AUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS. Piotr Malicki

AUSLANDER-REITEN THEORY FOR FINITE DIMENSIONAL ALGEBRAS Piotr Malicki CIMPA, Mar del Plata, March 2016

3. Irreducible morphisms and almost split sequences A algebra, L, M, N modules in mod A A homomorphism f : L M in mod A is called left minimal if every h End A (M) s.t. hf = f is an isomorphism. A homomorphism g : M N in mod A is called right minimal if every e End A (M) s.t. ge = g is an isomorphism. A homomorphism f : L M in mod A is called left almost split if: f is not a section for every homomorphism u : L U which is not a section there exists u : M U s.t. u f = u. A homomorphism g : M N in mod A is called right almost split if: f is not a retraction for every homomorphism v : V N which is not a retraction there exists v : V M s.t. gv = v. 1

A homomorphism f : L M in mod A is called left minimal almost split if it is both left minimal and left almost split. A homomorphism g : M N in mod A is called right minimal almost split if it is both right minimal and right almost split. Proposition 3.1. (1) Let f : L M and f : L M be left minimal almost split homomorphisms in mod A. Then there exists an isomorphism h : M M in mod A s.t. f = hf. (2) Let g : M N and g : M N be right minimal almost split homomorphisms in mod A. Then there exists an isomorphism e : M M in mod A s.t. g = g e. Proof. 2

Lemma 3.2. (1) Let f : L M be a left almost split homomorphism in mod A. Then the module L is indecomposable. (2) Let g : M N be a right almost split homomorphism in mod A. Then the module N is indecomposable. Proof. A homomorphism f : X Y in mod A is said to be irreducible provided: (1) f is neither a section nor a retraction (2) if f = f 1 f 2, then either f 1 is a retraction or f 2 is a section. Lemma 3.3. Let f : X Y be an irreducible homomorphism in mod A. Then f is either a proper monomorphism or a proper epimorphism. 3

Lemma 3.4. (Auslander-Reiten) Let P be an indecomposable projective module in mod A, and u : rad P P be the inclusion homomorphism. Then (1) u is right minimal almost split in mod A (2) u is irreducible in mod A. Lemma 3.5. (Auslander-Reiten) Let I be an indecomposable injective module in mod A, and v : I I/ soc I the canonical epimorphism. Then (1) v is left minimal almost split in mod A (2) v is irreducible in mod A. Lemma 3.6. (Bautista) Let X, Y be indecomposable modules in mod A, and f Hom A (X, Y ). Then f is an irreducible homomorphism iff f rad A (X, Y ) \ rad 2 A (X, Y ). 4

Lemma 3.7. Let 0 L f M g N 0 be a nonsplit short exact sequence in mod A. (1) The homomorphism f is irreducible iff for every homomorphism v : V N, there exists v 1 : V M s.t. v = gv 1 or v 2 : M V s.t. g = vv 2. (2) The homomorphism g is irreducible iff for every homomorphism u : L U, there exists u 1 : M U s.t. u = u 1 f or u 2 : U M s.t. f = u 2 u. Corollary 3.8. (1) Let f : L M be an irreducible monomorphism in mod A. Then N = Coker f is indecomposable. (2) Let g : M N be an irreducible epimorphism in mod A. Then L = Ker g is indecomposable. 5

Theorem 3.9. (1) Let f : L M be a nonzero left minimal almost split homomorphism in mod A. Then (a) f is irreducible in mod A. (b) A homomorphism f : L M in mod A is irreducible iff M 0 and there exists a direct sum decomposition M = M M and a homomorphism f : L M s.t. [ f f ] : L M M is left minimal almost split homomorphism in mod A. (2) Let g : M N be a nonzero right minimal almost split homomorphism in mod A. Then (a) g is irreducible in mod A. (b) A homomorphism g : M N in mod A is irreducible iff M 0 and there exists a direct sum decomposition M = M M and a homomorphism g : M N s.t. [ g g ] : M M N is right minimal almost split homomorphism in mod A. 6

