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The Structure of AS-regular Algebras

Izuru Mori

Department of Mathematics, Shizuoka University

Noncommutative Algebraic Geometry Shanghai Workshop 2011, 9/12

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Noncommutative algebraic geometry

Classify noncommutative projective schemes ⇓ Classify finitely generated graded algebras Classify quantum projective spaces ⇓ Classify AS-regular algebras

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

For simplicity, we assume that k = k, and A is a graded right coherent algebra over k. gr A = the abelian category of finitely presented graded right A-modules. tors A = the full subcategory of finite dimensional modules. Definition (Artin-Zhang) The noncommutative projective scheme associated to A is defined by tails A := gr A/ tors A.

Izuru Mori The Structure of AS-regular Algebras

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AS-regular algebras

Definition (Artin-Schelter) An N-graded algebra A is AS-regular of dimension d and of Gorenstein parameter ℓ if A0 = k (connected graded), gldim A = d, and Exti

A(k, A) ∼

=    if i = d k(ℓ) if i = d. A quantum projective space is a noncommutative projective scheme associated to an AS-regular algebra.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Zhang) Every AS-regular algebra of dimension 2 and of Gorenstein parameter ℓ is isomorphic to kx1, . . . , xn/(

n

• i=1

xiσ(xn+1−i)) where n ≥ 2, deg x1 ≤ · · · ≤ deg xn, deg xi + deg xn+1−i = ℓ for all i, and σ ∈ Autk kx1, . . . , xn.

Izuru Mori The Structure of AS-regular Algebras

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Theorem (Artin-Tate-Van den Bergh) Quadratic AS-regular algebras of dimension 3 and of finite GKdimension were classified by geometric triples (E, σ, L) where E ⊂ P2, σ ∈ Autk E, and L ∈ Pic E.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Representation theory

Classify finite dimensional algebras

• Classify finite dimensional algebras of finite global

dimensions

• Classify Fano algebras

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Gabriel) Every finite dimensional algebra of global dimension 1 is Morita equivalent to a path algebra of a finite acyclic quiver. Example Q = 1

α

• β

2

kQ ∼ =

• ke1

kα + kβ ke2

• Q = 1

α

2

β

3

kQ ∼ =    ke1 kα k(αβ) ke2 kβ ke3   

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

The double Q of a quiver Q is defined by Q0 = Q0 Q1 = {α : i → j, α∗ : j → i | α ∈ Q1}. The preprojective algebra of Q is defined by ΠQ := kQ/(

α∈Q1 αα∗ − α∗α).

Example Q = 1

α

• β

2

Q = 1

α

• β
• 2

α∗

• β∗
• ΠQ = kQ/(αα∗ + ββ∗, α∗α + β∗β).

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Fano algebras

Let R be a finite dimensional algebra. D := Db(mod R) has a standard t-structure D≥0 := {M ∈ D | hi(M) = 0 for all i < 0} D≤0 := {M ∈ D | hi(M) = 0 for all i > 0}. For s ∈ Autk D, we define Ds,≥0 := {M ∈ D | si(M) ∈ D≥0 for all i ≫ 0} Ds,≤0 := {M ∈ D | si(M) ∈ D≤0 for all i ≫ 0}.

Izuru Mori The Structure of AS-regular Algebras

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Definition (Minamoto) s ∈ Autk D is ample if si(R) ∈ D≥0 ∩ D≤0 ∼ = mod R for all i ≥ 0, and (Ds,≥0, Ds,≤0) is a t-structure for D. Theorem (Minamoto) If s ∈ Autk D is ample, then (R, s) is ample for H := Ds,≥0 ∩ Ds,≤0 in the sense of Artin-Zhang.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Definition (Minamoto) An algebra R is Fano of dimension d if gldim R = d, and − ⊗L

R ω−1 R ∈ Autk D is ample where

DR := Homk(R, k) and ωR := DR[−d]. The preprojective algebra of a Fano algebra R is defined by ΠR := TR(ω−1

R ).

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Example R is a Fano algebras of dimension 0 ⇔ R is a semi-simple algebra In this case, ΠR ∼ = R[x] Example R is a basic Fano algebras of dimension 1 ⇔ R ∼ = kQ where Q is a finite acyclic non-Dynkin quiver. In this case, ΠR ∼ = ΠQ.

Izuru Mori The Structure of AS-regular Algebras

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Interactions

Definition For a graded algebra A = ⊕i∈ZAi and r ∈ N+, we define the r-th quasi-Veronese algebra of A by A[r] :=

• i∈Z

      Ari Ari+1 · · · Ari+r−1 Ari−1 Ari · · · Ari+r−2 . . . . . . ... . . . Ari−r+1 Ari−r+2 · · · Ari       .

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Definition The Beilinson algebra of an AS-regular algebra A of Gorenstein parameter ℓ is defined by ∇A := (A[ℓ])0 Lemma For any graded algebra A and r ∈ N+, gr A[r] ∼ = gr A. Lemma For any algebra R, R-R bimodule M and σ ∈ Autk R, gr TR(Mσ) ∼ = gr TR(M).

