ALGEBRAS OF SMALL HOMOLOGICAL DIMENSION Andrzej Skowro nski - PDF document

� � � � � � � � � � � • For each nonprojective indecomposable module Y in mod A , the quiver Γ A ad- mits a valued mesh { V 1 } ( d ′ V 1 Y ,d V 1 Y ) ( d V 1 Y ,d ′ � � V 1 Y ) � � � � � � � � � � � � � � � � � � � � � { V 2 } � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( d ′ � � � � � ( d V 2 Y ,d ′ � V 2 Y ,d V 2 Y ) � � � τ A { Y } = { τ A Y } V 2 Y ) � � { Y } . . � � � � � � � � � � . � � � � � � � � � � � � � � � � ( d VrY ,d ′ � � � � ( d ′ � � VrY ) � � � � � VrY ,d VrY ) � � � � { V r } such that there is in mod A an almost split sequence r d ′ � ViY 0 − → τ A Y − → − → Y − → 0 . V i i =1 • For each noninjective indecomposable mod- ule X in mod A , the quiver Γ A admits a valued mesh { U 1 } ( d ′ XU 1 ,d XU 1 ) ( d XU 1 ,d ′ � � � XU 1 ) � � � � � � � � � � � � � � � � � � � { U 2 } � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( d ′ � � � ( d XU 2 ,d ′ � XU 2 ,d XU 2 ) { τ − 1 A X } = τ − 1 � � � XU 2 ) � { X } � A { X } . � � � . � � � � � � � � � � . � � � � � � � � � � � ( d XUs ,d ′ � � � � XUs ) � � ( d ′ � � � � � � � � XUs ,d XUs ) � � � { U s } � such that there is in mod A an almost split sequence s � d XUj → τ − 1 0 − → X − → U − A X − → 0 . j j =1 13

� � � � � � • For each nonsimple projective indecom- posable module P in mod A , the quiver Γ A admits a valued subquiver { R 1 } ( d R 1 P ,d ′ R 1 P ) � � � � � � � � � � � { R 2 } � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( d R 2 P ,d ′ � � � � � { P } R 2 P ) � . . � � � � � � � � . � � � � � � � ( d RtP ,d ′ � � � � RtP ) � � � � { R t } such that t d ′ � rad P ∼ RiP R . = i i =1 • For each nonsimple injective indecompos- able module I in mod A , the quiver Γ A admits a valued subquiver { T 1 } ( d IT 1 ,d ′ IT 1 ) � � � � � � � � � � { T 2 } � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( d IT 2 ,d ′ � � � � { I } � � � IT 2 ) � . � � . � � � � � � � � . � � � � � � ( d ITm ,d ′ � � � � ITm ) � � � { T m } such that m � d ITj I/ soc I ∼ T . = j j =1 14

� � � • Assume A is an algebra of finite repre- ( d XY ,d ′ XY ) sentation type and X − − − − − − − − → Y is an arrow of Γ A . Then d XY = 1 or d ′ XY = 1 . • Assume A is an algebra over an algebra- ( d XY ,d ′ XY ) ically closed field K and X − − − − − − − − → Y is an arrow of Γ A . Then d XY = d ′ XY . In particular, d XY = d ′ XY = 1 if A is of finite representation type. In representation theory of finite dimensional algebras over an algebraically closed field K , instead of a valued arrow X ( m,m ) − − − − → Y of an Auslander-Reiten quiver Γ A , usually one writes a multiple arrow . X Y . . consisting of m arrows from X to Y . 15

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 ( d xy ,d ′ xy ) − − − − − − → y x 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 ( d xy ,d ′ ( d ′ xy ) xy ,d xy ) ( i, x ) − − − − − − → ( i, y ) , ( i +1 , y ) − − − − − − → ( i, x ) , ( d xy ,d ′ xy ) i ∈ Z , for all arrows x − − − − − − → 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 16

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � 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 ∆. Examples 1 (1 , 2) (4 , 3) � 2 ∆ : 3 Z ∆ of the form (1 , 1) (0 , 1) ( − 1 , 1) ( − 2 , 1) � � � � � � � � � � � � (1 , 2) (2 , 1) (1 , 2) (2 , 1) (1 , 2) (2 , 1) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � · · · ( − 1 , 2) · · · (1 , 2) (0 , 2) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (3 , 4) (4 , 3) (3 , 4) (4 , 3) (3 , 4) (4 , 3) � � � � � � � � � � � (1 , 3) (0 , 3) ( − 1 , 3) ( − 2 , 3) N ∆ of the form (3 , 1) (2 , 1) (1 , 1) (0 , 1) � � � � � � � � � � � � (1 , 2) (2 , 1) (1 , 2) (2 , 1) (1 , 2) (2 , 1) (1 , 2) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � · · · (3 , 2) (2 , 2) (1 , 2) (0 , 2) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (3 , 4) (4 , 3) � (3 , 4) (4 , 3) � (3 , 4) (4 , 3) � (3 , 4) � � � � � � � � (3 , 3) (2 , 3) (1 , 3) (0 , 3) ( − N )∆ of the form (0 , 1) ( − 1 , 1) ( − 2 , 1) ( − 3 , 1) � � � � � � � � � � (1 , 2) (2 , 1) (1 , 2) (2 , 1) (1 , 2) (2 , 1) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � · · · (0 , 2) ( − 1 , 2) ( − 2 , 2) � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (3 , 4) (4 , 3) (3 , 4) (4 , 3) (3 , 4) (4 , 3) � � � � � � � � � � � (0 , 3) ( − 1 , 3) ( − 2 , 3) ( − 3 , 3) 17

� � � � � � � � � � � � � � � � � � � 1 � 2 � 3 � . . . A ∞ : 0 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 . All vertices of ZA ∞ / ( τ r ) are τ -periodic of period r 18

� � � � � � � � � � � � � � � � � � � � � � A stable tube of rank 3 is of the form τx 1 τ 2 x 1 • • x 1 • • τ 2 x 2 τx 2 x 2 • • • • τ 2 x 3 x 3 τx 3 • • x 4 � � � � � � � � τ 2 x 4 � � τx 4 � � � � � � • . � . � • � � . . � � � � � � � � � � . . � � � � � � � � � � � � � � � � . � . � . � . � � � . . . . . . . . 19

A algebra C component of Γ A is regular if C contains neither a projective module nor an injec- tive module (equivalently, τ A and τ − 1 are A defined on all vertices of C ) Theorem ( Liu , Zhang ) . Let A be an alge- bra and C be a regular component of Γ A . The following equivalences hold. (1) C contains an oriented cycle if and only if C is a stable tube ZA ∞ / ( τ r ) , for some r ≥ 1 . (2) C is acyclic if and only if C is of the form Z ∆ for a connected, locally finite, acyclic, valued quiver ∆ . A component C of Γ A is postprojective ( pre- projective ) if C is acyclic and each module in C is of the form τ − m P for a projective A 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 τ m A I for an injective module I in C and some m ≥ 0. 20

A finite dimensional K -algebra over a field K C , D components of Γ A We write Hom A ( C , D ) = 0 if Hom A ( X, Y ) = 0 for all modules X in C and Y in D C and D are orthogonal if Hom A ( C , D ) = 0 and Hom A ( D , C ) = 0 In general, if C � = D , then Hom A ( X, Y ) = rad ∞ A ( X, Y ) for all modules X in C and Y in D . A component C of Γ A is called generalized standard if rad ∞ A ( X, Y ) = 0 for all modules X and Y in C . • C postprojective or preinjective compo- nent of Γ A , then C is generalized stan- dard • A of finite representation type, C com- ponent of Γ A , then C is generalized stan- dard • C is generalized standard component of Γ A , X and Y modules in C , then every nonzero homomorphism f ∈ rad A ( X, Y ) is a sum of compositions of irreducible homomorphisms between indecomposable modules from C . 21

A component C of Γ A is called almost periodic if all but finitely many τ A -orbits in C are periodic. Theorem ( Skowro´ nski ) . Let A be an alge- bra and C be an almost periodic component of Γ A . Then, for each natural number d ≥ 1 , C contains at most finitely many modules of dimension d . Theorem ( Skowro´ nski ) . Let A be an alge- bra and C be a generalized standard compo- nent of Γ A . Then C is almost periodic. Theorem ( Skowro´ nski ) . Let A be an alge- bra. Then all but finitely many generalized standard components of Γ A are stable tubes. C regular, generalized standard component of Γ A , then • C a stable tube, or • C = Z ∆, for a connected, finite , acyclic, valued quiver ∆. 22

