Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Relative Auslander Correspondence
via
Exact Categories
Haruhisa Enomoto
Advisor: Prof. Osamu Iyama

January 31, 2018
(20 )

(30 )

(1)

(2)

(3)

Λ

E

Γ

Haruhisa Enomoto

∪

∈

∈

∈

∪

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

About My Thesis
My Thesis contains some new results,
and is based on my papers:[E1] H. Enomoto, Classifying exact categories via Wakamatsu
tilting, J. Algebra 485 (2017), 1–44.
[E2] H. Enomoto, Classifications of exact structures andCohen-Macaulay-finite algebras, arXiv:1705.02163.
Our Goal is to givean analogue of Auslander Correspondence
for CM Representation Theory
by using Exact Categories.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

About My Thesis
My Thesis contains some new results,
and is based on my papers:[E1] H. Enomoto, Classifying exact categories via Wakamatsu
tilting, J. Algebra 485 (2017), 1–44.
[E2] H. Enomoto, Classifications of exact structures andCohen-Macaulay-finite algebras, arXiv:1705.02163.
Our Goal is to givean analogue of Auslander Correspondence
for CM Representation Theory
by using Exact Categories.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Outline1

Introduction
Representation Theory of Algebras

2

Auslander Correspondence and CM Rep. Theory
Auslander Correspondence
CM Representation Theory

3

Exact Category and Main Results
Exact Category and Motivating Problems
Main Results

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Representation Theory of Algebras

Outline1

Introduction
Representation Theory of Algebras

2

Auslander Correspondence and CM Rep. Theory
Auslander Correspondence
CM Representation Theory

3

Exact Category and Main Results
Exact Category and Motivating Problems
Main Results

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Representation Theory of Algebras

What is Rep. Theory of Algebras?
k : a field.
Λ: a finite-dimensional k -algebra.
mod Λ: the category of f.d. Λ-modules.

General Motivation
Want to study the structure of mod Λ!
Krull-Schmidt Theorem
Every object in mod Λ is a finite direct sum ofindecomposable objects (in a unique way).
suffices to study indec. Λ-modules

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Representation Theory of Algebras

What is Rep. Theory of Algebras?
k : a field.
Λ: a finite-dimensional k -algebra.
mod Λ: the category of f.d. Λ-modules.

General Motivation
Want to study the structure of mod Λ!
Krull-Schmidt Theorem
Every object in mod Λ is a finite direct sum ofindecomposable objects (in a unique way).
suffices to study indec. Λ-modules

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Representation Theory of Algebras

Representation-Finite Algebras
Definition
Λ is representation-finite:⇔ mod Λ has only finitely many indec. objects (up to isom).
Examplek is rep-fin. (indec is only k ).
k [X ]/(X n ). (Jordan canonical form of nilpotent matrix)
Path alg. of Dynkin quivers (related to Lie theory)

Auslander Correspondence:
an important result about rep-fin. algebra.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Representation Theory of Algebras

Representation-Finite Algebras
Definition
Λ is representation-finite:⇔ mod Λ has only finitely many indec. objects (up to isom).
Examplek is rep-fin. (indec is only k ).
k [X ]/(X n ). (Jordan canonical form of nilpotent matrix)
Path alg. of Dynkin quivers (related to Lie theory)

Auslander Correspondence:
an important result about rep-fin. algebra.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Outline1

Introduction
Representation Theory of Algebras

2

Auslander Correspondence and CM Rep. Theory
Auslander Correspondence
CM Representation Theory

3

Exact Category and Main Results
Exact Category and Motivating Problems
Main Results

