, San Diego
November 9, 2015 1 / 43
Overview
1 PreliminariesPre-talk catchupNoncommutative things
2 Noncommutative projective schemes
3 Z-graded ringsPrior workGeneralized Weyl algebras
4 Future direction
November 9, 2015 2 / 43
What you missed in the pre-talk
• Throughout, k = k, char(k) = 0• Γ an abelian group• A Γ-graded k-algebra A has k-space decomposition
A =⊕γ∈Γ
Aγ
such that AγAδ ⊆ Aγ+δ
• A graded right A-module M:
M =⊕γ∈Γ
Mγ
such that Mγ · Aδ ⊆Mγ+δ
Preliminaries November 9, 2015 3 / 43
What you missed in the pre-talk
• A graded by N (or Z), the graded module category gr-A:• Objects: finitely generated graded right A-modules• Hom sets (degree 0) graded module homomorphisms:
homgr-A(M,N) = {f ∈ Hommod-A(M,N) | f (Mi) ⊆ Ni}
• The shift functor:
S i : gr-A −→ gr-AM 7→M〈i〉
−3 −2 −1 0 1 2 3M M−3 M−2 M−1 M0 M1
M〈1〉 M−3 M−2 M−1 M0 M1M〈2〉 M−3 M−2 M−1 M0 M1
Preliminaries November 9, 2015 4 / 43
The Picard group
• For a category of graded modules (e.g. gr-A) we define the Picardgroup, Pic(gr-A) to be the group of autoequivalences of gr-Amodulo natural isomorphism.
• (If R is a commutative k-algebra, Pic(mod-R) ∼= Pic(R), theisomorphism classes of invertible unitary R-bimodules)
• For a noncommutative ring, A, Pic(gr-A) can be nonabelian.
Preliminaries November 9, 2015 5 / 43
Morita equivalence
• Two rings R and S are called Morita equivalent if mod-R isequivalent to mod-S (if and only if R-mod equivalent to S-mod).
• For R and S commutative, then R is Morita equivalent to S if andonly if R ∼= S.
ExampleR a ring, then Mn(R) (the ring of n× n matrices) is Moritaequivalent to R.
• R and S are graded Morita equivalent if gr-R is equivalent to gr-S.
Preliminaries November 9, 2015 6 / 43
Noncommutative is not commutative
• Commutative graded ring: R←→ Proj R• Noncommutative ring: 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 fourhomogeneous prime ideals (namely (0), (x), (y), and (x, y)).
Preliminaries November 9, 2015 7 / 43
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.
Preliminaries November 9, 2015 8 / 43
Sheaves to the rescue
• All you need is modules
TheoremLet X = Proj R for a commutative, f.g. k-algebra R generated indegree 1.
(1) Every coherent sheaf on X is isomorphic to M for some f.g.graded R-module M.
(2) M ∼= N as sheaves if and only if there is an isomorphismM≥n ∼= N≥n.
• Let gr-R be the category of f.g. R-modules. Take the quotientcategory
qgr-R = gr-R/fdim-R
• The above says qgr-R ≡ coh(Proj R).
Preliminaries November 9, 2015 9 / 43
Noncommutative projective schemes
• A (not necessarily commutative) connected graded k-algebra A is
A = k⊕ A1 ⊕ A2 ⊕ · · ·
such that dimk Ai <∞ and A is a f.g. k-algebra.
Definition (Artin-Zhang, 1994)The noncommutative projective scheme ProjNC A is the triple
(qgr-A,A,S)
where A is the distinguished object and S is the shift functor.
• Idea: Use geometry to study ProjNC A = qgr-A to study A.
Noncommutative projective schemes November 9, 2015 10 / 43
All you need is modules
• Can attempt to do any geometry that only relies on the modulecategory.
• If X is a commutative projective k-scheme, for each x ∈ X, there isthe skyscraper sheaf k(x) ∈ coh(X).
• So the simple objects of coh(X) correspond to points of X.• A point module is a graded right module M such that M is cyclic,
generated in degree 0, and has dimk Mn = 1 for all n.• Fact: If A is f.g. connected graded noetherian k-algebra generated
in degree 1, then the point modules are simple objects of qgr-A.• “Points” of ProjNC A = simple modules.
Noncommutative projective schemes November 9, 2015 11 / 43
Twisted homogeneous coordinate rings
• We can go from rings to schemes via Proj.
R Proj R
• Can we go back? (Certainly not uniquely: Proj R(d) ∼= Proj R.)• X a projective scheme, L a line bundle on X• The homogeneous coordinate ring is
B(X,L) = k⊕⊕n≥1
H0(X,L⊗n).
Theorem (Serre)Assume L is ample. Then
(1) B = B(X,L) is a f.g. graded noetherian k-algebra.
