The Structure of AS-regular Algebras · Izuru Mori The Structure of AS-regular Algebras. Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics.. Fano

Noncommutative Algebraic GeometryRepresentation Theory

InteractionsRelated Topics

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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



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.Noncommutative algebraic geometry

Classify noncommutative projective schemes

⇓Classify finitely generated graded algebras

Classify quantum projective spaces

⇓Classify AS-regular algebras




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.

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Definition (Artin-Zhang)

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The noncommutative projective scheme associated to

A is defined by tails A := gr A/ tors A.




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.AS-regular algebras

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Definition (Artin-Schelter)

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An N-graded algebra A is AS-regular of dimension d

and of Gorenstein parameter ` if

A0 = k (connected graded),

gldim A = d, and

ExtiA(k, A) ∼=

0 if i 6= d

k(`) if i = d.

A quantum projective space is a noncommutative

projective scheme associated to an AS-regular algebra.




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Theorem (Zhang)

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Every AS-regular algebra of dimension 2 and of

Gorenstein parameter ` is isomorphic to

k〈x1, . . . , xn〉/(n∑

i=1

xiσ(xn+1−i))

where

n ≥ 2,

deg x1 ≤ · · · ≤ deg xn,

deg xi + deg xn+1−i = ` for all i, and

σ ∈ Autk k〈x1, . . . , xn〉.




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Theorem (Artin-Tate-Van den Bergh)

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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.




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.Representation theory

Classify finite dimensional algebras⋃

Classify finite dimensional algebras of finite global

dimensions⋃

Classify Fano algebras




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Theorem (Gabriel)

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Every finite dimensional algebra of global dimension 1 is

Morita equivalent to a path algebra of a finite acyclic

quiver.

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Example

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Q = 1α //

β// 2 kQ ∼=

(ke1 kα + kβ

0 ke2

)

Q = 1α // 2

β // 3 kQ ∼=

ke1 kα k(αβ)

0 ke2 kβ

0 0 ke3




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∈Q1αα∗ − α∗α).

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Example

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Q = 1α //

β// 2 Q = 1

α //β //

2α∗oo

β∗oo

ΠQ = kQ/(αα∗ + ββ∗, α∗α + β∗β).




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.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 À 0}Ds,≤0 := {M ∈ D | si(M) ∈ D≤0 for all i À 0}.




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Definition (Minamoto)

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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.

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Theorem (Minamoto)

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If s ∈ Autk D is ample, then (R, s) is ample for

H := Ds,≥0 ∩ Ds,≤0 in the sense of Artin-Zhang.




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Definition (Minamoto)

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An algebra R is Fano of dimension d if

gldim R = d, and

− ⊗LR 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−1R ).




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Example

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R is a Fano algebras of dimension 0 ⇔ R is a

semi-simple algebra

In this case, ΠR ∼= R[x]

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Example

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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.




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.Interactions

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Definition

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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

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Definition

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The Beilinson algebra of an AS-regular algebra A of

Gorenstein parameter ` is defined by

∇A := (A[`])0

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Lemma

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For any graded algebra A and r ∈ N+, gr A[r] ∼= gr A.

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Lemma

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For any algebra R, R-R bimodule M and σ ∈ Autk R,

gr TR(Mσ) ∼= gr TR(M).




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Theorem (Minamoto-Mori)

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If A is an AS-regular algebra of dimension d ≥ 1, then

S := ∇A is a Fano algebra of dimension d − 1.

A[`] ∼= TS((ω−1S )σ) for some σ ∈ Autk S.

gr A ∼= gr A[`] ∼= gr TS((ω−1S )σ) ∼= gr ΠS.

Db(tails A) ∼= Db(tails ΠS) ∼= Db(mod S).

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Example (Beilinson)

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Applying to A = k[x1, . . . , xn], deg xi = 1,

Db(coh Pn−1) ∼= Db(tails A) ∼= Db(mod ∇A).




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Let A, B be AS-regular algebras.

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1 The following are equivalent:

gr A ∼= gr B.

∇A ∼= ∇B.

Π(∇A) ∼= Π(∇B).grΠ(∇A) ∼= grΠ(∇B).

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2 The following are equivalent:

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




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Example

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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 = • //

²²

•

²²• •oo

(extended Dynkin)

Q is a reduced McKay quiver of⟨(ξ 0

0 ξ3

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

root of unity.




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Example

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A = k〈x, y, z〉/(xz + y2 + zx)

deg x = 1, deg y = 2, deg z = 3


⇒ ∇A ∼= kQ is a Fano algebra of dimension 1

⇒ Q = • //

²² ÃÃ@@@

@@@@

•

²²~~~~~~

~~~

• •oo

(not extended Dynkin)

Q is a reduced McKay quiver of⟨

ξ 0 0

0 ξ2 0

0 0 ξ3

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

4-th root of unity.




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Example

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A = k〈x, y〉/(x2y − yx2, xy2 − y2x),

deg x = deg y = 1


⇒ ∇A ∼= kQ/I is a Fano algebra of dimension 2

⇒ Q = • //// • //

// • //// •

Q is a reduced McKay quiver of

⟨(ξ 0

0 ξ

)⟩≤ GL(2, k)

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




AS-regular algebras (of dimension 2) can be classified

by (reduced) McKay quivers of a finite cyclic subgroups

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




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.Generalizations

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Definition (Minamoto-Mori)

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A graded algebra A is AS-regular over R of dimension

d and of Gorenstein parameter ` if

A0 = R, gldim R < ∞,

gldim A = d, and

ExtiA(R, A) ∼=

0 if i 6= d

(DR)(`) if i = d.

An AS-regular algebra A is symmetric if

ωA := D Hdm(A) ∼= A(−`) as graded A-A bimodules.




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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[`] ∼= TS((ω−1S )σ) for some σ ∈ Autk S.

gr A ∼= gr ΠS.

Db(tails A) ∼= Db(mod S).

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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.




{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

lClassifying Fano algebras of

dimension d − 1 up to isomorphism.




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.Graded Frobenius Algebras

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Definition

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A finite dimensional graded algebra A is graded

Frobenius of Gorenstein parameter ` if DA ∼= A(`) as

graded A-modules.

It is graded symmetric if DA ∼= A(`) as graded A-A

bimodules.

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Example

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The trivial extension of R is defined by

∆R := R ⊕ DR = TR(DR)/TR(DR)≥2.




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A is a trivial extensions of finite dimensional algebras

⇔ A is a graded symmetric algebras of Gorenstein

parameter 1.

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Definition

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The Beilinson algebra of a graded Frobenius algebra A

of Gorenstein parameter ` is defined by

∇A := (A[`])0.




{graded Frobenius algebras}∇ ↓↑ ∆

{finite dimensional algebras}

gr ∆(∇A) ∼= gr A

∇(∆S) ∼= S

Classifying graded Frobenius algebras

up to graded Morita equivalence

lClassifying finite dimensional algebras

up to isomorphism.


The Structure of AS-regular Algebras · Izuru Mori The Structure of AS-regular Algebras. Noncommutative Algebraic Geometry Representation Theory Interactions Related Topics.. Fano

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