A short exact sequence 0 L f M g N 0 in mod A is called an almost split sequence (Auslander-Reiten sequence) provided: f is left minimal almost split g is right minimal almost split Remarks. (1) An almost split sequence is never split (because f is not a section and g is not a retraction). (2) The modules L and N are indecomposable (it follows from Lemma 3.2). (3) L is not injective and N is not projective. 7

Lemma 3.10. Let 0 L f M g N 0 and 0 L f M g N 0 be two almost split sequences in mod A. TFAE (1) The two sequences are isomorphic. (2) The modules L and L are isomorphic. (3) The modules N and N are isomorphic. Proof. 8

Lemma 3.11. Let 0 L f M g N 0 u 0 L f M g N 0 be a commutative diagram in mod A, where the rows are exact and not split. Then (1) If L is indecomposable and w is an isomorphism, then u and hence v are isomorphisms. (2) If N is indecomposable and u is an isomorphism, then w and hence v are isomorphisms. v w Proof. 9

Theorem 3.12. Let 0 L f M g N 0 be a short exact sequence in mod A. TFAE (1) The given sequence is almost split sequence. (2) L is indecomposable, and g is right almost split. (3) N is indecomposable, and f is left almost split. (4) f is left minimal almost split. (5) g is right minimal almost split. (6) L and N are indecomposable, and f, g are irreducible. Some implications of the proof. (1) (4), (1) (5) by definition of almost split sequence (1) (2), (1) (3) by Lemma 3.2 (1) (6) by Lemma 3.2 and Theorem 3.9. 10

4. The Auslander-Reiten theorems A algebra proj A inj A full subcategory of mod A consisting of all projective modules full subcategory of mod A consisting of all injective modules Consider the contravariant functor (following Auslander and Bridger) ( ) t = Hom A (, A) : mod A mod A op then the functor ( ) t induces the duality proj A ( )t proj A op ( ) t 11

Remarks. Let e be an idempotent in A. (1) (ea) t = Hom A (ea, A) = Ae = ea op. (2) Every module in proj A is a direct sum of the modules of the form ea, where e is primitive. (3) Every module in proj A op is a direct sum of the modules of the form Ae = ea op, where e is primitive. Let M mod A and p ( ) P 1 p 1 P 0 0 M 0 be a minimal projective presentation of M in mod A. For ( ) we have in mod A op the induced exact sequence 0 M t p t 0 P t 0 p t 1 P t 1 Coker pt 1 0. Then Coker p t 1 we denoted by Tr M and called transpose of M. Remark. Tr(M) is uniquely determined by M, up to isomorphism. 12

Proposition 4.1. Let M be an indecomposable module in mod A. Then (1) Tr(M) has no nonzero projective direct summands. (2) If M is not projective, then the sequence P0 t p t 1 P1 t Tr M 0 induced from ( ) is a minimal projective presentation of Tr M in mod A op. (3) M is projective iff Tr M = 0. (4) If M is not projective, then Tr M is indecomposable and Tr(Tr M) = M. (5) If M, N are indecomposable nonprojective, then M = N iff Tr M = Tr N. Remark. Tr does not define a duality. 13

For M, N mod A we define two ideals P A and I A of mod A. P A (M, N) = f Hom A (M, N) f = f 2 f 1, f 1 Hom A (M, P ), f 2 Hom A (P, N), P proj A I A (M, N) = g Hom A (M, N) g = g 2 g 1, g 1 Hom A (M, I), g 2 Hom A (I, N), I inj A Now, we can define the projectively stable category: moda = mod A/P A objects of moda = objects of mod A K-vector space of morphisms from M to N in moda is the quotient vector space Hom A (M, N) = Hom A (M, N)/P A (M, N) the composition of morphisms in moda is induced from the composition of homomorphisms in mod A 14

Similarly, we define the injectively stable category: moda = mod A/I A. Proposition 4.2. The transpose Tr induces Tr a duality moda moda op Tr Recall that we have the standard duality D = Hom K (, K) : mod A mod A op mod A D mod A op D proj A D D inj A op So, D induces a duality between the stable categories D : moda moda op In particular, we have the equivalences of the categories τ A = D Tr : moda moda, = Tr D : moda moda τ 1 A called the Auslander-Reiten functors. 15