(2) X = Proj B so qgr-B ≡ coh(X).
Noncommutative projective schemes November 9, 2015 12 / 43
Twisted homogeneous coordinate rings
• X a (commutative) projective scheme, L a line bundle, andσ ∈ Aut(X). Define
Lσ = σ∗L and Ln = L ⊗ Lσ ⊗ · · · ⊗ Lσn−1.
• The twisted homogeneous coordinate ring is
B(X,L, σ) = k⊕⊕n≥1
H0(X,Ln).
• B(X,L, σ) is not necessarily commutative
Theorem (Artin-Van den Bergh, 1990)Assume L is σ-ample. Then
(1) B = B(X,L, σ) is a f.g. graded noetherian k-algebra.
(2) qgr-B ≡ coh(X).
Noncommutative projective schemes November 9, 2015 13 / 43
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 November 9, 2015 14 / 43
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 X, anautomorphism σ and invertible sheaf L such that up to a f.dvector space
A = B(X,L, σ)
• As a corollary, qgr-A ≡ coh(X).• Or “noncommutative projective curves are commutative”.
Noncommutative projective schemes November 9, 2015 15 / 43
Noncommutative surfaces
• The “right” definition of a noncommutative polynomial ring?
DefinitionA a f.g. connected graded k-algebra is Artin-Schelter regular if
(1) gldim A = d <∞
(2) GKdim A <∞ and
(3) ExtiA(k,A) = δi,dk.
• Behaves homologically like a commutative polynomial ring.• k[x1, . . . , xm] is AS-regular of dimension m.• Noncommutative P2s should be qgr-A for A AS-regular of
dimension 3.
Noncommutative projective schemes November 9, 2015 16 / 43
Noncommutative surfaces
Theorem (Artin-Tate-Van den Bergh, 1990)Let A be an AS-regular ring of dimension 3 generated in degree1. Either(a) A = B(X,L, σ) for X = P2 or X = P1 × P1 or(b) A� B(E,L, σ) 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 November 9, 2015 17 / 43
Z-graded rings
• Throughout, “graded” really meant N-graded• Way back to our first example:
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• Simple noetherian domain of GK dim 2• Exists an outer automorphism ω, reversing the grading
ω(x) = y ω(y) = −x
Z-graded rings November 9, 2015 18 / 43
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 November 9, 2015 19 / 43
Sierra (2009)
• Computed Pic(gr-A1)
• Shift functor, S :
• Autoequivalence ω:
Z-graded rings November 9, 2015 20 / 43
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 November 9, 2015 21 / 43
Smith (2011)
• Paul Smith, A quotient stack related to the Weyl algebra
• Proves that Gr-A1 ≡ Qcohχ• 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) ≡ Qcohχ
Z-graded rings November 9, 2015 22 / 43
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 = x2
m + m | m,n ∈ Z)
∼= k[z |√
z− n]
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 November 9, 2015 23 / 43
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) ≡ Qcoh(χ).
Z-graded rings November 9, 2015 24 / 43
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 November 9, 2015 25 / 43
The main object
• 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 November 9, 2015 26 / 43
The main object
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 November 9, 2015 27 / 43
Questions and strategy
For these generalized Weyl algebras 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 )?Strategy:
• First quadratic f• Distinct, non-congruent roots• Congruent roots• Double root
Z-graded rings November 9, 2015 28 / 43
Distinct, non-congruent roots
Let f (z) = z(z + α) for some α ∈ k \ Z.
A = A(f ) ∼=k〈x, y, z〉(
xy = z(z + α) yx = (z− 1)(z + α− 1)xz = (z + 1)x yz = (z− 1)y
)• We still have A is simple, K.dim(A) = gl.dim(A) = 1• Still exists an outer automorphism ω reversing the grading
ω(x) = y ω(y) = x ω(z) = 1 + α− z
Z-graded rings November 9, 2015 29 / 43
Distinct, non-congruent roots
Graded simple modules:
−2
α− 2
−1
α− 1
0
α
1
α+1
2
• One simple graded module Mλ for each λ ∈ k \ (Z ∪ Z + α)
• For each n ∈ Z two simple modules X0〈n〉 and Y0〈n〉• For each n ∈ Z two simple modules Xα〈n〉 and Yα〈n〉• A nonsplit extension of X0〈n〉 by Y0〈n〉 and vice versa• A nonsplit extension of Xα〈n〉 by Yα〈n〉 and vice versa
Z-graded rings November 9, 2015 30 / 43
Distinct, non-congruent roots
Theorem (W)There exist numerically trivial autoequivalences, ι(n,∅)permuting X0〈n〉 and Y0〈n〉 and fixing all other simple modules.Similarly, there exist ι(∅,n) permuting Xα〈n〉 and Yα〈n〉.
