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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
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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.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 GeometryRepresentation Theory
InteractionsRelated 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.
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
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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〉.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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.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 GeometryRepresentation Theory
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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
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated 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∈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).
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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}.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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 ).
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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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Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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).
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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).
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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Theorem (Minamoto-Mori)
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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).
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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 is an AS-regular algebra of dimension 2
⇒ ∇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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
.
Example
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A = k〈x, 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 0
0 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 GeometryRepresentation Theory
InteractionsRelated Topics
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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
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Theorem (Minamoto-Mori)
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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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Theorem (Minamoto-Mori)
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
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{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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
InteractionsRelated Topics
.
Theorem (Minamoto-Mori)
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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.
Izuru Mori The Structure of AS-regular Algebras
Noncommutative Algebraic GeometryRepresentation Theory
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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
lClassifying finite dimensional algebras
up to isomorphism.
Izuru Mori The Structure of AS-regular Algebras