For M mod A we have well-defined modules in mod A: τ A M = D Tr(M) and τ 1 M = Tr D(M) A called the Auslander-Reiten translations of M. Proposition 4.3. mod A. Then Let M be a module in (1) pd A M 1 iff Hom A (D(A), τ A M) = 0. (2) id A M 1 iff Hom A (τa 1 M, A) = 0. Theorem 4.4. (Auslander-Reiten) Let M, N mod A. Then there exist isomorphisms of K-vector spaces DHom A (τa 1 N, M) = Ext 1 A (M, N) = DHom A (N, τ A M). 16

Corollary 4.5.Let M, N mod A. (1) If pd A M 1, then there exists a K-linear isomorphism Ext 1 A (M, N) = D Hom A (N, τ A M). (2) If id A M 1 then there exists a K-linear isomorphism Ext 1 A (M, N) = D Hom A (τ 1 N, M). A Proof. Theorem 4.6. (Auslander-Reiten) (1) For any indecomposable nonprojective module M mod A, there exists an almost split sequence in mod A 0 τ A M E M 0. (2) For any indecomposable noninjective module N mod A, there exists an almost split sequence in mod A 0 N F τ 1 A N 0. 17

Proposition 4.7. (1) Let M be an indecomposable nonprojective module in mod A. Then there exists an irreducible morphism f : X M iff there exists an irreducible morphism f : τ A M X. (2) Let N be an indecomposable noninjective module in mod A. Then there exists an irreducible morphism g : N Y iff there exists an irreducible morphism g : Y τa 1 N. Proof. 18

Corollary 4.8. (1) Let S be a simple projective noninjective module in mod A. If f : S M is irreducible, then M is projective. In particular, we have in mod A the almost split sequence 0 S P τ 1 A S 0, where P is projective. (2) Let S be a simple injective nonprojective module in mod A. If g : M S is irreducible, then M is injective. In particular, we have in mod A the almost split sequence 0 τ A S I S 0, where I is injective. 19

Proposition 4.9. Let P be a nonsimple indecomposable projective-injective module. Then the sequence 0 rad P [q,u]t (rad P/ soc P ) P [ j,v] P/ soc P 0, where u, j are the inclusion homomorphisms and q, v are the canonical epimorphisms, is an almost split sequence in mod A. Example 4.10. Let A = KQ/I, where 1 4 λ Q : 3 µ β 2 5 and I is the ideal of KQ generated by βλ, σµ, αβ γσ. σ α γ 6 S 1 - simple projective noninjective summand of rad P 3 S 2 - simple projective noninjective summand of rad P 3 20

by Corollary 4.8 (1) we have almost split sequences 0 S 1 P 3 P 3 /S 1 0, P 3 /S 1 = τ 1 A S 1 0 S 2 P 3 P 3 /S 2 0, P 3 /S 2 = τ 1 A S 2 S 6 - simple injective nonprojective by Corollary 4.8 (2) we have an almost split sequence 0 P 6 /S 3 I 4 I 5 S 6 0, P 6 /S 3 = τ A S 6 P 6 = I 3 - projective-injective by Proposition 4.9 we have an almost split sequence 0 rad P 6 S 4 S 5 P 6 P 6 /S 3 0, where S 4 S 5 = rad P 6 / soc P 6 21

5. The Auslander-Reiten quiver of an algebra A finite dimensional K-algebra over a field K Z indecomposable module in mod A End A (Z) local K-algebra F Z = End A (Z)/ rad End A (Z) = End A (Z)/ rad A (Z, Z) division K-algebra X, Y indecomposable modules in mod A irr A (X, Y ) = rad A (X, Y )/ rad 2 A (X, Y ) the space of irreducible homomorphisms from X to Y irr A (X, Y ) is an F Y -F X -bimodule by (h+rad A (Y, Y ))(f+rad 2 A (X, Y )) = hf+rad2 A (X, Y ) (f+rad 2 A (X, Y ))(g+rad A(X, X)) = fg+rad 2 A (X, Y ) for f rad A (X, Y ), g End A (X), h End A (Y ) d XY = dim FY irr A (X, Y ) d XY = dim F X irr A (X, Y ) 22

