Relative homology and maximal l-orthogonal modules

RELA TIVE HOMOLOGY AND MAXI MA L l -OR THOGONAL MODULES MA GD ALINI LAD A Intr oduction Maximal l - orthogonal mo dules for artin algebras w ere in tro duced b y Iy ama in [10] and [11], and we re used to define a natural setting for dev eloping a ‘higher dimensional’ Auslander-Reiten theory . In the sec- ond o f these papers Iy ama conjectures that the endomorphism ring of an y t w o maximal l -orthogonal mo dules, M 1 and M 2 , are deriv ed equiv- alen t. He pro ves the conjecture f or l = 1, and for l > 1 he giv es some orthogonality condition on M 1 and M 2 , suc h that the End Λ ( M 2 ) op - End Λ ( M 1 )-bimo dule Hom Λ ( M 2 , M 1 ) is tilting, whic h implies that t he rings End Λ ( M 2 ) and End Λ ( M 1 ) are deriv ed e quiv a len t (see [9]). The purp ose of this pap er is to c haracterize tilting mo dules of the form Hom Λ ( M 2 , M 1 ) in terms o f the r elat ive theories induced by the Λ- mo dules M 1 and M 2 , th us getting a generilization of Iy ama’s result. Relativ e homological algebra, whic h w e use throughout this pap er, w as dev elop ed b y M. Auslander and Ø. Solb erg in a series of three pa- p ers [1], [2], [3] and was use d to study the rep resen tatio n theory o f artin algebras. Iy a ma ’s conjec ture fo r maximal l -orthogonal mo dules is motiv a ted b y the connection bet w een maximal l -o r t hogonal mo dules and non- comm uta tiv e crepan t resolutions, wh ic h w as show n in [11]. In part icu- lar Iy ama pro v ed that if R is a complete regular lo cal ring o f dimension d ≥ 2 and Λ is an R -order whic h is not an isolated singularity , then a Cohen-Macaulay Λ - mo dule M , whic h is a generator a nd cogenerator for mo d Λ, gives a non-comm utativ e crepant resolution if and only if M is maximal ( d − 2)-orthog onal. Considering maximal l -orthogonal mo dules as a nalogs of mo dules giving crepan t resolutions, Iy a ma’s con- jecture for maximal l - orthogonal mo dules is the analog of the Bondal- Orlo v-V an den Bergh conjecture fo r crepan t resolutions ( see [5], [13]). Let us fix some notation that w e will use in the r est of this pap er. By Λ we will denote an artin algebra a nd b y mod Λ the category of finitely generated left Λ-mo dules. If M is in mo d Λ , we denote b y add M the full sub catego r y of mo d Λ, consisting of summands of direct sums of M . A gener ator-c o gener ator for mo d Λ, is a Λ-mo dule M Date : O ctobe r 27, 2018 . Thanks .. for supp o rt. 1 2 LADA suc h that add M contains all the indecomp osable pro jectiv e and the indecomp o sable injectiv e Λ-mo dules. Maxim al l -or t hogonal modules ha v e this prop erty . In the first section,w e recall some concepts from relativ e homo lo gical algebra and w e study further the relativ e theories induced b y generator- cogenerators for mo d Λ. In the second section w e g ive Iy a ma’s defini- tion for maximal l -ortho g onal mo dules a nd state his conjecture. W e pro v e our main theorem whic h is, as w e already men t io ned, a c hara c- terization of tilting mo dules of the form Hom Λ ( M 2 , M 1 ) a nd w e apply it to maximal l -orthogonal mo dules. In this w a y w e a r e able to g iv e a conditio n on tw o maximal l -orthogo nal mo dules, so that the conjec- ture is true. W e pro ve that Iy ama ’s orthogonality condition implies our condition and giv e an example whic h shows that our condition is actually we ak er. 1. Rela tive and ab solute homology The aim of t his section is first to recall some basic definitions and re- sults on relat ive homology , and second to lo ok esp ecially at the relative theory induced by a generator-cogenerator for mo d Λ. In particular w e relate the global dimension