Z-Graded Noncommutative Projective Geometry Algebra Seminar Pre-Talk

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Z-graded noncommutative projective geometry Algebra Seminar Pre-Talk

Robert Won University 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 char k = 0 • Γ an abelian group • A Γ-graded k-algebra A has k-space decomposition M A = Aγ γ∈Γ

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 polynomials graded by Z2

Remark Usually, “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 M = Mi i∈Z

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

Example Multiplication by x, k[x] → k[x] is a graded module homomorphism of degree 1

Graded rings and things November 9, 2015 5 / 20 Graded modules and morphisms

M • A graded submodule N = N ∩ Mi i∈Z • If N ⊆ M a graded submodule M/N is a graded factor module with M M M/N = (M/N)i = (Mi + N)/N. i∈Z i∈Z • Given a graded module M, deﬁne the shift operator

(Mhni)i = Mi−n

−3 −2 −1 0 1 2 3 M M−3 M−2 M−1 M0 M1 Mh1i M−3 M−2 M−1 M0 M1 Mh2i 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. Example Hom(−, 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(f ) F(X) / F(Y)

ηX ηY G(f ) G(X) / G(Y)

• An equivalence of categories is a functor F : C → D and a functor ∼ ∼ G : 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:

Si : gr-A −→ gr-A M 7→ Mhii

Abstract nonsense November 9, 2015 11 / 20 Commutative algebraic geometry

2 • Let Spec k[x, y] = Ak be the set of prime ideals of k[x, y]. 2 2 • Ak contains a point (maximal ideal) for each point of k

(a, b) (x − a, y − b)

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

2 2 y = x (x − y) y = x (x − y)

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

2 V(a) = {p ∈ Ak | 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 geometry R Spec R prime ideals points I ⊆ J V(I) ⊇ V(J) ring homomorphisms R → S morphisms Spec S → Spec R factor 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

• = L Commutative graded ring R i∈N Ri • Projective scheme Proj R the set of homogeneous prime ideals excluding 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 = khx, yi/(xy − yx − 1).

If you like physics, position and momentum operators don’t commute in 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,

−1 −1 −1 −1 r1s1 r2s2 = r1r2s1 s2

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

Deﬁnition S 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

khx, yi/(xy − yx − 1) is a noncommutative analogue of k[x, y] but is simple.

The quantum polynomial ring

khx, yi/(xy − qyx) is a “noncommutative P1” but for qn 6= 1 has only three homogeneous 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

Theorem Let X = Proj R for a commutative, f.g. k-algebra R generated in degree 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 isomorphism ∼ M≥n = N≥n.

• Philosophy: To understand R (Proj R), understand the graded R-modules (coherent sheaves on Proj R) • Analogous to: Understand G by its representations

Noncommutative is not commutative November 9, 2015 20 / 20