Auslander-Reiten-quivers of functorially finite subcategories

Transcription

1 Auslander-Reiten-quivers of functorially finite subcategories Matthias Krebs University of East Anglia August 13, 2012

2 Preliminaries

3 Preliminaries Let K be an algebraically closed field,

4 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity,

5 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity, A-mod the category of all finitely generated left A-modules,

6 Preliminaries Let K be an algebraically closed field, A a finite dimensional associative K-algebra with multiplicative identity, A-mod the category of all finitely generated left A-modules, Ω a full subcategory of A-mod closed under direct sums, direct summands and isomorphisms.

7 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y

8 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0

9 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0 Then X τ(y ) = τ Ω (Y ) I such that I is an Ext-injective module and τ Ω (Y ) is indecomposable, but not Ext-injective.

10 Approximations Definition A right Ω-approximation of a module Y is a morphism f Y : X Y Y where X Y is in Ω such that for all Z in Ω and g : Z Y, g factors through f Y. Z g f Y X Y Y 0 τ(y ) X Y 0 Then X τ(y ) = τ Ω (Y ) I such that I is an Ext-injective module and τ Ω (Y ) is indecomposable, but not Ext-injective. Definition τ Ω (Y ) is the relative Auslander-Reiten translate of Y in Ω.

11 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation,

12 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation,

13 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation, functorially finite, if it is both contravariantly and covariantly finite.

14 Functorially finite subcategories Definition Ω is called contravariantly finite, if every A-module has a right Ω-approximation, covariantly finite, if every A-module has a left Ω-approximation, functorially finite, if it is both contravariantly and covariantly finite. If Ω is contravariantly finite and closed under extensions, then Ω has an Auslander-Reiten-quiver.

15 Resolving subcategories Definition Ω is called resolving if

16 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions,

17 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms,

18 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω.

19 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω. If T is a generalized cotilting module, then Ext i A (, T ) = 0 is a contravariantly finite resolving subcategory.

20 Resolving subcategories Definition Ω is called resolving if Ω is closed under extensions, Ω is closed under kernels of epimorphisms, A is in Ω. If T is a generalized cotilting module, then Ext i A (, T ) = 0 is a contravariantly finite resolving subcategory. Let Ω be a functorially finite resolving subcategory.

21 Brauer-Thrall conjectures

22 Brauer-Thrall conjectures Brauer-Thrall 1 Ω is representation finite if and only if the Jordan-Hölder length of its indecomposable modules is bounded.

23 Brauer-Thrall conjectures Brauer-Thrall 1 Ω is representation finite if and only if the Jordan-Hölder length of its indecomposable modules is bounded. Brauer-Thrall 1.5 Let Ω be a functorially finite resolving subcategory such that there exist a positive integer n such that there are N N 0 non-isomorphic indecomposable modules of Jordan-Hölder length n in Ω, where N 0 is the cardinality of an infinite, countable set. Then there are infinitely many positive integers n 1, n 2,... with N non-isomorphic indecomposable modules of length n i in Ω for all i N.

24 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2.

25 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]

26 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]

27 Sectional paths Definition A path X 0 X 1 X n 1 X n in the Auslander-Reiten-quiver is called sectional if X j τ Ω (X j+2 ) for all j = 0,..., n 2. [P 1 ] [P I 4 ] S 3 X S 2 Y S 3 N 2 M 2 N 3 M 3 N 2 S 1 3 S 4 3 S 1 [P 2 ] [P 3 ]

28 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω.

29 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω. Moreover, the cosets of all sectional paths from X 0 to X n in A-mod such that all modules along these paths other than X 0 and X n are not in Ω are linearly independent in rad Ω (X 0, X n )/rad 2 Ω (X 0, X n ).

