Auslander correspondence fortriangulated categories
Norihiro Hanihara

2018/09/19

Norihiro Hanihara

Auslander correspondence

2018/09/19

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Contents

1Background2

Finite case3[1]-finite case4

Application

Norihiro Hanihara

Auslander correspondence

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Background1

Background2

Finite case3[1]-finite case4

Application

Norihiro Hanihara

Auslander correspondence

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Background

Background

Theorem (Auslander correspondence)
There exists a bijection between
1Finite abelian categories.

2Finite dimensional algebras with gl. dim ≤ 2 ≤ dom. dim.

A relationship between categorical and homological properties.
Aim
Give a triangulated analogue.
=Give a homological characterization of ‘finite’ triangulated categories.

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Background

Notations and setup

k: field.
‘category’=k-linear, Hom-finite, Krull-Schmidt category.
a category C is finite if ] ind C < ∞for a finite category C = add M,
EndC (M): the Auslander algebra of C.
Finiteness for triangulated categories
1Finite.

2‘[1]-finite’.

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Finite case

1Background2

Finite case3[1]-finite case

4Application

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Auslander correspondence

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Finite case

Finite case
A homological characterization of Auslander algerbras of finite triangulated
categories:
Theorem 1
k: perfect field. The following are equivalent for a basic finite dimensionalk-algebra A:
1
2

A is the basic Auslander algebra of a finite triangulated category.
A is self-injective and Ω3 ' (−)α on mod A for some automorphism αof A.

Remark
Saying nothing about the triangle structures.

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Finite case

Proof of (1) ⇒ (2)A is self-injective since T is.
Each triangle

/YX

/Z

/in T yields an exact sequence(−, Z [−1]) / (−, X )?

/ (−, Y )/ (−, Z )

M[−1]/ (−, X [1])?

Min mod T ' mod A, so Ω3 M ' M[−1] in mod A.
[1] can induce an automorphism of A since it is basic.

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Finite case

Sketch of (2) ⇒ (1)
We want to show that proj A admits a structure of a triangulated category.
Proposition (Amiot)
Let A be a k-linear category such that mod A is naturally Frobenius. Let
S be an automorphism of A and extend this to mod A → mod A (byM 7→ M ◦ S −1 ). Assume there exists an exact sequence
0/1/ X0/ X1/ X2/S/0of exact functors from mod A to mod A such that the X i ’s values in A.
Then, A has a structure of a triangulated category with suspension S.
The triangles are given by X 0 M → X 1 M → X 2 M → SX 0 M withM ∈ mod A.

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Finite case

Sketch of (2) ⇒ (1)Such an exact sequence of functors can be obtained by considering
bimodules as functors.
Proposition (a variant of Green-Snashall-Solberg)
Let A be a ring-indecomposable non-semisimple finite dimensional k-alebrasuch that A/JA is separable over k and n > 0. Then, the following are
equivalent.
123Ωn (A/JA ) ' A/JA .
A is self-injective and Ωn ' (−)α on mod A for some automorphism αof A.
There exists an automorphism of α of A such that ΩnAe (A) ' 1 Aα .

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Finite case

Sketch of (2) ⇒ (1)Assume A is self-injective and Ω3 ' (−)α . Then, Ω3Ae (A) ' 1 Aσ for some
automorphism σ, so there exists an exact sequence
0/A/ P0/ P1/ P2/ 1 A −1
σ/0

in mod Ae with P i ∈ proj Ae , which gives a desired exact sequence offunctors.

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[1]-finite case

1Background2

Finite case3[1]-finite case4

Application

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Auslander correspondence

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[1]-finite case[1]-finite triangulated categories
Another finiteness for triangulated categories:
Definition
A triangulated category T is [1]-finite if
1

There exists M ∈ T such that T = add{M[i] | i ∈ Z}.

2

For any X , Y ∈ T , HomT (X , Y [i]) = 0 for almost all i ∈ Z.

In this case, we say M is a [1]-additive generator for T .
Example
Λ: representation-finite hereditary algebra with mod Λ = add M.
⇒ D b (mod Λ) is [1]-finite, M is a [1]-additive generator for D b (mod Λ).

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[1]-finite case

Known results
By the results of Xiao-Zhu and Riedtmann’s ‘knitting’ argument, we know
Proposition
T : [1]-finite triangulated category over an algebraically closed field.
1

The AR-quiver of T is ZQ for some Dynkin quiver Q.