Theorem (W)
Pic(gr-A) ∼= (Z/2Z)(Z) o D∞
Z-graded rings November 9, 2015 31 / 43
Distinct, non-congruent roots
• Define a Zfin × Zfin graded ring C:
C = k[an, bn | n ∈ Z]
modulo the relations
a2n + n = a2
m + m and a2n = b2
n + α for all 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 November 9, 2015 32 / 43
Multiple root
Let f (z) = z2.
A = A(f ) ∼=k〈x, y, z〉(
xy = z2 yx = (z− 1)2
xz = (z + 1)x yz = (z− 1)y
)• K.dim(A) = 1• Exists an outer automorphism ω reversing the grading
ω(x) = y ω(y) = x ω(z) = 1− z
• Now gl.dim(A) =∞
Z-graded rings November 9, 2015 33 / 43
Multiple root
The graded simple modules
−3 −2 −1 0 1 2 3
• For each λ ∈ k \ Z, Mλ
• For each n ∈ Z, X〈n〉 and Y〈n〉
X Y
• Nonsplit extensions of X〈n〉 by Y〈n〉 and vice versa. Alsoself-extensions of X〈n〉 by X〈n〉 and Y〈n〉 and Y〈n〉.
Z-graded rings November 9, 2015 34 / 43
Multiple root
Theorem (W)There exist numerically trivial autoequivalences, ιn permutingX〈n〉 and Y〈n〉 and fixing all other simple modules.
Theorem (W)
Pic(gr-A) ∼= (Z/2Z)(Z) o D∞
Z-graded rings November 9, 2015 35 / 43
Multiple root
Define a Zfin graded ring B:
B =⊕
J∈Zfin
homA(A, ιJA) ∼=k[z][an | 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 November 9, 2015 36 / 43
Two congruent rootsLet f (z) = z(z + m) for some m ∈ N.
A = A(f ) ∼=k〈x, y, z〉(
xy = z(z + m) yx = (z− 1)(z + m− 1)xz = (z + 1)x yz = (z− 1)y
)What do we lose?
• A no longer simple• There exist new finite-dimensional simple modules• Now gl.dim(A) = 2
What remains?• Still have K.dim(A) = 1• Still exists an outer automorphism ω reversing the grading
ω(x) = y ω(y) = x ω(z) = 1 + m− z
Z-graded rings November 9, 2015 37 / 43
Two congruent roots
The graded simple modules• For each λ ∈ k \ Z, Mλ
• For each n ∈ Z, X〈n〉, Y〈n〉, and Z〈n〉
X YZ
• Z〈n〉 is finite dimensional• proj.dim Z〈n〉 = 2
• Leads to “more” finite length modules than in previous cases
Z-graded rings November 9, 2015 38 / 43
Two congruent roots
Theorem (W)If F is an autoequivalence of gr-A then there exists a = ±1 and b ∈ Zsuch that
{F(X〈n〉),F(Y〈n〉)} ∼= {X〈an + b〉,Y〈an + b〉}
andF(Z〈n〉) ∼= Z〈an + b〉.
Theorem (W)There exist numerically trivial autoequivalences, ιn permuting X〈n〉and Y〈n〉 and fixing all other simple modules.
Z-graded rings November 9, 2015 39 / 43
Two congruent roots
• In this case, finite dimensional modules Z〈n〉.• 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 November 9, 2015 40 / 43
The upshot
In all cases,• There exist numerically trivial autoequivalences permuting X and
Y and fixing all other simples.• Pic(gr-A(f )) ∼= (Z2)Z o D∞.• There exists a Zfin-graded commutative ring R such that
qgr-A(f ) ≡ gr(R,Zfin).
Z-graded rings November 9, 2015 41 / 43
Questions
• Zfin-grading on B gives an action of SpeckZfin on Spec B
χ =
[Spec B
SpeckZfin
]• What are the properties of χ?• Other Z-graded domains of GK dimension 2?• GWAs defined by non-quadratic f ? Other base rings D?
• U(sl(2))• Quantum Weyl algebra• Simple Z-graded domains
Future direction November 9, 2015 42 / 43
Questions
Theorem (Smith, 2011)A1 a Z-graded domain of GK dim 2. Stack χ such that
Gr-A1 ≡ Gr(C,Zfin) ≡ Qcoh(χ).
• For A a Z-graded domain of GK dim 2, is there always a stack?• Construction of B: Γ ⊆ Pic(gr-A)
B =⊕F∈Γ
Homqgr-A(A,FA)
• Take Γ = 〈S〉 = Z then B = A.• Opposite view: O a quasicoherent sheaf on χ:⊕
n∈ZHomQcoh(χ)(O,SnO)
Future direction November 9, 2015 43 / 43
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