Introduction
Classifications of Exact Structures
Applications

Classifications of Exact Structures
and Cohen-Macaulay-finite Algebras
Haruhisa Enomoto
Graduate School of Mathematics, Nagoya University

October 9, 2017

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Outline1

Introduction
Auslander Correspondence for CM-finite IG Algebras?

2

Classifications of Exact Structures
Exact Categories
Categories of Finite Type
Main Results

3

Applications
Classification of CM-finite IG Algebras
Other Appications

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

Outline1

Introduction
Auslander Correspondence for CM-finite IG Algebras?

2

Classifications of Exact Structures
Exact Categories
Categories of Finite Type
Main Results

3

Applications
Classification of CM-finite IG Algebras
Other Appications

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

Categories of Finite Type = Algebras
k : a field.
Proposition
There exists a bijection between:1

Hom-finite k-categories E of finite type(:⇔ categories with finitely many indecomposables).

2

Finite-dimensional k -algebra Γ(we call Γ an Auslander algebra of E).

Idea
1

Categorical properties of E and2

Homological behavior of its Auslander alg Γshould be related!
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

Categories of Finite Type = Algebras
k : a field.
Proposition
There exists a bijection between:1

Hom-finite k-categories E of finite type(:⇔ categories with finitely many indecomposables).

2

Finite-dimensional k -algebra Γ(we call Γ an Auslander algebra of E).

Idea
1

Categorical properties of E and2

Homological behavior of its Auslander alg Γshould be related!
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

Auslander correspondence for rep-fin. algebras
Theorem (Auslander 1971)
There exists a bijection between:
1

Rep-fin. algebras Λ.

2

Abelian k-categories E of finite type.

3

Algebras Γ satisfying a certain homological condition(gl.dim Γ ≤ 2 ≤ dom.dim Γ).

{ Rep-fin algebras }mod{ Cats of fin-type }E := mod ΛΛHaruhisa Enomoto

1-1{Algebras}ΓClassifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

@ Auslander Correspondence for CM-fin IG Alg?
The same method doesn’t work for CM-finite IG alg:CM
{ CM-fin algebras }{ Cats of fin-type }1-1{Algebras}

CM Λ

Λ
Λ0

Γ

The map “CM” is not injective!
i.e. ∃ non-Morita-equivalent alg Λ and Λ0 s.t. CM Λ ' CM Λ0 .
HoweverΛ can be recovered from CM Λ
together with the exact structure on it!
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

@ Auslander Correspondence for CM-fin IG Alg?
The same method doesn’t work for CM-finite IG alg:CM
{ CM-fin algebras }{ Cats of fin-type }1-1{Algebras}

CM Λ

Λ
Λ0

Γ

The map “CM” is not injective!
i.e. ∃ non-Morita-equivalent alg Λ and Λ0 s.t. CM Λ ' CM Λ0 .
HoweverΛ can be recovered from CM Λ
together with the exact structure on it!
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Auslander Correspondence for CM-finite IG Algebras?

Auslander correspondence for CM-fin algebras?

CM-finite
algebras
Λ

CM




Cats of fin-type+ Exact str. on itCM Λ andnatural exact str.

?

Algebras+ some infoΓ

Our Aimis To Construct Bijection “?” above, i.e.
To Classify exact structures on a given additive category
using its Auslander algebra.

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Outline1

Introduction
Auslander Correspondence for CM-finite IG Algebras?

2

Classifications of Exact Structures
Exact Categories
Categories of Finite Type
Main Results

3

Applications
Classification of CM-finite IG Algebras
Other Appications

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Exact Category
f

g

0→X →
− Y −
→ Z → 0 in E is a kernel-cokernel pair
if f = ker g and g = coker f .
Definition (Quillen 1973)
An exact category consists of a pair (E, F ), where
E is an additive category, andF is a class of ker-coker pairs in E
satisfying some conditions.
ExampleΛ: Iwanaga-Gorenstein alg. (⇔ id ΛΛ = id Λ Λ < ∞),CM Λ := {X ∈ mod Λ| Ext>0Λ (X , Λ) = 0} is naturally an exact cat.
and ΛΛ is the progenerator (w.r.t. F ).
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Exact Category
f