A prominent role is played by the following Lemma ( Skowro´ nski ) . Let A be a finite di- mensional K -algebra and n be the rank of K 0 ( A ) . Assume M = M 1 ⊕ · · · ⊕ M r is a module in mod A such that • M 1 , . . . , M r are pairwise nonisomorphic and indecomposable • Hom A ( M, τ A M ) = 0 . Then r ≤ n . A finite dimensional K -algebra C component of Γ A � ann A C = ann A X annihilator of C X ∈ C (two-sided ideal of A ) ann A ( X ) = { a ∈ A | Xa = 0 } annihilator of A -module X C a faithful component of Γ A if ann A C = 0 In general, C is a faithful component of Γ A/ ann A C C faithful ⇒ Γ A is sincere (for any indecom- posable projective A -module P there exists a module X in C with Hom A ( P, X ) � = 0) 23

3. Homological dimensions A finite dimensional K -algebra over a field K M a module in mod A pd A M projective dimension of M in mod A pd A M = m ∈ N if there exists a projective resolution 0 → P m → P m − 1 → · · · → P 1 → P 0 → M → 0 of M in mod A and M has no projective resolution in mod A of length < m . pd A M = ∞ if M does not admit a finite projective resolution in mod A id A M injective dimension of M in mod A id A M = m ∈ N if there exists an injective resolution 0 → M → I 0 → I 1 → · · · → I m − 1 → I m → 0 of M in mod A and M has no injective reso- lution in mod A of length < m . id A M = ∞ if M does not admit a finite injective resolution in mod A 24

• pd A M = m ∈ N if and only if Ext m +1 ( M, − ) A = 0 and Ext m A ( M, − ) � = 0. • pd A M = ∞ if and only if Ext n A ( M, − ) � = 0 for all n ∈ N . • id A M = m ∈ N if and only if Ext m +1 ( − , M ) A = 0 and Ext m A ( − , M ) � = 0. • id A M = ∞ if and only if Ext n A ( − , M ) � = 0 for all n ∈ N . Moreover, we have the following useful facts • pd A M ≤ 1 if and only if Hom A ( D ( A A ) , τ A M ) = 0. • id A M ≤ 1 if and only if Hom A ( τ − 1 A M, A A ) = 0. For modules M and N in mod A , we have • If pd A M ≤ 1, then A ( M, N ) ∼ Ext 1 = D Hom A ( N, τ A M ) as K -vector spaces. • If id A M ≤ 1, then A ( M, N ) ∼ = D Hom A ( τ − 1 Ext 1 A N, M ) as K -vector spaces. For a faithful module M in mod A , we have • If Hom A ( M, τ A M ) = 0, then pd A M ≤ 1. • If Hom A ( τ − 1 A M, M ) = 0, then id A M ≤ 1. 25

� r . gl . dim A = max { pd A M | M modules in mod A } right global dimension of A l . gl . dim A = max { pd A op N | N modules in mod A op } left global dimension of A D � mod A op mod A D D standard duality of mod A pd A M = id A op D ( M ) id A M = pd A op D ( M ) for all modules M in mod A Hence, l . gl . dim A = max { id A M | M modules in mod A } � id A op N | N modules in mod A op � r . gl . dim A = max Theorem ( Auslander ) . A finite dimensional K -algebra over a field K . Then r . gl . dim = { pd A S | S simple right A -modules } . 26

Hence • r . gl . dim A minimal m ∈ N ∪{∞} such that Ext m +1 ( M, N ) = 0 for all modules M, N A in mod A � � � � M injective mo- � • r . gl . dim A = max id A M � dules in mod A = l . gl . dim A gl . dim A = r . gl . dim A = l . gl . dim A global dimension of A • A algebra with acyclic valued quiver Q A , then gl . dim A ≤ ∞ (gl . dim A ≤ length of longest path in Q A ) Theorem ( Skowro´ nski-Smalø-Zacharia ) . Let A be a finite dimensional K -algebra with gl . dim A = ∞ . Then there exists an inde- composable module M in mod A such that pd A M = ∞ and id A M = ∞ . 27

A finite dimensional K -algebra gl . dim A < ∞ �− , −� A : K 0 ( A ) × K 0 ( A ) − → Z Euler nonsymmetric Z -bilinear form ∞ � ( − 1) i dim K Ext i � [ M ] , [ N ] � A = A ( M, N ) i =0 for modules M, N in mod A q A : K 0 ( A ) − → Z Euler quadratic form ∞ � ( − 1) i dim K Ext i q A ([ M ]) = A ( M, M ) i =0 for a module M in mod A 28

Semisimple algebras A finite dimensional K -algebra over a field K M a module in mod A is semisimple if M is a direct sum of simple right A -modules. • M semisimple if and only if M rad A = 0 Theorem. A finite dimensional K -algebra. The following conditions are equivalent: (1) A A is semisimple. (2) Every module in mod A is semisimple. (3) rad A = 0 . (4) Every module in mod A op is semisimple. (5) A A is semisimple. A semisimple algebra if A A and A A are semi- simple modules 29

Theorem ( Wadderburn ) . A finite dimensional K -algebra over a field K . The following con- ditions are equivalent: (1) A is a semisimple algebra. (2) gl . dim A = 0 . (3) There exist positive integers n 1 , . . . , n r and division K -algebras F 1 , . . . , F r such that A ∼ = M n 1 ( F 1 ) × · · · × M n 1 ( F 1 ) . Observe that • A is a semisimple algebra if and only if the Auslander-Reiten quiver Γ A consists of the isolated vertices { S 1 } { S 2 } { S r } . . . corresponding to a complete set S 1 , S 2 , . . . , S r of pairwise nonisomorphic simple (equivalently, indecomposable) modules in mod A . 30

4. 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. Let A be a finite dimensional K - algebra over a field K . The following condi- tions are equivalent: (1) A is right hereditary. (2) Every right A -submodule of a projective module in mod A is projective. (3) The radical rad P of any indecomposable projective module P in mod A is projec- tive. (4) gl . dim A ≤ 1 . (5) The socle factor I/ soc I of any indecom- posable injective module I in mod A is injective. (6) Every factor module of an injective mo- dule in mod A is injective. (7) A is left hereditary. 31

A is hereditary if A is left and right hereditary Examples. K a field (1) Q finite acyclic quiver (arrows with trivial valuation) A = KQ the path algebra of Q over K A finite dimensional hereditary K -algebra Q A = Q (2) F, G finite dimensional division K -algebras F M G F - G -bimodule K acts centrally on F M G dim K ( F M G ) < ∞   � � � �   F 0 f 0 ; f ∈ F , g ∈ G , A = = F M G G m g   m ∈ F M G finite dimensional hereditary K -algebra Q A the valued quiver 2 (dim F ( F M G ) , dim G ( F M G )) − − − − − − − − − − − − − − − − − − − → 1 For example, � � � � � � � � 0 0 0 0 R R R R , , , C C C R H H H R R real numbers, C complex numbers, H quaternions, are hereditary R -algebras 32

(3) F 1 , F 2 , . . . , F n family of finite dimensional division K -algebras i M j F i - F j -bimodules, i, j ∈ { 1 , . . . , n } K acts centrally on i M j , dim K ( i M j ) < ∞ Consider the valued quiver Q : 1 , 2 , . . . , n vertices of Q There is an arrow j → i in Q ⇐ ⇒ i M j � = 0 ( d ij ,d ′ ij ) − − − − − → i Then we have the valued arrow j d ′ d ij = dim F i ( i M j ) , ij = dim F j ( i M j ) Assume that the valued quiver Q is acyclic n n � � F = F i , M = i M j , i =1 i,j =1 M is an F - F -bimodule, dim K M < ∞ ∞ � M ( n ) tensor algebra A = T F ( M ) = of M over F n =0 M (0) = F, M (1) = M, M ( n ) = M ⊗ F · · ·⊗ F M n -times, for n ≥ 2 Q acyclic ⇒ M ( r ) = 0 for large r A finite dimensional hereditary K -algebra Q A = Q 33

Theorem. Let be an indecomposable A finite dimensional hereditary K -algebra over a field K . The following conditions are equiv- alent: (1) The Euler form q A is positive definite. (2) The valued graph G A of A is one of the following Dynkin graphs . . . A m : • • • • ( m vertices), m ≥ 1 (1 , 2) . . . B m : • • • • ( m vertices), m ≥ 2 (2 , 1) . . . C m : • • • • ( m vertices), m ≥ 3 • � � � � � . . . � D m : • • • ( m vertices), m ≥ 4 � � � � � • � • E 6 : • • • • • • E 7 : • • • • • • • E 8 : • • • • • • • (1 , 2) F 4 : • • • • (1 , 3) G 2 : • • 34

Theorem. Let be an indecomposable A finite dimensional hereditary K -algebra over a field K . The following conditions are equiv- alent: (1) The Euler form q A is positive semidefinite but not positive definite. (2) The valued graph G A of A is one of the Euclidean graphs (1 , 4) � A 11 : • • (2 , 2) � A 12 : • • . . . • • ������ � ( m + 1 vertices), � � � � � • • A m : � � m ≥ 4 � ������ � � � . . . � • • ( m + 1 vertices), (1 , 2) (2 , 1) � . . . B m : • • • • m ≥ 2 ( m + 1 vertices), (2 , 1) (1 , 2) � . . . C m : • • • • m ≥ 2 ( m + 1 vertices), (1 , 2) (1 , 2) � . . . BC m : • • • • m ≥ 2 35