30 Sectional paths theorem Theorem Let X 0 f 1 X 1 f 2 f n 1 X n 1 f n X n be a sectional path in the Auslander-Reiten-quiver of A-mod such that X 0 and X n are in Ω while X 1,..., X n 1 are not in Ω. Then f n f 1 is irreducible in Ω. Moreover, the cosets of all sectional paths from X 0 to X n in A-mod such that all modules along these paths other than X 0 and X n are not in Ω are linearly independent in rad Ω (X 0, X n )/rad 2 Ω (X 0, X n ). Does the converse hold as well, i.e is a morphism in Ω given by a non-sectional path in A-mod reducible?

31 Sectional paths X n X 0

32 Sectional paths X n X 0

33 Sectional paths Y 1 Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 X n Y 2 Y 3 X 0 Y 4

34 Sectional paths Y 1 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Y 2 Y 3 X 0 Y 4

35 Sectional paths Y 1 Y 2 Y 3 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Every morphism g : X 0 X n factors through f X 0 X 0 g f Y X n Y 4

36 Sectional paths Y 1 Y 2 Y 3 X n Let Y = Y 1 Y 2 Y 3 Y 4 There is a short exact sequence 0 X 0 Y f X n 0 Y is in Ω Every morphism g : X 0 X n factors through f X 0 X 0 g f Y X n There are no irreducible morphisms from X 0 to X n Y 4

37 Sectional paths X n X 0

38 Sectional paths Y 1 Let Y = Y 1 Y 2 Y 3 X n X 0 Y 2 Z Y 3

39 Sectional paths X 0 Y 2 Y 1 X n Let Y = Y 1 Y 2 Y 3 There is a commutative diagram 0 Z Y X n 0 f Z 0 X Z f Y X Y X n 0 Z Y 3

40 Sectional paths X 0 Z Y 2 Y 3 Y 1 X n Let Y = Y 1 Y 2 Y 3 There is a commutative diagram 0 Z Y X n 0 f Z f Y 0 X Z X Y X n 0 If X Z contains only one direct summand that is not Ext-injective, then there are no Ω-irreducible morphisms from X 0 to X n.

41 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ]

42 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.

43 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.

44 A counterexample [P 1 S 3 S 2 I 1 ] [P 2 τ 2 (I 2 ) τ(i 2 ) I 2 ] [P 3 τ 2 (I 3 ) τ(i 3 ) I 3 ] [P 4 τ 2 (I 4 ) τ(i 4 ) I 4 ] [P 5 τ 2 (I 5 ) τ(i 5 ) I 5 ] T = P 1 P 4 P 5 S 2 I 2 is a generalized cotilting module and the functorially finite resolving subcategory Ω = Ext i A (, T ) = 0 contains P 2, P 3 and I 1 in addition to the direct summands of T.

45 A counterexample [P 1 ] [P 2 I 2 ] [P 3 S 2 I 1 ] [P 4 [P 5

46 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω.

47 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω. Definition A sectional subgraph Σ is a connected subgraph of Γ Ω such that all subpaths in Σ are sectional. Σ is called full, if any connected subgraph Σ of Γ Ω such that Σ Σ is not a sectional subgraph. The undirected graph Σ associated to Σ is called the type of Σ.

48 Sectional subgraphs Let Γ Ω denote the Auslander-Reiten-quiver of Ω. Definition A sectional subgraph Σ is a connected subgraph of Γ Ω such that all subpaths in Σ are sectional. Σ is called full, if any connected subgraph Σ of Γ Ω such that Σ Σ is not a sectional subgraph. The undirected graph Σ associated to Σ is called the type of Σ. For certain left or right stable components the type of a full sectional subgraph is independent from the choice of a path. In these components we define the left and right subgraph type of a component as the type of an arbitrary full sectional subgraph respectively.

49 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent:

50 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite.

51 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite. The left subgraph types of all connected components of Γ l are given by Dynkin diagrams.

52 The main result Let Γ l and Γ r denote the subquivers of an Auslander-Reiten-quiver consisting of all left stable and right stable modules in Ω respectively. Theorem Let Ω be a functorially finite resolving subcategory. Then the following is equivalent: Ω is representation finite. The left subgraph types of all connected components of Γ l are given by Dynkin diagrams. The right subgraph types of all connected components of Γ r are given by Dynkin diagrams.