of the endomorphism ring of a generator- cogenerator M in mo d Λ, with t he r elat ive - with resp ect to M - global dimension of Λ (it will be defined pre cisely later). Moreov er w e will compare the relativ e homolo g y induced b y suc h a mo dule, with the ordinary absolute homology . F or unexplained terminology and results, w e refer to [1] and [2 ]. Let F b e a n a dditiv e sub-bifunctor of Ext 1 Λ ( − , − ) : (mo d Λ) op × mo d Λ → Ab, where Ab denotes the category o f ab elian g roups. A short exact sequence ( η ) : 0 → A → B → C → 0, in mo d Λ, is called F -exact if η is in F ( C , A ). A Λ-mo dule P is called F -pr oje ctive if for an y F - exact seque nce 0 → A → B → C → 0, the sequence Hom Λ ( P , B ) → Hom Λ ( P , C ) → 0 is exact. Dually , a Λ-mo dule I is called F -inje ctive if for an y F -exact sequence 0 → A → B → C → 0, the se quence Hom Λ ( B , I ) → Hom Λ ( A, I ) → 0 is exact. W e denote b y P ( F ) the full sub category of mod Λ consisting of all F - pro jectiv e Λ-mo dules and b y I ( F ) the one consisting of all F -injective Λ-mo dules. Note that if P (Λ) and I (Λ) denote the sub categories of pro jective and inj ective mo dules in mo d Λ respectiv ely , w e ha v e that P (Λ) ⊆ P ( F ) and I (Λ) ⊆ I ( F ). In this pap er w e w or k with sub-bifunctors of Ext 1 Λ ( − , − ) of the fol- lo wing sp ecial form. Let M b e in mo d Λ. F or eac h pair of Λ-mo dules A and C w e define F M ( C , A ) = { 0 → A → B → C → 0 | Hom Λ ( M , B ) → Hom Λ ( M , C ) → 0 is ex act } RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 3 and dually F M ( C , A ) = { 0 → A → B → C → 0 | Hom Λ ( B , M ) → Hom Λ ( A, M ) → 0 is exact } . It w as shown in [1], that the ab o v e assignmen ts giv e additiv e sub- bifunctors of Ext 1 Λ ( − , − ). It is straigh tforw ard that P ( F M ) = P (Λ) ∪ add M and I ( F M ) = I (Λ ) ∪ a dd M . Moreo v er, it is kno wn that for an y short exact sequence 0 → A → B → C → 0 and any Λ-mo dule M , the sequenc e Hom Λ ( M , B ) → Ho m Λ ( M , C ) → 0 is exact if and only if the sequence Hom Λ ( B , DT r M ) → Hom Λ ( A, DT r M ) → 0 is exact. Using this fact it is easy to see that F M = F DT r M , so I ( F M ) = I (Λ ) ∪ add DT r M , and F M = F T r D M , so P ( F M ) = P (Λ) ∪ add T rD M . Hence w e hav e a complete picture of the F -pro jectiv es and F -injectiv es for sub-bifunctors of the abov e form. Another nice prop ert y of a sub-bifunctor F = F M of Ext 1 Λ ( − , − ) is that it has enough pr oje ctives , in the sense that fo r an y Λ- mo dule C , there exists an F -exact sequence 0 → K → P → C → 0, with P in P ( F ). Note that the map P → C is nothing but a righ t P ( F )- appro ximation of C whic h w e kno w is an epimorphism since P (Λ) ⊆ P ( F ). Dually w e see that F = F M has enough inje ctives . Let F = F M and C in mo d Λ. An F - p r oje ctive r es o lution of C is an exact sequence · · · → P l f l − → P l − 1 → · · · → P 1 f 1 − → P 0 f 0 − → C → 0 where P i is in P ( F ) and eac h short exact sequence 0 → Im f i +1 → P i → Im f i → 0 is F - exact. Note that suc h a se quence exists for an y Λ-mo dule since F = F M has enough pro j ectiv es. The sequence is called a minima l F -pr oje c tive r esolution if in additio n eac h P i → Im f i is a minimal map. The F-pr oje ctive dimension of C , whic h w e denote b y p d F C , is defined to b e the smallest n suc h that there ex ists an F -pro jectiv e resolution 0 → P n → · · · → P 1 → P 0 → C → 0 . If suc h n does not exist w e set p d F C = ∞ . Dually , w e c an define the notion of a ( m inimal) F - i n je ctive r esolution a nd the F-inje ctive dimension , id F C , of C . Then, the F-glob al dimension of Λ is defined as: gldim F Λ = sup { p d F C | C ∈ mo d