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Minamoto-Mori) If A is an AS-regular algebra of dimension d ≥ 1, then S := ∇A is a Fano algebra of dimension d − 1. A[ℓ] ∼ = TS((ω−1

S )σ) for some σ ∈ Autk S.

gr A ∼ = gr A[ℓ] ∼ = gr TS((ω−1

S )σ) ∼

= gr ΠS. Db(tails A) ∼ = Db(tails ΠS) ∼ = Db(mod S). Example (Beilinson) Applying to A = k[x1, . . . , xn], deg xi = 1, Db(coh Pn−1) ∼ = Db(tails A) ∼ = Db(mod ∇A).

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Minamoto-Mori) Let A, B be AS-regular algebras.

1

The following are equivalent:

gr A ∼ = gr B. ∇A ∼ = ∇B. Π(∇A) ∼ = Π(∇B). gr Π(∇A) ∼ = gr Π(∇B).

2

The following are equivalent:

Db(tails A) ∼ = Db(tails B). Db(mod ∇A) ∼ = Db(mod ∇B).

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Example A = k[x, y], deg x = 1, deg y = 3 ⇒ A is an AS-regular algebra of dimension 2 ⇒ ∇A ∼ = kQ is a Fano algebra of dimension 1 ⇒ Q = •

• (extended Dynkin)

Q is a reduced McKay quiver of

• ξ

ξ3

• ≤ SL(2, k) where ξ ∈ k is a primitive 4-th

root of unity.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Example A = kx, y, z/(xz + y2 + zx) deg x = 1, deg y = 2, deg z = 3 ⇒ A is an AS-regular algebra of dimension 2 ⇒ ∇A ∼ = kQ is a Fano algebra of dimension 1 ⇒ Q = •

• (not extended Dynkin)

Q is a reduced McKay quiver of    ξ ξ2 ξ3   

• ≤ GL(3, k) where ξ ∈ k is a primitive

4-th root of unity.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Example A = kx, y/(x2y − yx2, xy2 − y2x), deg x = deg y = 1 ⇒ A is an AS-regular algebra of dimension 3 ⇒ ∇A ∼ = kQ/I is a Fano algebra of dimension 2 ⇒ Q = •

Q is a reduced McKay quiver of

• ξ

ξ

• ≤ GL(2, k)

where ξ ∈ k is a primitive 4-th root of unity.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

AS-regular algebras (of dimension 2) can be classified by (reduced) McKay quivers of a finite cyclic subgroups

• f GL(n, k) up to graded Morita equivalence.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Generalizations

Definition (Minamoto-Mori) A graded algebra A is AS-regular over R of dimension d and of Gorenstein parameter ℓ if A0 = R, gldim R < ∞, gldim A = d, and Exti

A(R, A) ∼

=    if i = d (DR)(ℓ) if i = d. An AS-regular algebra A is symmetric if ωA := D Hd

m(A) ∼

= A(−ℓ) as graded A-A bimodules.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Minamoto-Mori) If A is an AS-regular algebra over R of dimension d ≥ 1, then S := ∇A is a Fano algebra of dimension d − 1. A[ℓ] ∼ = TS((ω−1

S )σ) for some σ ∈ Autk S.

gr A ∼ = gr ΠS. Db(tails A) ∼ = Db(mod S). Theorem (Minamoto-Mori) A is a preprojective algebras of Fano algebras of dimension d ⇔ A is a symmetric AS-regular algebras of dimension d + 1 and of Gorenstein parameter 1.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

{AS-regular algebras over R of dimension d} ∇ ↓↑ Π {Fano algebras of dimension d − 1} gr Π(∇A) ∼ = gr A ∇(ΠS) ∼ = S Classifying AS-regular algebras over R of dimension d ≥ 1 up to graded Morita equivalence

• Classifying Fano algebras of

dimension d − 1 up to isomorphism.

Izuru Mori The Structure of AS-regular Algebras

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Graded Frobenius Algebras

Definition A finite dimensional graded algebra A is graded Frobenius of Gorenstein parameter ℓ if DA ∼ = A(ℓ) as graded A-modules. It is graded symmetric if DA ∼ = A(ℓ) as graded A-A bimodules. Example The trivial extension of R is defined by ∆R := R ⊕ DR = TR(DR)/TR(DR)≥2.

Izuru Mori The Structure of AS-regular Algebras

Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics

Theorem (Minamoto-Mori) A is a trivial extensions of finite dimensional algebras ⇔ A is a graded symmetric algebras of Gorenstein parameter 1. Definition The Beilinson algebra of a graded Frobenius algebra A

• f Gorenstein parameter ℓ is defined by

∇A := (A[ℓ])0.

Izuru Mori The Structure of AS-regular Algebras

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{graded Frobenius algebras} ∇ ↓↑ ∆ {finite dimensional algebras} gr ∆(∇A) ∼ = gr A ∇(∆S) ∼ = S Classifying graded Frobenius algebras up to graded Morita equivalence

• Classifying finite dimensional algebras

up to isomorphism.

Izuru Mori The Structure of AS-regular Algebras