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Statement of Auslander Correspondence
Theorem (Auslander 1971)
There exists a bijection between:(1) Representation-finite algebras Λ.
(2) Abelian categories E with finitely many indec. objects.
(3) Algebras Γ satisfying a certain homological condition(gl.dim Γ ≤ 2 ≤ dom.dim Γ).
This relates rep-fin alg Λ to another class of alg. Γ!
1{ Rep-fin alg }
Λ

mod

2

3

{ Abelian cat }{Algebras}

E := mod ΛΓHaruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Bijections Between (1) and (2): Morita Theory1{ Rep-fin. alg }mod2{ Abelian cat with fin. many indec. }E := mod ΛΛThis bijection is related to:Morita theory
Characterize the module category of algebraby the categorical property.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory(2) and (3): Cat. of Finite Type “=” Algebras
Proposition
There exists a bijection between:(20 ) Additive categories E of finite type(:⇔ categories with finitely many indec. objects).
(30 ) Algebra Γ (an Auslander alg. of E).
20{ Cats of fin. type }

E

1-1

30
{Algebras}

Γ: Aus. alg. of E
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory(2) and (3): Cat. of Finite Type “=” Algebras
Proposition
There exists a bijection between:(20 ) Additive categories E of finite type(:⇔ categories with finitely many indec. objects).
0(3 ) Algebra Γ (an Auslander alg. of E).
20{ Cats of fin. type }

E

2

∪

1-1

30
{Algebras}∪ ΓAbelian catof fin. typeHaruhisa Enomoto: Aus. alg. of E3

Alg. satisfying
homological cond.
Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Auslander Correspondence in a Big Diagram
Bijections between (1), (2) and (3) are summerized as:
30

20{ Cats of fin. type }1
Rep-finite
algebras

Λ

mod

2

∪

Abelian catof fin. type

E := mod ΛHaruhisa Enomoto

1-1{Algebras}3∪Some classof algsΓ

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Auslander Correspondence in a Big Diagram
Bijections between (1), (2) and (3) are summerized as:
30

20{ Cats of fin. type }1
Rep-finite
algebras

mod

2

∪

Abelian catof fin. type1-1

{Algebras}3∪Some classof algs

Morita theoryΛE := mod ΛHaruhisa EnomotoΓ

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Auslander Correspondence in a Big Diagram
Bijections between (1), (2) and (3) are summerized as:

20

Auslander alg.

{ Cats of fin. type }1
Rep-finite
algebras

mod

2

∪

Abelian catof fin. type

1-1

30

{Algebras}3∪Some classof algs

Morita theoryΛE := mod ΛHaruhisa EnomotoΓ

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Motivation for CM Rep. Theory
More detailed and interesting results are obtained
if we restrict the class of modules we consider.
That is,Study “good” subcategory of mod Λ (instead of mod Λ itself).
Cohen-Macaulay Rep. Theory
Today: CM rep. theory, which studies cat. CM Λ ofCohen-Macaulay modules.
Origin: Commutative rings, and their CM rep. theory is
closely related to algebraic geometry.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Motivation for CM Rep. Theory
More detailed and interesting results are obtained
if we restrict the class of modules we consider.
That is,Study “good” subcategory of mod Λ (instead of mod Λ itself).
Cohen-Macaulay Rep. Theory
Today: CM rep. theory, which studies cat. CM Λ ofCohen-Macaulay modules.
Origin: Commutative rings, and their CM rep. theory is
closely related to algebraic geometry.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

CM Rep. Theory for Iwanaga-Gorenstein Algebras
Definition
Algebra Λ is Iwanaga-Gorenstein (IG):⇔ id ΛΛ and id Λ Λ are finite.
CM Λ := {X ∈ mod Λ | Ext>0Λ (X , Λ) = 0},
the category of Cohen-Macaulay Λ-modules.
IG alg Λ is CM-finite if CM Λ is of finite type.
ExampleAlgebra Λ with finite gl.dim.
Λ is CM-fin. IG withCM Λ = { f.g. projective Λ-modules }.
Self-injective algebra ΛHaruhisa Enomoto

Λ is IG, and CM Λ = mod Λ.