g

0→X →
− Y −
→ Z → 0 in E is a kernel-cokernel pair
if f = ker g and g = coker f .
Definition (Quillen 1973)
An exact category consists of a pair (E, F ), where
E is an additive category, andF is a class of ker-coker pairs in E
satisfying some conditions.
ExampleΛ: Iwanaga-Gorenstein alg. (⇔ id ΛΛ = id Λ Λ < ∞),CM Λ := {X ∈ mod Λ| Ext>0Λ (X , Λ) = 0} is naturally an exact cat.
and ΛΛ is the progenerator (w.r.t. F ).
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Auslander Algebras of Categories of Finite Type
From now on, fix a field k andAlgebra = finite-dimensional k -algebra.
Category = idempotent-complete Hom-finite k -category.
E: an idem-comp Hom-fin k -category of finite type(:⇔ # ind C is finite).
Definition
An Auslander algebra Γ of E is defined by Γ := EndE (G), where
G is the additive generator of E (E = add G).
{ Cats of fin-type }E
E := proj ΓHaruhisa Enomoto

1-1{Algebras}Γ := EndE (G)ΓClassifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Auslander Algebras of Categories of Finite Type
From now on, fix a field k andAlgebra = finite-dimensional k -algebra.
Category = idempotent-complete Hom-finite k -category.
E: an idem-comp Hom-fin k -category of finite type(:⇔ # ind C is finite).
Definition
An Auslander algebra Γ of E is defined by Γ := EndE (G), where
G is the additive generator of E (E = add G).
{ Cats of fin-type }E
E := proj ΓHaruhisa Enomoto

1-1{Algebras}Γ := EndE (G)ΓClassifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Projectivization
Γ := EndE (G): the Auslander algebra of E.
Proposition (Auslander’s “Projectivization”)
proj Γ

P(−)
∼
∼

P (−)

∼

E

mod Γ

HomΓ (−, Γ)proj Γop

HomΓ (−, Γ)mod Γop

We have equivalences:∼P(−) := E(G, −) : E −→ proj Γ∼

P (−) := E(−, G) : E −→ proj ΓopHaruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Coker pair in E in terms of Γ-module
E: cat of fin. type, Γ: its Auslander algebra.
Propositionfg

Let 0 → X →− Y −→ Z → 0 be a complex in E,
M := Coker(PY → PZ ) in mod Γ. Then it is a ker-coker pair ⇔1

The following is exact in mod Γ

pd MΓ ≤ 2g◦f◦

0 → PX −→ PY −→ PZ → M → 0.
2

The following is exact in mod Γop◦g

Ext0,1
Γ (M, Γ) = 0..

◦f

0 → P Z −→ P Y −→ P X → Ext2Γ (M, Γ) → 0

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Coker pair in E in terms of Γ-module
E: cat of fin. type, Γ: its Auslander algebra.
Propositionfg

Let 0 → X →− Y −→ Z → 0 be a complex in E,
M := Coker(PY → PZ ) in mod Γ. Then it is a ker-coker pair ⇔1

The following is exact in mod Γ

pd MΓ ≤ 2g◦f◦

0 → PX −→ PY −→ PZ → M → 0.
2

The following is exact in mod Γop◦g

Ext0,1
Γ (M, Γ) = 0..

◦f

0 → P Z −→ P Y −→ P X → Ext2Γ (M, Γ) → 0

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Cok pairs in E ↔ Objects in C2 (Γ)

Definition
The subcat C2 (Γ) ⊂ mod Γ consists of Γ-modules M satisfying
1

pd MΓ ≤ 2.

2

ExtΓ0,1 (M, Γ) = 0.

0→X →Y →Z →0

M := Coker(PY → PZ )Ker-cok pairs in E

Obj in C2 (Γ)

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Cok pairs in E ↔ Objects in C2 (Γ)

Definition
The subcat C2 (Γ) ⊂ mod Γ consists of Γ-modules M satisfying
1

pd MΓ ≤ 2.

2

ExtΓ0,1 (M, Γ) = 0.

0→X →Y →Z →0

M := Coker(PY → PZ )Ker-cok pairs in E

Obj in C2 (Γ)

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Cok pairs in E ↔ Objects in C2 (Γ)
Definition
The subcat C2 (Γ) ⊂ mod Γ consists of Γ-modules M satisfying
1

pd MΓ ≤ 2.

2

ExtΓ0,1 (M, Γ) = 0.