Γ A Auslander Reiten quiver of A valued translation quiver defined as follows: The vertices of Γ A are the isoclasses {X} of indecomposable modules X in mod A For two vertices {X} and {Y }, there exists an arrow {X} {Y } iff irr A (X, Y ) 0. Then we have in Γ A the valued arrow {X} (d XY,d XY ) {Y } τ A translation of Γ A defined on each nonprojective vertex {X} of Γ A by τ A {X} = {τ A X} = {D Tr X} τ 1 A translation of Γ A defined on each noninjective vertex {X} of Γ A by τa 1 1 {X} = {τ X} = {Tr DX} A Tr - transpose, D - standard duality We identify a vertex {X} of Γ A with the indecomposable module X and write X (d XY,d XY ) Y instead of {X} (d XY,d XY ) {Y } and X Y instead of X (1,1) Y 23

X, Y indecomposable modules in mod A (vertices of Γ A ) d XY = multiplicity of Y in the codomain of a minimal left almost split homomorphism in mod A with the domain X X f M = Y d XY M without direct summand isomorphic to Y M Remarks. (1) We know that there exists a minimal left almost split homomorphism X M in mod A if X is injective, then by Lemma 3.5 we have X X/ soc X = M if X is not injective, then by Theorem 3.12 (4) and Theorem 4.6 (2) we have almost split sequence 0 X f M τ 1 X 0, where f is minimal left almost split. (2) By Theorem 3.9, for any irreducible homomorphism X Y m we get m d XY. A 24

d XY = multiplicity of X in the domain of a minimal right almost split homomorphism in mod A with the codomain Y N N X d XY = N g Y without direct summand isomorphic to X Remarks. (1) We know that there exists a minimal right almost split homomorphism N Y in mod A if Y is projective, then by Lemma 3.4 we have N = rad Y Y if Y is not projective, then by Theorem 3.12 (5) and Theorem 4.6 (1) we have almost split sequence 0 τ A Y f N Y 0, where f is minimal left almost split. (2) By Theorem 3.9, for any irreducible homomorphism X n Y we get n d XY. 25

Proposition 5.1. Let X, Y be indecomposable modules in mod A, and assume that there exists an irreducible homomorphism from X to Y. Then (1) If Y is nonprojective, then d τ A Y,X = d XY. (2) If X is noninjective, then d Y,τ 1 A X = d XY. Remarks. Assume X (d XY,d XY ) Y is an arrow in Γ A (1) If A is an algebra of finite representation type, then d XY = 1 or d XY = 1. (2) If A is an algebra over an algebraically closed field K, then d XY = d XY (because F X = K = F Y ). In particular, d XY = d XY finite representation type. = 1 if A is of 26

Component of Γ A = connected component of the quiver Γ A Shapes of components of Γ A give important information on A and mod A locally finite valued quiver without loops and multiple arrows 0 set of vertices of 1 set of arrows of d, d : 1 0 the valuation maps x (d xy,d xy ) y Z valued translation quiver (Z ) 0 = Z 0 = { } (i, x) i Z, x 0 set of vertices of Z. (Z ) 1 set of arrows of Z consists of the valued arrows (i, x) (d xy,d xy ) (i, y), (i + 1, y) (d xy,d xy) (i, x), i Z, for all arrows x (d xy,d xy ) y in 1. The translation τ : Z 0 Z 0 is defined by τ(i, x) = (i + 1, x) for all i Z, x 0. Z stable valued translation quiver 27