• ( m + 1 vertices), � (1 , 2) � � � . . . � � • • • BD m : � � � m ≥ 3 � � � � • • � ( m + 1 vertices), (2 , 1) � � � � . . . � • • • CD m : � � m ≥ 3 � � � � � • • • � ( m + 1 vertices), � � � � � � � � � . . . � � D m : • • � � m ≥ 4 � � � � � � � � � � • • � • � • E 6 : • • • • • • � E 7 : • • • • • • • • � E 8 : • • • • • • • • (1 , 2) � F 41 : • • • • • (2 , 1) � F 42 : • • • • • (1 , 3) � G 21 : • • • (3 , 1) � G 22 : • • • 36

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 • A wild type, then there exists an inde- composable module M in mod A such that q A ([ M ]) = dim K End A ( M ) − dim K Ext 1 A ( M, M ) < 0 37

Theorem. Let A be an indecomposable finite dimensional hereditary K -algebra over a field K , and Q = Q A the valued quiver of A . Then the Auslander-Reiten quiver Γ A has the following shape � � � � � � � � � � � � . . . Q ( A ) � P ( A ) . . . � � R ( A ) � � � � � � � � � � � � � � � � � • P ( A ) is the postprojective component con- taining all indecomposable projective A - modules • Q ( A ) is the preinjective component con- taining all indecomposable injective A -mo- dules • R ( A ) is the family of all regular compo- nents 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 , Q ( A ) ∼ = N Q op and R ( A ) is an infinite family of stable tubes, all but fi- nitely many of them of rank one. (3) If A is of wild type, then P ( A ) ∼ = ( − N ) Q op , Q ( A ) ∼ = N Q op , and R ( A ) is an infinite family of components of type ZA ∞ . 38

� � � A indecomposable hereditary not of Dynkin type, then • Hom A ( P ( A ) , R ( A )) � = 0, Hom A ( R ( A ) , P ( A )) = 0, • Hom A ( R ( A ) , Q ( A )) � = 0, Hom A ( Q ( A ) , R ( A )) = 0, • Hom A ( P ( A ) , Q ( A )) � = 0, Hom A ( Q ( A ) , P ( A )) = 0, × � � � � � � � � � � � � � � � � P ( A ) . . . . . . Q ( A ) � � R ( A ) � � � � � � � � � � � � � � � � � � � � � � × × 39

� � � A hereditary of Euclidean type, then R ( A ) is an infinite family ( T A λ ) λ ∈ Λ of pairwise orthog- onal generalized standard stable tubes sepa- rating P ( A ) form Q ( A ): for any homomor- phism f : X → Y with X in P ( A ) and Y in Q ( A ) there exists a module Z in R ( A ) and a factorization f X Y g � � � � � � � � � � � � h � � � Z A hereditary of Euclidean type, then A ) 3 = 0 (rad ∞ A ) m � = 0 A hereditary of wild type, then (rad ∞ for all m ≥ 1 40

5. Tilted algebras A finite dimensional K -algebra over a field K A module T in mod A is a tilting module if the following conditions are satisfied: (T1) pd A T ≤ 1; (T2) Ext 1 A ( T, T ) = 0; (T3) T is a direct sum of n pairwise noniso- morphic indecomposable modules, where n = rank of K 0 ( A ). ( Brenner-Butler , Happel-Ringel , Bongartz ) B = End A ( T ) tilted algebra of A 41

We have the torsion pairs ( F ( T ) , T ( T )) in mod A with torsion-free part F ( T ) = { X ∈ mod A | Hom( T, X ) = 0 } = Cogen τ A T torsion part � � X ∈ mod A | Ext 1 T ( T ) = A ( T, X ) = 0 = Gen T and ( Y ( T ) , X ( T )) in mod B with torsion-free part � � Y ∈ mod B | Tor B Y ( T ) = 1 ( T, Y ) = 0 = Gen τ − 1 B D ( B T ) torsion part X ( T ) = { Y ∈ mod B | Y ⊗ B T = 0 } = Cogen D ( B T ) 42

� � Theorem ( Brenner-Butler ) . Let A be a finite dimensional K -algebra over a field K , T a tilting module in mod A , and B = End A ( T ) . Then (1) B T is a tilting module in mod B op and there is a canonical isomorphism of K - algebras A → End B op ( B T ) op . (2) The functors Hom A ( T, − ) : mod A → mod B and − ⊗ B T : mod B → mod A induce mu- tually inverse equivalences ∼ T ( T ) − → Y ( T ) (3) The functors Ext 1 A ( T, − ) : mod A → mod B and Tor B 1 ( T, − ) : mod B → mod A induce mutually inverse equivalences ∼ F ( T ) − → X ( T ) Γ A F ( T ) T ( T ) � � � ������������������ � � � � � � � � � Ext 1 � � Hom A ( T, − ) � A ( T, − ) � � � � � ��������� � � � � � � � Tor B � � ������������������ 1 ( T, − ) � −⊗ B T � � � � � � � � � � � � � � � � � Γ B Y ( T ) X ( T ) inj A ⊆ T ( T ) , proj B ⊆ Y ( T ) , 43

A finite dimensional K -algebra, T a tilting module in mod A , and B = End A ( T ). Then • | gl . dim A − gl . dim B | ≤ 1. • There is a canonical isomorphism f : K 0 ( A ) → K 0 ( B ) of Grothendieck groups such that f ([ M ]) = [Hom A ( T, M )] − [Ext 1 A ( T, M )] for any module M in mod A . Moreover, if gl . dim A < ∞ , then � [ M ] , [ N ] � A = � f ([ M ]) , f ([ N ]) � B for all modules M, N in mod A . • If gl . dim A < ∞ then the Euler forms q A of A and q B of B are Z -equivalent. 44

A hereditary finite dimensional K -algebra T tilting module in mod A B = End A ( T ) tilted algebra ( of type G A (valued graph of A )) Then • gl . dim B ≤ 2; • For every indecomposable module Y in mod B , we have pd B Y ≤ 1 or id B Y ≤ 1; • The torsion pair ( Y ( T ) , X ( T )) in mod B is splitting : every module from ind B be- longs to Y ( T ) or X ( T ). 45

Moreover, the images Hom A ( T, I ) of the in- decomposable injective modules I in mod A via the functor Hom A ( T, − ) : mod A → mod B belong to one component C T of Γ B , and form a faithful section ∆ T ∼ = Q op of C A ∆ T � � � � � � � � � � � � � � Y ( T ) ∩ C T � C T ∩ X ( T ) � � � � � � � � � � � � � � � � � � � C T C T connecting component of Γ B determined by T (connects the torsion-free part with the torsion part of Γ B : every predecessor of a module Hom A ( T, I ) from ∆ T in ind B lies in Y ( T ) and every successor of a module τ − 1 Hom A ( T, I ) in ind B lies in X ( T )) B ∆ T section : acyclic, convex in C , and inter- sects each τ Λ -orbit of C exactly once ∆ T faithful : the direct sum of all modules lying on ∆ is a faithful B -mo- dule (has zero annihilator in B ) C T faithful generalized standard compo- nent of Γ A with a section ∆ T 46

Theorem ( Ringel ) . Let A be a hereditary algebra, T a tilting module in mod A , B = End A ( T ) and C T the connecting component of Γ B determined by T . Then (1) C T contains a projective B -module if and only if T admits a preinjective indecom- posable direct summand. (2) C T contains an injective B -module if and only if T admits a postprojective inde- composable direct summand. (3) C T is regular if and only if T is regular (belongs to add R ( A ) ). Theorem ( Ringel ) . Let A be a hereditary algebra. Then there is a regular tilting mod- ule in mod A if and only A is of wild type and K 0 ( A ) is of rank ≥ 3 . 47

Handy criterion for a tilted algebra Theorem ( Liu , Skowro´ nski ) . Let B be a finite dimensional K -algebra over a field K . Then B is a tilted algebra if and only if Γ B admits a component C with a faithful sec- tion ∆ such that Hom B ( X, τ B Y ) = 0 for all modules X, Y from ∆ . Moreover, in this case, if T ∗ is the direct sum of all modules lying on ∆ , then • T ∗ is a tilting module in mod B . • A = End B ( T ∗ ) is a hereditary K -algebra of type ∆ op . • T = D ( A T ∗ ) is a tilting module in mod A . • B ∼ = End A ( T ) . Theorem ( Liu , Skowro´ nski ) . Let B be a finite dimensional K -algebra over a field K . Then B is a tilted algebra if and only if Γ B admits a faithful generalized standard com- ponent C with a section ∆ . 48