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

CM Rep. Theory for Iwanaga-Gorenstein Algebras
Definition
Algebra Λ is Iwanaga-Gorenstein (IG):⇔ id ΛΛ and id Λ Λ are finite.
CM Λ := {X ∈ mod Λ | Ext>0Λ (X , Λ) = 0},
the category of Cohen-Macaulay Λ-modules.
IG alg Λ is CM-finite if CM Λ is of finite type.
ExampleAlgebra Λ with finite gl.dim.
Λ is CM-fin. IG withCM Λ = { f.g. projective Λ-modules }.
Self-injective algebra ΛHaruhisa Enomoto

Λ is IG, and CM Λ = mod Λ.

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Naive Approach to CM Auslander Corresp. Fails
1{ CM-fin. IG alg }CM

20{ Cats of fin. type }30{Algebras}

CM Λ

Γ

Λ
Λ0
The map “CM” is not injective!
HoweverΛ can be recovered from CM Λ
together with the structure of exact category on it.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Auslander Correspondence
CM Representation Theory

Naive Approach to CM Auslander Corresp. Fails
1{ CM-fin. IG alg }CM

20{ Cats of fin. type }30{Algebras}

CM Λ

Γ

Λ
Λ0
The map “CM” is not injective!
HoweverΛ can be recovered from CM Λ
together with the structure of exact category on it.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Outline1

Introduction
Representation Theory of Algebras

2

Auslander Correspondence and CM Rep. Theory
Auslander Correspondence
CM Representation Theory

3

Exact Category and Main Results
Exact Category and Motivating Problems
Main Results

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Exact Cat. = Additive Cat. + Short Exact Seq.
Definition (Quillen 1973)

An exact category consists of a pair (E, F ), where
E is an additive category, andF is a class of short exact sequences in E
satisfying some conditions.
ExampleCM Λ (and other good subcat of mod Λ) naturally has
the structure of exact categories.

Λ can be recovered from CM Λ + exact str. on it!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Exact Cat. = Additive Cat. + Short Exact Seq.
Definition (Quillen 1973)

An exact category consists of a pair (E, F ), where
E is an additive category, andF is a class of short exact sequences in E
satisfying some conditions.
ExampleCM Λ (and other good subcat of mod Λ) naturally has
the structure of exact categories.

Λ can be recovered from CM Λ + exact str. on it!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Modified Approach for CM Auslander Corresp.
Modified Approach
Want to seek bijections between:(1) CM-finite IG algebras Λ.
(2) Categories E of finite type satisfying some conditions+ exact structure on it.
(3) Algebra Γ satisfying some conditions+ some information.
To this aim, we should consider:

Problem A (1) and (2): Morita theory for exact cat.
Problem B (2) and (3): Exact str. via Auslander alg.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Modified Approach for CM Auslander Corresp.
Modified Approach
Want to seek bijections between:(1) CM-finite IG algebras Λ.
(2) Categories E of finite type satisfying some conditions+ exact structure on it.
(3) Algebra Γ satisfying some conditions+ some information.
To this aim, we should consider:

Problem A (1) and (2): Morita theory for exact cat.
Problem B (2) and (3): Exact str. via Auslander alg.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Two Problems in a Big Diagram
0
30
2


Cat. of fin. type 1-1
Algebra+ exact str.
+ some info?

1CM-finite
IG alg

Λ

CM

2


∪


Exact catsatifying ???

E := CM Λ+ natural exact str.

Haruhisa Enomoto

3∪Some alg+some info

Γ + some info

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Two Problems in a Big Diagram
0
30
2


Cat. of fin. type 1-1
Algebra+ exact str.
+ some info?

1CM-finite
IG alg

CM

2


∪


Exact catsatifying ???

3∪Some alg+some info

Problem A: What’s the image of CM?
ΛE := CM Λ+ natural exact str.

Haruhisa Enomoto

Γ + some info

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Two Problems in a Big Diagram
Problem B: Construct this bij.

0
30
2


Cat. of fin. type 1-1
Algebra+ exact str.
+ some info?