For a subset I of Z, I is the full translation subquiver of Z given by the set of vertices (I ) 0 = I 0. In particular, we have the valued translation subquivers N and ( N) of Z. Example 5.2. Let : 1 (1,3) (4,2) 2 3 Z of the form (1, 1) (1,3)(3,1) (1, 2) (0, 1) (1,3)(3,1) (0, 2) ( 1, 1) (1,3)(3,1) ( 1, 2) ( 2, 1) (1, 3) (4,2)(2,4) (0, 3) (4,2)(2,4) ( 1, 3) (4,2)(2,4) ( 2, 3) N of the form (3, 1) (1,3)(3,1) (2, 1) (1,3)(3,1) (1, 1) (1,3)(3,1) (0, 1) (3, 2) (2, 2) (1, 2) (3, 3) (4,2)(2,4) (2, 3) (4,2)(2,4) (1, 3) (4,2)(2,4) (0, 3) ( N) of the form (0, 1) (1,3) (3,1) ( 1, 1) (1,3) (3,1) ( 2, 1) (1,3) (3,1) ( 3, 1) (0, 2) ( 1, 2) ( 2, 2) (0, 3) (4,2) (2,4) ( 1, 3) (4,2) (2,4) ( 2, 3) (4,2) (2,4) ( 3, 3) 28

A : 0 1 2 3... ZA is the translation quiver (i + 1, 0) (i, 0) (i 1, 0) (i 2, 0)... (i + 1, 1) (i, 1) (i 1, 1)...... (i + 1, 2) (i, 2)...... τ(i, j) = (i + 1, j) for all i Z, j N..... For r 1, we may consider the translation quiver ZA /(τ r ) obtained from ZA by identifying each vertex x with τ r x and each arrow x y with τ r x τ r y. ZA /(τ r ) stable tube of rank r. 29

A algebra C component of Γ A is regular if C contains neither a projective module nor an injective module (equivalently, τ A and τa 1 are defined on all vertices of C ) Theorem 5.3. (Liu, Zhang) Let C be a regular component of Γ A. The following equivalences hold. (1) C contains an oriented cycle iff C is a stable tube ZA /(τ r ), for some r 1. (2) C is acyclic iff C is of the form Z for a connected, locally finite, acyclic, valued quiver. A component C of Γ A is postprojective if C is acyclic and each module in C is of the form τa m P for a projective module P in C and some m 0. A component C of Γ A is preinjective if C is acyclic and each module in C is of the form τa m I for an injective module I in C and some m 0. 30

Example 5.4. Let [ ] {[ R 0 a 0 A = = C C c b ] M 2 (C) a R, b, c C Then A is 5-dimensional R-algebra of finite representation type and Γ A is of the form } where e 1 = (1,2) P 1 [ 1R 0 0 0 P 2 (2,1) ], e 2 = I 2 I 1 (1,2) [ 0 0 0 1 C ], P 1 = e 1 A, P 2 = e 2 A, I 1 = D(Ae 1 ) = Hom R (Ae 1, R), I 2 = D(Ae 2 ) = Hom R (Ae 2, R), P 1 = [ R 0 0 0 ] = S 1 = e 1 A/e 1 rad A simple projective (rad A = [ 0 0 C 0 ], A/ rad A = R C) 31

[ 0 0 P 2 = C C ] S 2 = e 2 A/e 2 rad A = [ 0 0 0 C ] = C we have the valued arrow P 1 (d P1 P 2,d P 1 P 2 ) P 2 d P1 P 2 multiplicity of P 2 in a minimal left almost split homomorphism P 1 M One can show that we have an almost split sequence 0 P 1 P 2 I 1 0. Therefore, d P1 P 2 = 1. d P 1 P 2 multiplicity of P 1 in a minimal right almost split homomorphism N P 2 N = rad P 2 = [ 0 0 C 0 ] = Therefore, d P 1 P 2 = 2. [ 0 0 R Ri 0 ] = P 1 P 1 we get the valued arrow P 1 (1,2) P 2 32

Example 5.5. Let A = 2 C 0 0 0 0 C C 0 0 0 C 0 C 0 0 C 0 0 C 0 C 0 0 C R : 1 3 4 Then Γ A : (2,1) 5 P(A)... R(A)... Q(A) where P(A) = ( N) is a postprojective component containing all indecomposable projective A-modules of the form P 2 τa 1 P 2 P 1 P 3 τa 1 P 1 τ 1 A P 3 τa 2 P 1 P 4 τa 1 P 4 (2,1) P 5 (1,2) (2,1) τ 1 A P 5 (1,2) 33