� � � � � � � � � � � � � � � � � � � � � Example. Let B = KQ/I where Q is the quiver γ β σ α 1 ← − 2 ← − 3 ← − 4 ← − 5 and I is the ideal of KQ generated by αβγσ Γ B is of the form � � � S 1 = P 1 = K 0000 0 K 000= S 2 00 K 00 = S 3 00 K 00= S 4 0000 K = S 5 = I 5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 = KK 000 0 KK 00 00 KK 0 000 KK = I 4 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 3 = KKK 00 00 KKK = I 3 0 KKK 0 0 KKKK � � � � � � � � � P 5 = I 2 � � � � � � � � � ∆ P 4 = KKKK 0 = I 1 ∆ faithful section of C = Γ B T ∗ T ∗ T ∗ 1 = S 2 , 2 = 0 KK 00, 3 = 0 KKK 0, T ∗ T ∗ 4 = P 5 , 5 = P 4 T ∗ = T ∗ 1 ⊕ T ∗ 2 ⊕ T ∗ 3 ⊕ T ∗ 4 ⊕ T ∗ 5 , T ∗ faithful tilting B -module, Hom B ( T ∗ , τ B T ∗ ) = 0 A = End B ( T ∗ ) hereditary K -algebra K ∆ op , where ∆ op is of the form 4 � ������� 1 2 3 � � � � � � � 5 49

� � � � � � � � � � � � � � � � � � � � � � � � � � � � T = D ( A T ∗ ) tilting module in mod A T = T 1 ⊕ T 2 ⊕ T 3 ⊕ T 4 ⊕ T 5 T i = D ( T ∗ i ) for i ∈ { 1 , 2 , 3 , 4 , 5 } T 1 = 000 0 T 2 = KKKK T 3 = 0 KKK K K K T 4 = 00 KK T 5 = 000 K K 0 Γ A • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • T 2 • • • T 4 • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • T 1 T 3 T 5 A ( T, T ) ∼ Ext 1 = D Hom A ( T, τ A T ) = 0 End A ( T ) ∼ = B = KQ/I 50

A indecomposable hereditary finite dimensio- nal K -algebra T tilting module in mod A B = End A ( T ) • A of Dynkin type ⇒ A of finite representation type ⇒ B of finite representation type • B of finite representation type ⇒ Γ B = C T and finite ⇒ C T contains all indecomposable pro- jective modules and all indecomposa- ble injective modules ⇒ T has a postprojective and a preinjec- tive direct summand • A of Euclidean type, T has a postprojec- tive and a preinjective direct summand ⇒ B is of finite representation type 51

� � � � Concealed algebras A indecomposable hereditary of infinite representation type T postprojective tilting module in mod A , T ∈ add P ( A ) B = End A ( T ) concealed algebra of type G A P ( A ) ������� ���������� ���� ������� ���� � � � Γ A � � � P ( A ) ∩ T ( T ) R ( A ) Q ( A ) � � � � � � � � � � ������� ������� � � � � � � F ( T ) � � � � �� Ext 1 A ( T, − ) Hom A ( T, − ) �� ���� ������� ���������� ���� � � Γ B � � P ( B ) R ( B ) � � � � ∆ T � � � ������� � � � � � � � � X ( T ) � � � Q ( B ) = C T • P ( B ) = Hom A ( T, P ( A ) ∩ T ( T )) postpro- jective component of Γ B containing all indecomposable projective B -modules • Q ( B ) = C T = Hom A ( T, Q ( A )) ∪ X ( T ) preinjective component of Γ B containing all indecomposable injective B -modules • R ( B ) Hom A ( T, R ( A )) family of all = regular components of Γ B • A of Euclidean type ⇒ R ( B ) infinite fa- mily of pairwise orthogonal generalized standard stable tubes • A of wild type ⇒ R ( B ) infinite family of components of type ZA ∞ 52

� � � � T preinjective tilting module in mod A , T ∈ add Q ( A ) B = End A ( T ) Q ( A ) � � � � ������� ������� � T ( T ) � � � � � � � � � � � � Γ A � � � ���� P ( A ) R ( A ) Q ( A ) ∩ F ( T ) � � � � � � ������� ������� ������� ������� �� Hom A ( T, − ) Ext 1 A ( T, − ) �� � � � � ������� Y ( T ) � � � � � � � � � � � � Γ B � � ���� � R ( B ) Q ( B ) � � � ������� ������� ������� ∆ T P ( B ) = C T • P ( B ) = C T = Y ( T ) ∪ Ext 1 A ( T, P ( A )) post- projective component of Γ B containing all indecomposable projective B -modules • Q ( B ) = Ext 1 A ( T, Q ( A ) ∩ F ( T )) preinjecti- ve component of Γ B containing all inde- composable injective B -modules • R ( B ) = Ext 1 A ( T, R ( A )) family of all regu- lar components of Γ B • A of Euclidean type ⇒ R ( B ) infinite fa- mily of pairwise orthogonal generalized standard stable tubes • A of wild type ⇒ R ( B ) infinite family of components of type ZA ∞ B ∼ = End A ( T ) for a postprojective tilting A - ⇒ B ∼ = End A ( T ′ ) for a preinjec- module T ⇐ tive tilting A -module T ′ 53

Representation-infinite tilted algebras of Euclidean type A indecomposable hereditary of Euclidean type T tilting module in mod A without preinjec- tive direct summands B = End A ( T ) T = T pp ⊕ T rg , T pp ∈ add P ( A ), T rg ∈ add R ( A ) ⇒ T pp � = 0, C = End A ( T pp ) concealed algebra of Euclidean type C factor algebra of B ∆ T � � � � � � � � ������� ���������� � ����� � Γ B � P ( B ) = P ( C ) � � � � � � ������� � � � � � � X ( T ) � � Q ( B ) = C T T B • P ( B ) = Hom A ( T, T ( T ) ∩ P ( A )) = Hom A ( T pp , T ( T ) ∩ P ( A )) = P ( C ) post- projective component of Γ B containing all indecomposable projective C -modules • Q ( B ) = C T = Hom A ( T, Q ( A )) ∪X ( T ) pre- injective component of Γ B containing all indecomposable injective B -modules • T B = Hom A ( T, R ( A ) ∩ T ( T )) infinite fa- mily of pairwise orthogonal generalized standard ray tubes • T B contains a projective module ⇐ ⇒ T rg � = 0 54

T tilting module in mod A without postpro- jective direct summands B = End A ( T ) T = T rg ⊕ T pi , T rg ∈ add R ( A ), T pi ∈ add Q ( A ) ⇒ T pi � = 0, C = End A ( T pi ) concealed algebra of Euclidean type C factor algebra of B � � � � � � � � � � � ������� Y ( T ) � � � � � � � � � � � Γ B � � Q ( B ) = Q ( C ) � � � � ����� ������� ������� ∆ T P ( B ) = C T T B • P ( B ) = C T = Y ( T ) ∪ Ext 1 A ( T, P ( A )) post- projective component of Γ B containing all indecomposable projective B -modules Ext 1 • Q ( B ) = A ( T, F ( T ) ∩ Q ( A )) = Ext 1 A ( T pi , F ( T ) ∩ Q ( A )) = Q ( C ) preinjec- tive component of Γ B containing all in- decomposable injective C -modules • T B = Ext 1 A ( T, R ( A ) ∩ F ( T )) infinite fa- mily of pairwise orthogonal generalized standard coray tubes • T B contains an injective module ⇐ ⇒ T rg � = 0 55

Almost concealed algebras of wild type A indecomposable hereditary of wild type T tilting module in mod A T = T pp ⊕ T rg ⊕ T pi , T pp ∈ add P ( A ), T rg ∈ add R ( A ), T pi ∈ add Q ( A ) B = End A ( T ) B almost concealed if T pp = 0 or T pi = 0 The cases • T = T pp • T = T pi were considered above It remains to consider the cases • T = T rg • T = T pp ⊕ T rg , T pp � = 0, T rg � = 0 • T = T rg ⊕ T pi , T rg � = 0, T pi � = 0 56

� T = T rg regular tilting module, B = End A ( T ) ������ ������ Γ A � � � � P ( A ) R ( A ) Q ( A ) � � ������ ������ � � � ����������������������� � � � � � � � Ext 1 � Hom A ( T, − ) � A ( T, − ) � � � � � � � � � � � � ∆ T ������ ������ � � Γ B Ext 1 Y Γ B Hom A ( T, Q ( A )) � � X Γ B A ( T, P ( A )) � � ������ ������ C T = Z ∆ T = Z Q op A • C T regular connecting component • Y Γ B = Hom A ( T, T ( T ) ∩ R ( A )) contains all indecomposable projective B -modules and consist of – one postprojective component P ( B ) = P ( C ), for a wild concealed factor al- gebra C of B – an infinite family of components ob- tained from components of type ZA ∞ by ray insertions, containing at least one projective B -module • X Γ B = Ext 1 A ( T, F ( T ) ∩R ( A )) contains all indecomposable injective B -modules and consist of – one preinjective component Q ( B ) = Q ( C ′ ), for a wild concealed factor al- gebra C ′ of B – an infinite family of components ob- tained from components of type ZA ∞ by coray insertions, containing at least one injective B -module 57

