Z-graded noncommutative projective geometryAlgebra Seminar Pre-Talk

Robert WonUniversity of California, San Diego

November 9, 2015 1 / 20

Overview

1 Graded rings and things

2 Abstract nonsense

3 Commutative is commutative

4 Noncommutative is not commutative

November 9, 2015 2 / 20

Graded rings (k-algebras)

• Throughout, k = k and chark = 0• Γ an abelian 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 November 9, 2015 3 / 20

Graded rings (k-algebras)

Examples• Any ring R with trivial grading• k[x, y] graded by N• k[x, x−1] graded by Z• If you like physics: superalgebras graded by Z2

• Or combinatorics: symmetric and alternating polynomialsgraded by Z2

RemarkUsually, “graded ring” means N-graded.

Graded rings and things November 9, 2015 4 / 20

Graded modules and morphisms• A graded right A-module M:

M =⊕i∈Z

Mi

such that Mi · Aj ⊆Mi+j

• A graded module homomorphism, f : M→ N:

f (Mi) ⊆ Ni

• A graded module homomorphism of degree d, f : M→ N:

f (Mi) ⊆ Ni+d

ExampleMultiplication by x, k[x]→ k[x] is a graded modulehomomorphism of degree 1

Graded rings and things November 9, 2015 5 / 20

Graded modules and morphisms

• A graded submodule N =⊕i∈Z

N ∩Mi

• If N ⊆M a graded submodule M/N is a graded factor modulewith

M/N =⊕i∈Z

(M/N)i =⊕i∈Z

(Mi + N)/N.

• Given a graded module M, define the shift operator

(M〈n〉)i = Mi−n

−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

Graded rings and things November 9, 2015 6 / 20

Category theory

Abstract nonsense November 9, 2015 7 / 20

Category theory

A category, C , consists of• a collection of objects, Obj(C ),• for any two objects X,Y a class of morphisms HomC (X,Y),• and composition HomC (Y,Z)×HomC (X,Y)→ HomC (X,Z)

such that identity morphisms exist and composition associates

Examples• Set, whose objects are sets and morphisms are functions• Top, topological spaces and continuous functions• A directed graph (with loops), whose objects are vertices

and morphisms are paths

Abstract nonsense November 9, 2015 8 / 20

Category theory

• Given two categories. Morphisms between them?• A (covariant) functor, F : C → D associates to each:

• object X ∈ Obj(C ) an object F(X) ∈ Obj(D), and• morphism f ∈ HomC (X,Y) a morphism F(f ) ∈ HomD(X,Y)

preserving identities and compositions

Examples

• Identity functor IdC

• Forgetful functor Grp→ Set

• A contravariant functor reverses arrows.

ExampleHom(−,C): A→ B then Hom(B,C)→ Hom(A,C)

Abstract nonsense November 9, 2015 9 / 20

Category theory

• Given two functors. Morphisms between them?• Given F,G : C → D , a natural transformation η from F to G is

• for each object X ∈ Obj(C ) a morphism ηX : F(X)→ G(X)• that respects morphisms i.e. ηY ◦ F(f ) = G(f ) ◦ ηX

F(X)

ηX

��

F(f ) // F(Y)

ηY

��G(X)

G(f ) // G(Y)

• An equivalence of categories is a functor F : C → D and a functorG : D → C such that F ◦ G ∼= IdD and G ◦ F ∼= IdC .

Abstract nonsense November 9, 2015 10 / 20

The graded module category gr-A

• Objects: finitely generated graded right A-modules• Hom sets:

homgr-A(M,N) = {f ∈ Hommod-A(M,N) | f (Mi) ⊆ Ni}

• The shift functor:

S i : gr-A −→ gr-AM 7→M〈i〉

Abstract nonsense November 9, 2015 11 / 20

Commutative algebraic geometry

• Let Speck[x, y] = A2k be the set of prime ideals of k[x, y].

• A2k contains a point (maximal ideal) for each point of k2

(a, b) (x− a, y− b)

• Also contains a point for each subscheme (prime ideal)

y = x (x− y) y = x2 (x2 − y)

• A topological space with the Zariski topology. Closed sets:

V(a) = {p ∈ A2k | a ⊆ p}

for ideals a of k[x, y].• Replace k[x, y] with any commutative ring R for Spec R.

Commutative is commutative November 9, 2015 12 / 20

Commutative algebraic geometry

• Make Spec R an affine scheme by constructing the structure sheaf(localize at prime ideals).

• Interplay beteween algebra and geometry

algebra geometryR Spec R

prime ideals pointsI ⊆ J V(I) ⊇ V(J)

ring homomorphisms R→ S morphisms Spec S→ Spec Rfactor rings subschemes

• Contravariant functor Spec : CommRing→ AffSch• A scheme is glued together from afine schemes

Commutative is commutative November 9, 2015 13 / 20

Commutative algebraic geometry

• Commutative graded ring R =⊕

i∈N Ri

• Projective scheme Proj R the set of homogeneous prime idealsexcluding the irrelevant ideal R>0

• Zariski topology, closed sets of form

V(a) = {p ∈ Proj R | a ⊆ p}

for homogeneous ideals a of R• Structure sheaf: localize at homogeneous prime ideals

Commutative is commutative November 9, 2015 14 / 20

Noncommutative is not commutative

Noncommutative rings are ubiquitous

Examples• Ring R, nonabelian group G, the group algebra R[G]

• Mn(R), n× n matrices• Differential operators on k[t], generated by t· and d/dt is

isomorphic to

A1 = k〈x, y〉/(xy− yx− 1).

If you like physics, position and momentum operators don’t commutein quantum mechanics

Noncommutative is not commutative November 9, 2015 15 / 20

Noncommutative is not commutative

• Localization is different.• Given R commutative and S ⊂ R multiplicatively closed,

r1s−11 r2s−1

2 = r1r2s−11 s−1

2

• If R noncommutative, can only form RS−1 if S is an Ore set.

DefinitionS is an Ore set if for any r ∈ R, s ∈ S

sR ∩ rS 6= ∅.

Noncommutative is not commutative November 9, 2015 16 / 20

Noncommutative is not commutative

• Forget localization. Who needs it?• Even 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 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 November 9, 2015 17 / 20

Noncommutative is not commutative

• Forget prime ideals. Can we come up with a space?• We’re clever! Left prime ideals? Spec(R/[R,R])?• Even worse, there may not even be a set.

Theorem (Reyes, 2012)Suppose

F : Ring→ Set

extends the functor

Spec : CommRing→ Set.

Then for n ≥ 3, F(Mn(C)) = ∅.

Noncommutative is not commutative November 9, 2015 18 / 20

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 November 9, 2015 19 / 20

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.

• Philosophy: To understand R (Proj R), understand the gradedR-modules (coherent sheaves on Proj R)

• Analogous to: Understand G by its representations

Noncommutative is not commutative November 9, 2015 20 / 20