1CM-finite
IG alg

CM

2


∪


Exact catsatifying ???

3∪Some alg+some info

Problem A: What’s the image of CM?
ΛE := CM Λ+ natural exact str.
Haruhisa Enomoto

Γ + some info

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Result On Problem A
Problem A
Characterize CM category of IG alg by categorical conditions!
Theorem A [E1]For an exact category E, the following are equivalent:
E is equivalent to CM Λ for some IG alg Λ.
E is Frobenius category with progenerator
and higher kernels.
1CM-finite
IG algCM

2

Exact catsatifying above
E := CM ΛΛHaruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Result On Problem B
E: cat. of finite type, Γ: its Auslander algebra.
Theorem B [E2]
There are bijections between the following:

Exact structures on E.
Sets of 2-regular simple Γ-modules.
Sets of dotted arrows A in the graph Q(Γ).
20
Cat. of fin. type+ exact str.

30
1-1




Algebra+ set of dotted arrows(Γ, A)

E + exact str.
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Theorem B: Classifying Exact Structures

ExampleE: the module category of k [• ← • ← •].
∃ 3 dotted arrow, henceQ(Γ) :∃ 23 = 8 exact str. on E

A: Red arrows

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Auslander Correspondence for CM-finite IG Algebras
Corollary [E1, E2]
There exists a bijection between the following.

(1) CM-finite Iwanaga-Gorenstein algebras Λ.
(2) Frobenius exact cats E of finite typewith higer kernels.
(3) Pairs (Γ, A), where Γ is an algebra with finite gl.dim
and A is a sets of cycles of dotted arrows of Q(Γ).
ALL CM-finite IG algebras are obtained in this way.
This gives systematic method to construct such algebras.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Auslander Correspondence for CM-finite IG Algebras
Corollary [E1, E2]
There exists a bijection between the following.

(1) CM-finite Iwanaga-Gorenstein algebras Λ.
(2) Frobenius exact cats E of finite typewith higer kernels.
(3) Pairs (Γ, A), where Γ is an algebra with finite gl.dim
and A is a sets of cycles of dotted arrows of Q(Γ).
ALL CM-finite IG algebras are obtained in this way.
This gives systematic method to construct such algebras.

Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Corollary
ExampleΓ:with commutativity and zero relation
Γ has fin. gl.dim.
Thus Q(Γ) has 2 cycles of dotted arrows.
We obtain 22 = 4 CM-finite IG algebras Λ
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices.
All CM categories of Λ are the same (as additive cat)!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Corollary
Example:Q(Γ)Γ:

with commutativity and zero relation
Γ has fin. gl.dim.
Thus Q(Γ) has 2 cycles of dotted arrows.
We obtain 22 = 4 CM-finite IG algebras Λ
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices.
All CM categories of Λ are the same (as additive cat)!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Corollary
Example:Q(Γ)Γ:

with commutativity and zero relation
Γ has fin. gl.dim.
Thus Q(Γ) has 2 cycles of dotted arrows.
We obtain 22 = 4 CM-finite IG algebras Λ
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices.
All CM categories of Λ are the same (as additive cat)!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Corollary
Example:Q(Γ)Γ:

with commutativity and zero relation
Γ has fin. gl.dim.
Thus Q(Γ) has 2 cycles of dotted arrows.
We obtain 22 = 4 CM-finite IG algebras Λ
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices.
All CM categories of Λ are the same (as additive cat)!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories

Introduction
Auslander Correspondence and CM Rep. Theory
Exact Category and Main Results

Exact Category and Motivating Problems
Main Results

Example of Corollary
Example:Q(Γ)Γ:

with commutativity and zero relation
Γ has fin. gl.dim.
Thus Q(Γ) has 2 cycles of dotted arrows.
We obtain 22 = 4 CM-finite IG algebras Λ
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices.
All CM categories of Λ are the same (as additive cat)!
Haruhisa Enomoto

Relative Auslander Correspondence via Exact Categories