T = T pp ⊕ T rg , T pp � = 0, T rg � = 0 Γ B is of the form ∆ T ������ ������ � � � � � � Y Γ B Hom A ( T, Q ( A )) X Γ B � � � � � ������ ������ Ext 1 A ( T, P ( A ) ∩ F ( T )) C T • C T connecting component containing at least one injective module and no projec- tive modules • Y Γ B = Hom A ( T, T ( T ) ∩ ( P ( A ) ∪ R ( A ))) contains all indecomposable projective B - modules and consist of – one postprojective component P ( B ) = P ( C ), for a wild concealed factor al- gebra C of B – an infinite family of components ob- tained from components of type ZA ∞ by ray insertions, containing at least one projective B -module • X Γ B = Ext 1 A ( T, F ( T ) ∩ R ( A )) consists of preinjective components and components obtained from stable tubes or components of type ZA ∞ by coray insertions 58

T = T rg ⊕ T pi , T rg � = 0, T pi � = 0 Γ B of the form ∆ T ������ ������ Hom A ( T, Q ( A ) ∩ T ( T )) ��� � � Ext 1 Y Γ B � � A ( T, P ( A )) X Γ B � � � ������ ������ � C T • C T connecting component containing at least one projective module and no injec- tive modules • Y Γ B = Hom A ( T, T ( T ) ∩ ( R ( A ) ∪ Q ( A ))) consists of preprojective components and components obtained from stable tubes or components of type ZA ∞ by ray inser- tions • X Γ B = Ext 1 A ( T, F ( T ) ∩ ( R ( A ) ∪ Q ( A ))) contains all indecomposable injective B - modules and consist of – one preinjective component Q ( B ) = Q ( C ′ ), for a wild concealed factor al- gebra C ′ of B – an infinite family of components ob- tained from components of type ZA ∞ by coray insertions, containing at least one injective B -module 59

Tilted algebras of wild type – general case A indecomposable hereditary algebra of wild type T tilting module in mod A B = End A ( T ) Γ B is of the form ∆ (1) ∆ (1) ∆ T r l Y Γ B (1) D (1) � � � X Γ B (1) D (1) � � � � � � � � r l � r l � � � � � � � � � � � � � � � X Γ B (2) D (2) � � � � � r � � � r � � � ∆ (2) � � � � � � � � r Y Γ B (2) � D (2) � � � � � ������������������������� � � l � l � ∆ (2) � � � � � � l � • • • • • • � ∆ ( n ) � � � � � r � � ������ � X Γ B ( n ) � D ( n ) � � � � r � r � � ����� � � � � Y Γ B ( m ) D ( m ) � � l � l � ∆ ( m ) C T l where • C T connecting component of Γ B deter- minend by T , possibly C T = Γ B (if B is of finite representation type) • For each i ∈ { 1 , . . . , m } , ∆ ( i ) connected l valued subquiver of ∆ T of Euclidean or wild type, D ( i ) = N ∆ ( i ) full translation l l subquiver of C T closed under predeces- sors • For each j ∈ { 1 , . . . , n } , ∆ ( j ) connected r valued subquiver of ∆ T of Euclidean or wild type, D ( j ) = ( − N )∆ ( j ) full transla- r r tion subquiver of C T closed under succes- sors 60

• For each i ∈ { 1 , . . . , m } , there exists a tilted algebra B ( i ) ( T ( i ) = End A ( i ) ) l l l where A ( i ) is a hereditary algebra of type l ∆ ( i ) and T ( i ) is a tilting module in mod A ( i ) l l l without preinjective direct summands such that – B ( i ) is a factor algebra of B l – D ( i ) = Y ( T ( i ) ) ∩ C T ( i ) l l l – Y Γ B ( i ) family of all connected compo- l nents of Γ B ( i ) contained entirely in the l torsion-free part Y ( T ( i ) ) of mod B ( i ) l l 61

• For each j ∈ { 1 , . . . , n } , there exists a tilted algebra B ( j ) r ( T ( j ) = End A ( j ) ) r r where A ( j ) is a hereditary algebra of type r ∆ ( j ) and T ( j ) is a tilting module in mod A ( j ) r r r without postprojective direct summands such that – B ( j ) is a factor algebra of B r – D ( j ) = X ( T ( j ) ) ∩ C T ( j ) r r r – X Γ B ( j ) family of all connected compo- r nents of Γ B ( j ) contained entirely in the r torsion part X ( T ( j ) ) of mod B ( j ) r r • All but finitely many modules of C T are in D (1) ∪ · · · ∪ D ( m ) ∪ D (1) ∪ · · · ∪ D ( n ) r r l l 62

We know from the facts described before that • For each i ∈ { 1 , . . . , m } , the translation quiver Y Γ B ( i ) consists of l – one postprojective component P ( B ( i ) ) l – an infinite family of pairwise orthogo- nal generalized standard ray tubes, if ∆ ( i ) is an Euclidean quiver, or an in- l finite family of components obtained from components of type ZA ∞ by ray insertions, if ∆ ( i ) is a wild quiver l • For each j ∈ { 1 , . . . , n } , the translation quiver X Γ B ( j ) consists of r – one preinjective component Q ( B ( j ) ) r – an infinite family of pairwise orthogo- nal generalized standard coray tubes, if ∆ ( j ) is an Euclidean quiver, or an r infinite family of components obtained from components of type ZA ∞ by coray insertions, if ∆ ( j ) is a wild quiver r 63

Acyclic generalized standard Auslander- Reiten components Theorem ( Skowro´ nski ) . Let A be a finite dimensional K -algebra over a field K , C a component of Γ A and B = A/ ann A C . (1) C is generalized standard, acyclic, with- out projective modules if and only if B is a tilted algebra of the form End H ( T ) , where H is a hereditary algebra, T is a tilting module in mod H without preinjec- tive direct summands, and C is the con- necting component C T of Γ B determined by T . (2) C is generalized standard, acyclic, with- out injective modules if and only if B is a tilted algebra of the form End H ( T ) , where H is a hereditary algebra, T is a tilting module in mod H without postpro- jective direct summands, and C is the connecting component C T of Γ B deter- mined by T . (3) C is generalized standard, acyclic, regular if and only if B is a tilted algebra of the form End H ( T ) , where H is a hereditary algebra, T is a regular tilting module in mod H , and C is the connecting compo- nent C T of Γ B determined by T . 64

In general, an arbitrary acyclic generalized standard component C of Γ A is a glueing of • torsion-free parts Y ( T ( i ) ) ∩ C T ( i ) of the l l connecting components C T ( i ) of tilted al- l gebras B ( i ) ( T ( i ) = End A ( i ) ) of hereditary l l l algebras A ( i ) by tilting A ( i ) -modules T ( i ) l l l without preinjective direct summands • torsion parts X ( T ( j ) ) ∩ C T ( j ) of the con- r r necting components C T ( j ) of tilted alge- r bras B ( j ) r ( T ( j ) = End A ( j ) ) of hereditary r r algebras A ( j ) by tilting A ( j ) -modules T ( j ) r r r without postprojective direct summands along a finite acyclic part in the middle of C (and usually C does not admit a section) 65

6. Quasitilted algebras Abelian K -category H over a field K is said to be hereditary if, for all objects X and Y of H , the following conditions are satisfied • Ext 2 H ( X, Y ) = 0 • Hom H ( X, Y ) and Ext 1 H ( X, Y ) are finite dimensional K -vector spaces An object T of a hereditary abelian K -category H is said a tilting object if the following conditions are satisfied • Ext 1 H ( T, T ) = 0 • For an object X of H , Hom H ( T, X ) = 0 and Ext 1 H ( T, X ) = 0 force X = 0 • T direct sum of pairwise nonisomorphic indecomposable objects of H A finite dimensional hereditary K -algebra. Then • H = mod A hereditary abelian K -category • A module T in mod A is a tilting object of mod A if and only if T is a tilting module A quasitilted algebra is an algebra of the form End H ( T ), where T is a tilting object of an abelian hereditary K -category H . 66

A finite dimensional K -algebra over a field K A path in ind A is a sequence of homomor- phisms f 1 f 2 f t − → M 1 − → M 2 − → . . . − → M t − 1 − → M t M 0 in ind A with f 1 , f 2 , . . . , f t nonzero and noni- somorphisms M 0 predecessor of M t in ind A M t successor of M 0 in ind A Every module M in ind A is its own (trivial) predecessor and successor L A full subcategory of ind A formed by all modules X such that pd A Y ≤ 1 for every predecessor Y of X in ind A R A full subcategory of ind A formed by all modules X in ind A such that id A Y ≤ 1 for every successor Y of X in ind A L A closed under predecessors in ind A R A closed under successors in ind A 67