Q(A) = N is a preinjective component containing all indecomposable injective A-modules of the form τ A I 2 I 2 τ A I 1 τ A I 3 I 1 I 3 (1,2) τ A I 4 (1,2) I 4 (2,1) τ 2 A I 5 (2,1) τ A I 5 (2,1) I 5 R(A) is a family of all regular components containing one stable tube of rank 3 one stable tube of rank 2 infinitely many stable tubes of ranks 1 34

[ R 0 Example 5.6. Let A = H H ]. Then Γ A : : 1 (1,4) 2 P(A)... R(A)... Q(A) P(A) = ( N) is a postprojective comp. containing all indecomp. proj. A-modules of the form P 1 (1,4) P 2 (4,1) τ 1 A P 1 τa 1 P 2 (4,1) (1,4) τ 2 A P 1 (1,4) Q(A) = N is a preinjective comp. containing all indecomp. inj. A-modules of the form (1,4) τ 2 A I 2 (4,1) τ A I 1 (1,4) τ A I 2 (4,1) I 2 I 1 (1,4) R(A) is a family of all regular components containing infinitely many stable tubes of ranks 1. 35

6. Hereditary algebras A finite dimensional K-algebra over a field K A is right hereditary if any right ideal of A is a projective right A-module A is left hereditary if any left ideal of A is a projective left A-module Theorem 6.1. TFAE (1) gl. dim A 1. (2) A is right hereditary. (3) A is left hereditary. (4) Every right A-submodule of a projective module in mod A is projective. (5) Every factor module of an injective module in mod A is injective. (6) The radical rad P of any indecomposable projective module P in mod A is projective. (7) The socle factor I/ soc I of any indecomposable injective module I in mod A is injective. A is hereditary if A is left and right hereditary 36

Q A valued quiver of A 1, 2,..., n vertices of Q A there is an arrow i j in Q A if dim K Ext 1 A (S i, S j ) 0, and has the valuation (dim EndA (S j ) Ext1 A (S i, S j ), dim EndA (S i ) Ext1 A (S i, S j )) End A (S 1 ), End A (S 2 ),..., End A (S n ) are division K-algebras G A = Q A (underlying graph of Q A ) valued graph of A A hereditary K-algebra A is of Dynkin type if G A is a Dynkin graph A is of Euclidean type if G A is an Euclidean graph A is of wild type if G A is neither a Dynkin nor Euclidean graph 37

Theorem 6.2. Let A be an indecomposable finite dimensional hereditary K-algebra over a field K, and Q A the valued quiver of A. Then Γ A has the following shape P(A)... R(A)... Q(A) P(A) is the postprojective component containing all indecomposable projective A- modules Q(A) is the preinjective component containing all indecomposable injective A-modules R(A) is the family of all regular components Moreover, (1) If A is of Dynkin type, then P(A) = Q(A) is finite and R(A) is empty. (2) If A is of Euclidean type, then P(A) = ( N)Q op A, Q(A) = NQ op A and R(A) is an infinite family of stable tubes, all but finitely many of them of rank one. (3) If A is of wild type, then P(A) = ( N)Q op Q(A) = NQ op A, and R(A) is an infinite family of components of type ZA. 38 A,

7. The number of terms in the middle of almost split sequences A algebra, M mod A l(m) length of M (length of a composition series 0 = M 0 M 1... M l(m) = M, M i+1 /M i simple, i {0,..., l(m) 1}) A path X 1 X 2 X t 1 X t in Γ A is called sectional if for each i {1,..., t 2} we have X i = τa X i+2. Lemma 7.1. Let 0 X f r Y i i=1 g Z 0 be an almost split sequence in mod A, Y 1,..., Y r indecomposable modules, and l(y i ) < l(x) for any i {1,..., r}. Then any sectional path in Γ A ending with Z does not contain projective module. Proof. 39