0→X →Y →Z →0

M := Coker(PY → PZ )Ker-cok pairs in E

Obj in C2 (Γ)

Classes of ker-cok pairs

Subcat of C2 (Γ)

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Ker-Cok pairs in E ↔ Objects in C2 (Γ)
Definition
The subcat C2 (Γ) ⊂ mod Γ consists of Γ-modules M satisfying
1

pd MΓ ≤ 2.

2

ExtΓ0,1 (M, Γ) = 0.

0→X →Y →Z →0

M := Coker(PY → PZ )Ker-cok pairs in E

Obj in C2 (Γ)

Classes of ker-cok pairs

Subcat of C2 (Γ)∪∪

Exact str. on E
Haruhisa Enomoto???
Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

Main Result I
E: cat. of fin. type, Γ: its Auslander algebra.
We have a duality Ext2Γ (−, Γ) : C2 (Γ) ↔ C2 (Γop ).
Theorem (E)There exists a bijection between the following two classes.
1
2

Exact structures F on E.
Subcategories D of C2 (Γ) satisfying the following.
D is a Serre subcat. of mod Γ.
Ext2Γ (D, Γ) is a Serre subcat. of mod Γop .

D ⊂ mod Γ is Serre :⇔ D is closed under submodules,
factor modules and extensions.

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

2-Regular Condition
Serre subcats of mod Γ ! sets of simple Γ-modules.
Definition
A simple Γ-module S is called 2-regular :⇔1

S ∈ C2 (Γ), i.e, pd SΓ = 2 and Ext0,1
Γ (S, Γ) = 0.

2

Ext2Γ (S, Γ) is a simple Γop -module.

It’s a “regular version” of 2-Gorenstein condition.
2-regular simple Γ-mod correspond to AR ker-coker pairs in E:
0→X →Y →Z →0:
AR ker-cok pair in E

Haruhisa Enomoto

0 → PX → PY → PZ → S → 0
2-reg. simple Γ-mod S

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

2-Regular Condition
Serre subcats of mod Γ ! sets of simple Γ-modules.
Definition
A simple Γ-module S is called 2-regular :⇔1

S ∈ C2 (Γ), i.e, pd SΓ = 2 and Ext0,1
Γ (S, Γ) = 0.

2

Ext2Γ (S, Γ) is a simple Γop -module.

It’s a “regular version” of 2-Gorenstein condition.
2-regular simple Γ-mod correspond to AR ker-coker pairs in E:
0→X →Y →Z →0:
AR ker-cok pair in E

Haruhisa Enomoto

0 → PX → PY → PZ → S → 0
2-reg. simple Γ-mod S

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

AR Quivers and Main Result II
E: cat. of fin. type, Γ: its Auslander algebra.
Definition
The AR quiver Q(E) of E is the translation quiver defined by:
Quiver = the usual quiver of E (or Γ)
X L99 Z if ∃ an AR ker-cok pair 0 → X → Y → Z → 0 in E.
Theorem (E)There exists a bijection between the following classes.
1

Exact structures on E.

2

Sets of 2-regular simple Γ-modules.

3

Sets of dotted arrows in Q(E) (= Q(proj Γ) ).
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

AR Quivers and Main Result II
E: cat. of fin. type, Γ: its Auslander algebra.
Definition
The AR quiver Q(E) of E is the translation quiver defined by:
Quiver = the usual quiver of E (or Γ)
X L99 Z if ∃ an AR ker-cok pair 0 → X → Y → Z → 0 in E.
Theorem (E)There exists a bijection between the following classes.
1

Exact structures on E.

2

Sets of 2-regular simple Γ-modules.

3

Sets of dotted arrows in Q(E) (= Q(proj Γ) ).
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Exact Categories
Categories of Finite Type
Main Results

ExampleExampleE := mod k [• ← • ← •].
∃ 3 dotted arrow, hence
Q(E) :∃ 23 = 8 exact str. on E

Red arrows are chosen.
No arrows ↔ trivial exact str. of E (the smallest one).
All arrows ↔ usual exact str. of E (the largest one).

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Outline1

Introduction
Auslander Correspondence for CM-finite IG Algebras?

2

Classifications of Exact Structures
Exact Categories
Categories of Finite Type
Main Results

3

Applications
Classification of CM-finite IG Algebras
Other Appications

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Characterizing CM categories of IG algebrasCM-finiteIG-alg. ΛCMExact cats Eof finite type



1-1
X




Alg. Γ + sets ofdotted arrows

E: exact cat. of fin. type, Γ: its Auslander algebra.
Proposition
E ' CM Λ as exact cats for some IG algebra Λ ⇔1gl.dim Γ < ∞.