Theorem ( Happel-Reiten-Smalø ) . Let B be a finite dimensional K -algebra. The following conditions are equivalent: (1) B is a quasitilted algebra. (2) gl . dim B ≤ 2 and every module X in ind B satisfies pd B X ≤ 1 or id B X ≤ 1 . (3) L B contains all indecomposable projec- tive B -modules. (4) R B contains all indecomposable injective B -modules. Theorem ( Happel-Reiten-Smalø ) . Let B be a quasitilted K -algebra. Then (1) The quiver Q B of B is acyclic. (2) ind B = L B ∪ R B . (3) If B is of finite representation type, then B is a tilted algebra. 68

Theorem ( Skowro´ nski ) . Let B be an in- decomposable finite dimensional K -algebra. The following conditions are equivalent: (1) B is a tilted algebra. (2) gl . dim B ≤ 2 , ind B = L B ∪ R B and L B ∩ R B contains a directing module. A module M in ind B is directing if M does not lie on an oriented cycle in ind B . Theorem ( Coelho-Skowro´ nski ) . Let B be a quasitilted but not tilted algebra. Then every component of Γ B is semiregular. A component C of Γ B is semiregular if C does not contain simultaneously a projective module and an injective module. 69

Canonical algebras Special case: K a field m ≥ 2 natural number p = ( p 1 , . . . , p m ) m -tuple of natural numbers λ = ( λ 1 , . . . , λ m ) m -tuple of pairwise different elements of P 1 ( K ) = K ∪ {∞} , normali- sed such that λ 1 = ∞ , λ 2 = 0, λ 3 = 1 α 1 p 1 − 1 ◦ α 12 − ◦ α 13 ← ← − · · · ← − − − − ◦ α 11 α 1 p 1 ւ տ α 2 p 2 − 1 α 2 p 2 0 ◦ α 21 − ◦ α 22 − ◦ α 23 ← ← ← − · · · ← − − − − ◦ ← − ◦ ω ∆( p ) : . . . . . . տ . ւ . . α mpm α m 1 ◦ ← α m 2 ◦ ← − − α m 3 · · · − − ← α mpm − 1 ◦ − − − − C ( p , λ ) defined as follows. For m = 2, C ( p , λ ) = K ∆( p ) path algebra of ∆( p ) For m ≥ 3, C ( p , λ ) = K ∆( p ) /I ( p , λ ) I ( p , λ ) ideal of K ∆( p ) ge- nerated by α jp j . . . α j 2 α j 1 + α 1 p 1 . . . α 12 α 11 + λ j α 2 p 2 . . . α 22 α 21 for j ∈ { 3 , . . . , m } C ( p , λ ) canonical algebra of type ( p , λ ) p weight sequence, λ parameter sequence For K algebraically closed, these are all ca- nonical algebras (up to isomorphism) 70

General case (version of Crawley-Boevey) Let F and G be finite dimensional division al- gebras over a field K , F M G an F - G -bimodule with (dim F M )(dim M G ) = 4, K acting cen- trally on F M G . Denote � dim F M χ = , dim M G hence χ = 1 2 , 1, or 2. An M -triple is a triple ( F N, ϕ, N ′ G ), where F N is a finite dimensional nonzero left F -module, N ′ G a finite dimensional nonzero right G -module, and ϕ : F N ⊗ Z N ′ G → F M G an F - G -homomorphism such that • dim F N = χ , dim N ′ G • whenever F X and X ′ G are nonzero sub- modules of F N and N ′ G , respectively, with dim F N + dim X ′ ϕ ( X ⊗ Z X ′ ) = 0, then dim F X G < 1. dim N ′ G 71

� � � Two M -triples ( N 1 , ϕ 1 , N ′ 1 ) and ( N 2 , ϕ 2 , N ′ 2 ) are said to be congruent if there are isomor- phisms of modules Θ : F ( N 1 ) → F ( N 2 ) and Θ ′ : ( N ′ 1 ) G → ( N ′ 2 ) G such that the following diagram is commutative N 1 ⊗ Z N ′ 1 � ϕ 1 � � � � � � � � � � Θ ⊗ Θ ′ M � � � � � � ϕ 2 � � � � � N 2 ⊗ Z N ′ . 2 The middle D of an M -triple ( F N, ϕ, N ′ G ) is defined to be the set of pairs ( d, d ′ ), where d is an endomorphism of F N and d ′ is an endo- morphism of N ′ G such that ϕ ( d ⊗ 1) = ϕ (1 ⊗ d ′ ). Then D is a division K -algebra under componentwise addition and multiplication, N is an F - D -bimodule, N ′ a D - G -bimodule, and induces an F - G -homomorphism ϕ ϕ : F N ⊗ D N ′ G → F M G . 72

Let r ≥ 0 and n 1 , . . . , n r ≥ 2 be integers. A canonical algebra Λ of type ( n 1 , . . . , n r ) over a field K is an algebra isomorphic to a matrix algebra of the form   N 1 · · · N 1 N 2 · · · N 2 · · · N r · · · N r F M   N ′  D 1 · · · D 1   1     . . ... . . n 1 − 1   0 0 · · · 0 . .       N ′ 0 D 1    1   N ′ D 2 · · · D 2     2   . . ... n 2 − 1   . . 0 0 · · · 0   . .     N ′ 0 D 2   2   . . . . . .   . . . . . .   . . . . . .      N ′ D r · · · D r     r   . . ... n r − 1  . .  0 0 0 · · · . .       N ′ 0 D r r 0 0 0 0 0 G where F and G are finite dimensional division algebras over K , M = F M G an F - G -bimodule with (dim F M )(dim M G ) = 4 and K acting centrally on F M G , ( N 1 , ϕ 1 , N ′ 1 ) , . . . , ( N r , ϕ r , N ′ r ) are mutually noncongruent M -triples with the middles D 1 , . . . , D r , and the multiplication given by the actions of division algebras on bi- modules and the appropriate homomorphisms ϕ 1 , . . . , ϕ r . 73

� � � � � � � � � � � The valued quiver Q Λ of a canonical algebra Λ of type ( n 1 , . . . , n r ) is of the form · · · (1 , n 1 − 1) (1 , 1) (1 , 2) ( a 1 ,b 1 ) ( c 1 ,d 1 ) � ����������������� � �������������� ( a 2 ,b 2 ) ( c 2 ,d 2 ) · · · ω (2 , 1) (2 , 2) (2 , n 2 − 1) 0 � �������������� � ����������������� ( a r ,b r ) ( c r ,d r ) · · · ( r, 1) ( r, 2) ( r, n r − 1) a i = dim F N i , b i = dim( N i ) F i , c i = dim F i N ′ d i = dim( N ′ i , i ) G for i ∈ { 1 , . . . , r } Λ canonical algebra ⇒ gl . dim Λ ≤ 2 Hence the Euler form q Λ of Λ is defined Λ canonical algebra ⇒ • q Λ positive semidefinite of corank one or two, or • q Λ is indefinite 74

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � Theorem. Let Λ be a canonical algebra over a field K . The following conditions are equiv- alent: (1) q Λ is positive semidefinite of corank one. (2) Q Λ is of one of the following forms . . . • • (1 , 2) (2 , 1) . . . � � • • • • � � � � � � � � • • � � � � (2 , 1) (1 , 2) � � � � . . . . . . � � • • • • • • • • (1 , 2) (2 , 1) (2 , 1) (1 , 2) � ������������� � ������������� � ������������� � ������������� • • • • � ����� � ����� � ����� � ����� . . . . . . • • • • • • � ������������� � ������������� � ������������� � ������������� • • • • • • • � ������� � ����� � ������� � ����� . . . • • • • • • � ��������������������� � ������������� � ��������������������� � ������������� • • • • • • • • � � � � � � � � � � � � � � � � � � � � • • • • • • • • • � ������������� � ������������� � ������������� � ������������� • • • • � ������� � ������� � ������� � ������� (1 , 2) • • (2 , 1) (2 , 1) • • (1 , 2) (1 , 3) (3 , 1) (3 , 1) (1 , 3) • • • • • • 75

� � � � � � � � � � � � � � � � � � � � � � � Theorem. Let Λ be a canonical algebra over a field K . The following conditions are equiv- alent: (1) q Λ is positive semidefinite of corank two. (2) Q Λ is of one of the following forms (1 , 4) (4 , 1) (4 , 1) (1 , 4) • • • • • • (2 , 2) (2 , 2) (1 , 4) (2 , 2) • • • • • • (4 , 1) (2 , 2) (2 , 2) (1 , 4) (2 , 2) (4 , 1) • • • • • • • • • • • • (2 , 1) (1 , 2) (1 , 2) (2 , 1) (2 , 1) (1 , 2) � ������ � ������ � ������ � ������ � ������ � ������ • • • • • • � ������ � ������ � ������ � ������ � ������ � ������ (2 , 1) • (1 , 2) (1 , 2) • (2 , 1) (1 , 2) • (2 , 1) • • � ������ � ������ � ������ � ������ • • • • • • � ������ � ������ � ������ � ������ (2 , 1) • (1 , 2) (1 , 2) • (2 , 1) • • • � � � � � � � � • � ������ � � � ������ � � � � � ������ � ������ � � � � • • • • • • � ������ � ������ � ������ � � � ������ � • � � � � � � � • • � � � � � � � � • 76