Proposition 7.2. Let f : X 4 i=1 Y i be an irreducible homomorphism in mod A, X indecomposable, and Y 1,..., Y 4 indecomposable nonprojectives. Let 4 i=1 l(y i ) 2l(X). Then X has no projective predecessor in Γ A. Corollary 7.3. Let f : X 4 i=1 Y i be an irreducible epimorphism in mod A, X indecomposable, and Y 1,..., Y 4 indecomposable nonprojectives. Then X has no projective predecessor in Γ A. We have dual facts... 40

Lemma 7.4. Let 0 X f r Y i i=1 g Z 0 be an almost split sequence in mod A, Y 1,..., Y r indecomposable modules, and l(y i ) < l(z) for any i {1,..., r}. Then any sectional path in Γ A starting at X does not contain injective module. Proposition 7.5. Let g : 4 i=1 Y i Z be an irreducible homomorphism in mod A, Z indecomposable, and Y 1,..., Y 4 indecomposable noninjectives. Let 4 i=1 l(y i ) 2l(Z). Then Z has no injective successor in Γ A. Corollary 7.6. Let g : 4 i=1 Y i Z be an irreducible monomorphism in mod A, Z indecomposable, and Y 1,..., Y 4 indecomposable noninjectives. Then Z has no injective successor in Γ A. 41

Theorem 7.7. (Liu) Let 0 X f r Y i i=1 g Z 0 be an almost split sequence in mod A, and Y 1,..., Y r indecomposable modules. Assume that X has a projective predecessor in Γ A and Z has an injective successor in Γ A. Then r 4, and r = 4 implies that Y i is projectiveinjective for some i {1,..., 4} and Y j is not projective-injective for any j {1,..., 4} \ {i}. Corollary 7.8. (Bautista-Brenner) Let A an algebra of finite representation type, and 0 X f r Y i i=1 g Z 0 be an almost split sequence in mod A, where Y i is indecomposable for any i {1,..., r}. Then r 4, and r = 4 implies that one of the Y i is projective-injective. Remark. If A is of finite representation type, then any indecomposable module has a projective predecessor and an injective successor in Γ A. 42

Example 7.9. Let A = KQ/I, where α β 2 Q : 1 3 ε µ 5 and I is the ideal of KQ generated by εα µβ, µβ ωγ. γ 4 ω Then A is of finite representation type and Γ A is of the form P 2 S 2 I 2 P 1 P 3 R S 3 I 3 I 5 P 4 S 4 I 4 P 5 where P 1 = S 1, R = rad P 5, P 5 = I 1, I 5 = S 5. 43

Example 7.10. Let A = KQ, where 1 Q : 0 2. Then Γ A contains a component P(A) of the form n P 0 P 1 P 2. τa 1 P 0 τ 1 A P 1 τa 1 P 2.. P n τ 1 A P n and a component Q(A) of the form τ A I 1 I 1 = S 1 τ A I 0 τ A I 2 I 0 I 2 = S 2... τ A I n I n = S n 44

Conjecture 7.11. (Brenner-Butler) Let A be a tame finite dimensional K-algebra over an algebraically closed field K, 0 X f r Y i i=1 g Z 0 an almost split sequence in mod A, and X indecomposable nonprojective module. Then r 5. We give the affirmative answer in the case of cycle-finite algebras. Let A be a finite dimensional K-algebra. A cycle of indecomposable modules in mod A is a sequence X 0 f 1 X 1 X r 1 f r X r = X 0 of nonzero nonisomorphisms in mod A, where X i is indecomposable for i {1,..., r}, and such a cycle is said to be finite if the homomorphisms f 1,..., f r do not belong to rad A. 45

An algebra A is said to be cycle-finite if all cycles between indecomposable modules in mod A are finite. Theorem 7.12. (M.-de la Peña-Skowroński) Let A be a cycle-finite K-algebra, 0 X f r Y i i=1 g Z 0 an almost split sequence between indecomposable modules in mod A, and X nonprojective module. Then r 5, and r = 5 implies that Y i is projective-injective for some i {1,..., 5} and Y j is not projective-injective for any j {1,..., 5} \ {i}. Remark. For finite dimensional cycle-finite algebras over an algebraically closed field Theorem 7.12 has been proved by de la Peña and Takane, by application of spectral properties of Coxeter transformations of algebras and Liu s results. 46