2

Projective objects in E = Injective objects in E(⇔ E is a Frobenius exact cat)Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Characterizing CM categories of IG algebrasCM-finiteIG-alg. ΛCMExact cats Eof finite type



1-1
X




Alg. Γ + sets ofdotted arrows

E: exact cat. of fin. type, Γ: its Auslander algebra.
Proposition
E ' CM Λ as exact cats for some IG algebra Λ ⇔1gl.dim Γ < ∞.

2

Projective objects in E = Injective objects in E(⇔ E is a Frobenius exact cat)Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Corollary
There exists a bijection between the following.
1

CM-finite Iwanaga-Gorenstein algebras Λ.

2Pairs (Γ, A), where Γ is an algebra with gl.dim Γ < ∞ and
A is a set of dotted arrows of Q(proj Γ) which is union of
stable τ -orbits (i.e. A: disjoint union of S 1 ’s)(Γ, A) corresponds to Λ := EndΓ (P), where P is the direct sumof proj. Γ-modules which are not contained in A.
ALL CM-finite IG algebras are obtained by the following steps.
1

Take any algebra Γ with finite global dimension.

2

Draw the translation quiver Q(proj Γ).

3

For each union of stable τ -orbit of it, compute Λ.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Corollary
There exists a bijection between the following.
1

CM-finite Iwanaga-Gorenstein algebras Λ.

2Pairs (Γ, A), where Γ is an algebra with gl.dim Γ < ∞ and
A is a set of dotted arrows of Q(proj Γ) which is union of
stable τ -orbits (i.e. A: disjoint union of S 1 ’s)(Γ, A) corresponds to Λ := EndΓ (P), where P is the direct sumof proj. Γ-modules which are not contained in A.
ALL CM-finite IG algebras are obtained by the following steps.
1

Take any algebra Γ with finite global dimension.

2

Draw the translation quiver Q(proj Γ).

3

For each union of stable τ -orbit of it, compute Λ.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Corollary
There exists a bijection between the following.
1

CM-finite Iwanaga-Gorenstein algebras Λ.

2Pairs (Γ, A), where Γ is an algebra with gl.dim Γ < ∞ and
A is a set of dotted arrows of Q(proj Γ) which is union of
stable τ -orbits (i.e. A: disjoint union of S 1 ’s)(Γ, A) corresponds to Λ := EndΓ (P), where P is the direct sumof proj. Γ-modules which are not contained in A.
ALL CM-finite IG algebras are obtained by the following steps.
1

Take any algebra Γ with finite global dimension.

2

Draw the translation quiver Q(proj Γ).

3

For each union of stable τ -orbit of it, compute Λ.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:
Q(proj Γ):Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:
Q(proj Γ):: CM Λ

Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:
Q(proj Γ):: CM Λ

Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:
Q(proj Γ):: CM Λ

Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

ExampleQ:
Q(proj Γ):: CM Λ

Γ := kQ/(commutativity and zero relation)(two vertical arrows are identified).
⇒ the above is Q(proj Γ). Thus ∃ 2 stable τ -orbits.
We obtain 22 = 4 CM-fin IG algebras Λ.
A: Orange Dotted Arrows.
Corresponding CM-finite IG Λ is the End of Red vertices,
projective object in this exact structure.
Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

NON-Example

Q(proj Γ):

If A: Orange, then in the corresponding exact str,
Red: proj. objects,
Blue: injective objects.
Proj 6= Inj. (not Frobenius)(This exact cat. is ⊥ U for some cotilting Λ-module U)

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras

Introduction
Classifications of Exact Structures
Applications

Classification of CM-finite IG Algebras
Other Appications

Other Applications
For an exact category E of finite type,E has enough projectives and injectives (if k is a field).
the relation of the Grothendieck group K0 (E) is generated
by AR sequences in E.
Instead of CM-fin IG alg, a similar classification is available forcotilting Λ-modules U s.t. ⊥ U is of finite type.
Auslander-type correspondence for representation-finite
R-orders for dim R ≥ 2.
and so on...

Haruhisa Enomoto

Classifications of Exact Structures and CM-finite Algebras