� � � � � � � � � � � � � � � � � � � � � � � � � � � • • � ��������������� � ������������ � ��������������� � ������������ • • • • • • • • • � ����� � � � ����� � � � � � � � � • • • • • • • • • • � ������������ � ������������ � ������������ � ������������ • • • • � ����� � ����� � ����� � ����� (1 , 2) • • • (1 , 2) (2 , 1) • • • (2 , 1) • • • • � ������ � ������ � ������ � ������ • • • • � ������ � ������ � ������ � ������ (1 , 2) • • (1 , 2) (2 , 1) • • (2 , 1) (3 , 1) (1 , 3) (1 , 3) (3 , 1) • • • • • • • • • • � ������ � ������ � ������ � ������ • • • • � ������ � ������ � ������ � ������ (1 , 3) • (3 , 1) (3 , 1) • (1 , 3) 77

Λ canonical algebra over a field K Λ canonical algebra of Euclidean type : q Λ is positive semidefinite of corank one Λ canonical algebra of tubular type : q Λ is positive semidefinite of corank two Λ canonical algebra of wild type : q Λ is indefinite Q ∗ Λ the valued quiver obtained from the va- lued quiver Q Λ of Λ by removing the unique source and the arrows attached to it • Λ canonical algebra of Euclidean type if and only if Q ∗ Λ is a Dynkin valued quiver • Λ canonical algebra of tubular type if and only if Q ∗ Λ is a Euclidean valued quiver 78

Theorem ( Ringel ) . Let Λ be a canonical algebra of type ( n 1 , . . . , n r ) over a field K . Then the general shape of the Auslander- Reiten quiver Γ Λ of Λ is as follows Q Λ P Λ T Λ • P Λ is a family of components contain- ing a unique postprojective component P (Λ) and all indecomposable projective Λ -modules. • Q Λ is a family of components containing a unique preinjective component Q (Λ) and all indecomposable injective Λ -modules. • T Λ is an infinite family of faithful pair- wise orthogonal generalized standard sta- ble tubes, having stable tubes of ranks n 1 , . . . , n r and the remaining tubes of rank one. • T Λ separates P Λ from Q Λ . • pd Λ X ≤ 1 for all modules X in P Λ ∪ T Λ . • id Λ Y ≤ 1 for all modules Y in T Λ ∪ Q Λ . • gl . dim Λ ≤ 2 . 79

Let Λ be a canonical algebra of type ( n 1 , . . . , n r ) T tilting module in add P Λ C = End Λ ( T ) concealed canonical algebra of type Λ The general shape of Γ C is a as follows Q C P C T C • P C = Hom Λ ( T, T ( T ) ∩P Λ ) ∪ Ext 1 Λ ( T, F ( T )) is a family of components containing a unique postprojective component P ( C ) and all indecomposable projective C -modules. • Q C = Hom Λ ( T, Q Λ ) is a family of com- ponents containing a unique preinjective component Q ( C ) and all indecomposable injective C -modules. • T C = Hom Λ ( T, T Λ ) is an infinite family of faithful pairwise orthogonal general- ized standard stable tubes, having stable tubes of ranks n 1 , . . . , n r and the remain- ing tubes of rank one. • T C separates P C from Q C . • pd C X ≤ 1 for all modules X in P C ∪ T C . • id C Y ≤ 1 for all modules Y in T C ∪ Q C . • gl . dim C ≤ 2. 80

C ∼ = End Λ ( T ), T tilting module in add P Λ , if and only if C ∼ = End Λ ( T ′ ), T ′ tilting module in add Q Λ . Λ canonical algebra T tilting module in add( P Λ ∪ T Λ ) B = End Λ ( T ) almost concealed canonical algebra of type Λ The general shape of Γ B is as follows � � � � � � � P B Q B T B • P B = P C for a concealed canonical factor algebra C of B . • Q B a family of components containing a unique preinjective component Q ( B ) and all indecomposable injective B -modules. • T B an infinite family of pairwise ortho- gonal generalized standard ray tubes, separating P B from Q B . • pd B X ≤ 1 for all modules X in P B ∪ T B . • id B Y ≤ 1 for all modules Y in T B . • gl . dim B ≤ 2. 81

Λ canonical algebra T tilting module in add( T Λ ∪ Q Λ ) B = End Λ ( T ) The general shape of Γ B is as follows � � � � � � � P B Q B T B • P B a family of components containing a unique postprojective component P ( B ) and all indecomposable projective B -modules. • Q B = Q C for a concealed canonical factor algebra C of B . • T B an infinite family of pairwise orthogo- nal generalized standard coray tubes, separating P B from Q B . • pd B X ≤ 1 for all modules X in P B . • id B Y ≤ 1 for all modules Y in T B ∪ Q B . • gl . dim B ≤ 2. B ∼ = End Λ ( T ), T tilting module in add( T Λ ∪ Q Λ ), if and only if B op ∼ = End Λ ( T ′ ), T ′ tilting module in add( P Λ ∪ T Λ ) ( B op almost con- cealed canonical algebra) 82

Almost concealed canonical algebras of Euclidean type Theorem. (1) The class of concealed canon- ical algebras of Euclidean type coincides with the class of concealed algebras of Euclidean type. (2) The class of almost concealed canoni- cal algebras of Euclidean types coincides with the class of tilted algebras of the form End H ( T ) , where H is a hereditary algebra of a Euclidean type and T is a tilt- ing H -module without preinjective direct summands. (3) The class of the opposite algebras of almost concealed canonical algebras of Euclidean types coincides with the class of tilted algebras of the form End H ( T ) , where H is a hereditary algebra of a Eu- clidean type and T is a tilting H -module without postprojective direct summands. (4) An algebra A is a representation-infinite tilted algebra of a Euclidean type if and only if A is isomorphic to B or B op , for an almost concealed canonical algebra B of a Euclidean type. 83

Tubular algebra = almost concealed canonical algebra of tubular type Theorem. Let B be a tubular algebra. Then the Auslander-Reiten quiver Γ B of B is of the form � � � � � � �� �� �� � � � � � � �� �� �� � � � � � � � � � � � � � � � � � • • • • • • • • • � � � � � � � � � � � � � � � � � � � � P B T B q ∈ Q + T B T B Q B q ∞ 0 where P B is a postprojective component with a Euclidean section, Q B is a preinjective com- ponent with a Euclidean section, T B is an 0 infinite family of pairwise orthogonal gener- alized standard ray tubes containing at least one indecomposable projective B -module, T B ∞ is an infinite family of pairwise orthogonal generalized standard coray tubes containing at least one indecomposable injective B -module, q , for q ∈ Q + (the set of positive and each T B rational numbers) is an infinite family of pair- wise orthogonal faithful generalized standard stable tubes. 84

Quasitilted algebra of canonical type – an algebra A of the form End H ( T ), where T is a tilting object in an abelian hereditary K - category H whose derived category D b ( H ) of H is equivalent, as a triangulated cate- gory, to the derived category D b (mod Λ) of the module category mod Λ of a canonical algebra Λ over K . Theorem ( Happel-Reiten ) . Let be a A finite dimensional quasitilted K -algebra over a field K . Then A is either a tilted algebra or a quasitilted algebra of canonical type. 85

Theorem ( Lenzing-Skowro´ nski ) . Let A be a finite dimensional K -algebra over a field K . The following conditions are equivalent: (1) A is a representation-infinite quasitilted algebra of canonical type. (2) Γ A admits a separating family T A of pair- wise orthogonal generalized standard se- miregular (ray or coray) tubes. � � � � � � � � P A Q A T A • Hom A ( T A , P A ) = 0, Hom A ( Q A , T A ) = 0, Hom A ( Q A , P A ) = 0 • every homomorphism f : X → Y with X in P A and Y in Q A factorizes through a module Z from add T A Moreover, A admits factor algebras A l (left part of A ) and A r (right part of A ) such that • A l is almost concealed of canonical type and P A = P A l • A op is almost concealed of canonical type r and Q A = Q A r 86

� � � � � � � � Example. Let A = KQ/I where Q is the quiver σ (1 , 1) 4 � �������������������������������� � ������������������������������ α 1 α 2 β 1 β 2 β 3 ω (2 , 1) (2 , 2) 0 � ���������������� � ������������������ γ 1 γ 3 (3 , 1) (3 , 2) γ 2 ξ δ η ̺ � 6 � 9 5 7 ν 8 and I is the ideal of KQ generated by the elements α 2 α 1 + β 3 β 2 β 1 + γ 3 γ 2 γ 1 , α 2 σ, ξγ 1 , δγ 2 , ν̺ Then A is a quasitilted algebra of canonical type A l = KQ ( l ) /I ( l ) tubular algebra of type (3 , 3 , 3) Q ( l ) obtained from Q by removing the vertices 5 , 6 , 7 , 8 , 9 and the arrows ξ, η, δ, ̺, ν I ( l ) ideal of KQ ( l ) generated by α 2 α 1 + β 3 β 2 β 1 + γ 3 γ 2 γ 1 , α 2 σ 87