2

T is standard, hence T ' k(ZQ) as k-linear categories.

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ZQ for type A3

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[1]-finite case

Graded projectivization

How to build the ‘Auslander algebras’ for [1]-finite triangulated categories?
Recall
C: finite category with add M = C. Setting Γ = EndC (M), we have anequivalence'HomC (M, −) : C −→ proj Γ.
Γ does not depend on M up to Morita equivalence.

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[1]-finite case

Graded projectivization
Proposition
C: category with an automorphism
F . Assume C = add{F i M | i ∈ Z} forL
some M ∈ C. Set Γ = i∈Z HomC (M, F i M). Then, there exists anequivalenceM' /projZW Γ.
HomC (M, F i (−)) : C Yi∈ZF(1)

Γ does not depend on M up to graded Morita equivalence.
Here, Graded rings A, B are graded Morita equivalent if there exists anequivalence' /ModZY A
ModZY B .
(1)

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[1]-finite case[1]-Auslander algebra
Apply the above proposition to triangulated categories.
Definition
T : [1]-finite triangulated category with [1]-additive generator M. We callM
C=HomT (M, M[i])i∈Z

the [1]-Auslander algebra of T .
Proposition1

C is a finite dimensional algebra.

2

We have an equivalence T ' projZ C such that [1] ↔ (1).

3

C is self-injective and Ω3 ' (−1) on modZ C .

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[1]-finite caseProof of (3)C is self-injective since T is.
Each triangle/YX/Z/in T yields an exact sequence(−, Z [−1]) / (−, X )?

/ (−, Y )/ (−, Z )

M[−1]/ (−, X [1])?

Min mod T ' modZ C , so Ω3 M ' M(−1) in modZ C .

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[1]-finite case[1]-Auslander correspondence
Theorem 2
k: algebraically closed field. There exists a bijection between
123[1]-finite algebraic triangulated categories / triangle equivalence
Finite dimensional graded self-injective algebras such that Ω3 ' (−1)/ graded Morita equivalence
Disjoint union of Dynkin diagrams of type A, D, and E .

The correspondences are given by
From (1) to (2): taking the [1]-Auslander algebra.
From (2) to (1): C 7→ projZ C .
From (1) to (3): taking the tree type of the AR-quiver of T .
From (3) to (1): Q 7→ k(ZQ).
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[1]-finite case

Uniqueness of triangle structures(3) to (1) (or (2) to (1)) says:Proposition
Q: Dynkin quiver, k(ZQ): its mesh category. Then, k(ZQ) has theunique structure of an algebraic triangulated category up to equivalence.
On the other hand, k(ZQ) has a structure of an algebraic triangulated
category D b (mod kQ).
Corollary
Any [1]-finite algebraic triangulated category over an algebraically closed
field k is triangle equivalent to D b (mod kQ) for some Dynkin quiver Q.
RemarkThe uniqueness of algebraic triangle structures (up to equivalence) holds
for K b (proj Λ) for certain ring Λ.
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Application

1Background2

Finite case3[1]-finite case4

Application

Norihiro Hanihara

Auslander correspondence

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Application

Graded Iwanaga-Gorenstein algebras

A graded Noetherian algebra Λ is Iwanaga-Gorenstein if
inj. dim Λ < ∞ on each side.
We have the category
CMZ Λ = {X ∈ modZ Λ | Ext>0Λ (X , Λ) = 0}of graded Cohen-Macaulay Λ-modules.
CMZ Λ is naturally Frobenius, hence the stable category CMZ Λ isalgebraically triangulated.
Λ is CM-finite if CMZ Λ is finite up to degree shift.

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Application

The triangle equivalence
Λ=Li≥0 Λi : positively graded Iwanaga-Gorenstein algebra such thateach Λi is finite dimensional over k.
gl. dim Λ0 < ∞.
Assume Λ is CM-finite.
Theorem 3
CMZ Λ is [1]-finite, and therefore, if k is algebraically closed,
1

The AR-quiver of CMZ Λ is ZQ for some Dynkin quiver Q.

2

There exists a triangle equivalence CMZ Λ ' D b (mod kQ).

e.g.
(commutative) simple singularitiesfinite dimensional representation-finite self-injective algebras
representation-finite Gorenstein orders
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