A r = KQ ( r ) /I ( r ) almost concealed canonical algebra of wild type (2 , 3 , 8) Q ( r ) obtained from Q by removing the vertex 4 and the arrow σ I ( r ) ideal of KQ ( r ) generated by α 2 α 1 + β 3 β 2 β 1 + γ 3 γ 2 γ 1 , ξγ 1 , δγ 2 , ν̺ Γ A = P A ∨ T A ∨ Q A P A = P A l , Q A = Q A r T A semiregular family of tubes separating P A from Q A T A consists of a stable tube T A of rank 3 1 88

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S (2 , 1) S (2 , 2) M M � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (identifying along the dashed lines) consisting of indecomposable modules over the canonical algebra C = K ∆ /J , where ∆ is the full subquiver of Q given by the ver- tices 0, ω , (1 , 1), (2 , 1), (2 , 2), (3 , 1), (3 , 2) and J is the ideal of K ∆ generated by α 2 α 1 + β 3 β 2 β 1 + γ 3 γ 2 γ 1 K � ����������������������� � ����������������������� 1 1 M : 0 0 K K � �������������� � �������������� − 1 1 K K 1 89

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � a coray tube T A of the form 0 I 4 N � � � � � � � � � � � � � � � � � � � � � � � � � S (1 , 1) • N � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • � � � � � � � � � � � � � � � � � � � � � � � � � (identifying along the dashed lines) obtained from the stable tube T C of Γ C of 0 rank 2, with S (1 , 1) and N on the mouth, by one coray insertion 0 � ����������������������� � ����������������������� 1 1 1 N : K K K K � �������������� � �������������� − 1 1 K K 1 90

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � a ray tube T A of the form 2 P 8 R � � � � � � � � � � � � � � � � � � • • P 9 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • P 7 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • S (3 , 2) • • P 6 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • P 5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S (3 , 1) • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • • R � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � • • • • • � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � (identifying along the dashed lines) obtained from the stable tube T C of rank 3, 2 with S (3 , 1) , S (3 , 2) and R on the mouth, by 5 ray insertions K � ����������������������� � ����������������������� 1 1 − 1 1 1 R : K K K K � �������������� � �������������� 0 0 and the infinite family of stable tubes of rank 1, consisting of indecomposable C -modules 91

7. Double tilted algebras Theorem ( Happel–Reiten–Smalø ) . Let A be a finite dimensional K -algebra such that each indecomposable X in mod A satisfies pd A X ≤ 1 or id A X ≤ 1 . Then gl . dim A ≤ 3 . Following Coelho and Lanzilotta a finite dimenisional K -algebra A is said to be • shod (small homological dimension) if every indecomposable module X in mod A satisfies pd A X ≤ 1 or id A X ≤ 1. • strict shod if A is shod and gl . dim A = 3. Theorem ( Coelho–Lanzilotta ) . Let A be a finite dimensional K -algebra over a field K . The following conditions are equivalent: (1) A is a shod algebra. (2) ind A = L A ∪ R A . (3) There exists a splitting torsion pair ( Y , X ) in mod A such that pd A Y ≤ 1 , for each module Y ∈ Y (torsion-free part), and id A X ≤ 1 , for each module X ∈ X (tor- sion part). 92

� � � � � � � � � � Theorem. Let A be a shod algebra. The following conditions are equivalent: (1) A is a strict shod algebra. (2) L A \ R A contains an indecomposable in- jective A -module. (3) R A \L A contains an indecomposable pro- jective A -module. Example. A = KQ/I , Q the quiver β γ α σ 1 ← − − 2 ← − − 3 ← − − 4 ← − − 5 I ideal of KQ generated by βα and γβ . The Auslander-Reiten quiver Γ A is of the form � � � P 3 = I 2 � � � � � � � � � � � � � � � � � � � � � � � S 1 = P 1 S 2 S 3 S 4 I 5 = S 5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � L A � � � � � � � � � � � � � � P 2 = I 1 � P 4 I 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � R A � � � � � � � � P 5 = I 3 � � � � � � � � � � � � � � � � � � � � � 0 − → P 1 − → P 2 − → P 3 − → P 4 − → S 4 − → 0 minimal projective resolution of S 4 , so pd A S 4 = 3. A strict shod algebra 93

A finite dimensional K -algebra over a field K C a component of Γ A . A full translation subquiver ∆ of C is said to be a double section of C if the following conditions are satisfied: (a1) ∆ is acyclic. (a2) ∆ is convex in C . (a3) For each τ A -orbit O in C , we have 1 ≤ | ∆ ∩ O| ≤ 2. (a4) If O is a τ A -orbit O in C and | ∆ ∩ O| = 2 then ∆ ∩O = { X, τ A X } , for some module X ∈ C , and there exist sectional paths I → · · · → τ A X and X → · · · → P in C with I injective and P projective. A double section ∆ in C with | ∆ ∩O| = 2, for some τ A -orbit O in C , is said to be a strict double section of C . 94

A path X 0 → X 1 → · · · → X m , with m ≥ 2, in an Auslander-Reiten quiver Γ A is said to be almost sectional if there exists exactly one index i ∈ { 2 , . . . , m } such that X i − 2 ∼ = τ A X i . For a double section ∆ of C , we define the full subquivers of ∆:   there is an almost sectional     ∆ ′ l =  X ∈ ∆; path X → · · · → P with P  ,   projective   there is an almost sectional     ∆ ′ r =  X ∈ ∆; path I → · · · → X with I in-  ,   jective ∆ l = (∆ \ ∆ ′ r ) ∪ τ A ∆ ′ r , left part of ∆, l ) ∪ τ − 1 ∆ r = (∆ \ ∆ ′ A ∆ ′ l , right part of ∆. ∆ is a section if and only if ∆ l = ∆ = ∆ r 95

An indecomposable finite dimensional K -algebra B is said to be a double tilted algebra if the following conditions are satisfied: (1) Γ B admits a component C with a faithful double section ∆. (2) There exists a tilted quotient algebra B ( l ) of B (not necessarily indecomposable) such that ∆ l is a disjoint union of sections of the connecting components of the inde- composable parts of B ( l ) and the cate- gory of all predecessors of ∆ l in ind B coincides with the category of all prede- cessors of ∆ l in ind B ( l ) . (3) There exists a tilted quotient algebra B ( r ) of B (not necessarily indecomposable) such that ∆ r is a disjoint union of sections of the connecting components of the inde- composable parts of B ( r ) , and the cate- gory of all successors of ∆ r in ind B coin- cides with the category of all successors of ∆ r in ind B ( r ) . B is a strict double tilted algebra if the double section ∆ is strict B ( l ) left tilted algebra of B B ( r ) right tilted algebra of B B is a tilted algebra if and only if B = B ( l ) = B ( r ) 96

Theorem ( Reiten-Skowro´ nski ) . An indecom- posable finite dimensional K -algebra A is a double tilted algebra if and only if the quiver Γ A contains a component C with a faithful double section ∆ such that Hom A ( U, τ A V ) = 0 , for all modules U ∈ ∆ r and V ∈ ∆ l . Theorem ( Reiten-Skowro´ nski ) . Let A be an indecomposable finite dimensional K -algebra. The following conditions are equivalent: (1) A is a strict shod algebra. (2) A is a strict double tilted algebra. (3) Γ A admits a component C with a faithful strict double section ∆ such that Hom A ( U, τ A V ) = 0 , for all modules U ∈ ∆ r and V ∈ ∆ l . Corollary. An indecomposable finite dimen- sional K -algebra A is a shod algebra if and only if A is one of the following • a tilted algebra, • a strict double tilted algebra, • a quasitilted algebra of canonical algebra. 97

� � � � � � � � � � Example. A = KQ/I , Q the quiver β γ α σ 1 ← − 2 ← − 3 ← − 4 ← − 5 I ideal of KQ generated by βα and γβ Γ A is of the form � � � � � � � � � � � � P 3 = I 2 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � S 1 = P 1 � S 2 S 3 � S 4 I 5 = S 5 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ��������������� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 2 = I 1 ∆ � P 4 � I 4 � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P 5 = I 3 � � � � � � � � � ∆ faithful double section of C = Γ A ∆ ′ l = { P 2 , S 2 } ∆ ′ r = { S 3 , P 4 , P 5 } ∆ l = (∆ \ ∆ ′ r ) ∪ τ A ∆ ′ r = { P 2 , S 2 , P 3 } l ) ∪ τ − 1 ∆ r = (∆ \ ∆ ′ A ∆ ′ l = { P 3 , S 3 , P 4 , P 5 } A ( l ) left tilted algebra of A is hereditary of Dynkin type A 3 A ( r ) right tilted algebra of A is hereditary of Dynkin type A 4 98