Λ } (= sup { id F C | C ∈ mo d Λ } ) . In the sp ecial case where M is a g enerator -cogenerator fo r mo d Λ, w e ha v e the follo wing nice connection b etw een the global dimension of the endomorphism ring of M and the relativ e g lobal dimension of Λ. Prop osition 1.1. L et M b e a ge n er ator-c o gene r ator for mo d Λ . Then, for any p o sitive inte ger l , the fol lowin g ar e e quivalent: (a) gldim End Λ ( M ) ≤ l + 2 4 LADA (b) gldim F M Λ ≤ l (c) gldim F M Λ ≤ l . Pr o of. (a) ⇒ (b) Let N b e a Λ-mo dule and let · · · → M l → M l − 1 f l − 1 − − → · · · → M 1 → M 0 → N → 0 b e a minimal F M -pro jectiv e resolution of N . Set K l − 1 = K er f l − 1 . W e will sho w that K l − 1 is in add M . Applying the functor Hom Λ ( M , − ) to the ab o v e se quence, w e get the lo ng exact sequence η 1 : . . . → Hom Λ ( M , M l ) → Hom Λ ( M , M l − 1 ) Hom( M ,f l − 1 ) − − − − − − − → · · · → Hom Λ ( M , M 1 ) → Hom Λ ( M , M 0 ) → Hom Λ ( M , N ) → 0 . Note that Ker Ho m( M , f l − 1 ) = Hom( M , K l − 1 ). Let also 0 → N → M 0 f − → M 1 → · · · b e the b eginning of an F M -injectiv e resolution of N . Applying the functor Ho m Λ ( M , − ) to this sequen ce, w e get the long exact sequ ence η 2 : 0 → Hom Λ ( M , N ) → Hom Λ ( M , M 0 ) Hom Λ ( M ,f ) − − − − − − − → Hom Λ ( M , M 1 ) → X → , 0 where the End Λ ( M ) op -mo dule X is the cokerne l of the map Hom Λ ( M , f ). Com bining now the sequences η 1 and η 2 w e get the long exact sequence . . . → Ho m Λ ( M , M l ) → Hom Λ ( M , M l − 1 ) → · · · → Hom Λ ( M , M 1 ) → Hom Λ ( M , M 0 ) → Hom Λ ( M , M 0 ) → Hom Λ ( M , M 1 ) → X → 0 Since the Λ-mo dules M i for i = 0 , 1 , . . . and M j for j = 0 , 1 are in add M , the End Λ ( M ) op -mo dules Hom Λ ( M , M i ) , i = 0 , 1 , . . . and Hom Λ ( M , M j ) , j = 0 , 1 are pro jectiv e. So the End Λ ( M ) op -mo dule Hom Λ ( M , K l − 1 ) is an ( l + 2)-th syzygy of the End Λ ( M ) op -mo dule X . Since gldim End Λ ( M ) ≤ l + 2, the mo dule Hom Λ ( M , K l − 1 ) has to b e a pro jectiv e End Λ ( M ) op -mo dule, so K l − 1 is in add M . Th us gldim F M Λ ≤ l . (b) ⇒ (a) Let X b e an End Λ ( M ) op -mo dule and let . . . → Ho m Λ ( M , M l +2 ) d l +2 − − → Hom Λ ( M , M l +1 ) → · · · → Hom Λ ( M , M 2 ) d 2 − → Hom Λ ( M , M 1 ) d 1 − → Hom Λ ( M , M 0 ) → X → 0 b e a pro jectiv e resolution of X . Since M is a generator for mo d Λ, the functor Hom Λ ( M , − ) is full and faithfull hence w e ha v e that Ker d i = Hom Λ ( M , K i ) where K i is the kerne l of the morphism f i : M i → M i − 1 , with Hom Λ ( M , f ) = d i , for i > 0. W e will show that Ker d l +1 is a pro jectiv e End Λ ( M ) op -mo dule. The sequence · · · → M l +2 f l +2 − − → M l +1 → · · · → M 2 → K 1 → 0 RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 5 is an F M -pro jectiv e r esolution of K 1 . Since gldim F M Λ ≤ l , w e ha v e that K l +1 is in add M and hence Hom Λ ( M , K l +1 ) is a pro jectiv e End Λ ( M ) op - mo dule. So gldim End Λ ( M ) ≤ l + 2. Th us w e hav e prov ed the equiv alence of ( a ) and ( b ). The pro of of the equiv alence of ( a ) and ( c ) is sym metric.  Note that t he ab ov e result w a s basically a lready kno wn and can be found, f or example in [8], in a differen t form. But, b esides the fact that using relativ e homology helps in having a simpler statemen t, t here is also the adv a n tage of getting extra information a b out coresolutions of Λ-mo dules in add DT r M and res olutions of Λ- mo dules in add T rD M , since F M = F DT r M and F M = F T r D M . In part icular, for selfinjectiv e algebras, we hav e the follo wing conseque nce. Corollary 1.2. L et Λ b e a se lfinje ctive artin algebr a and X b e in mo d Λ . Then, for any p ositive inte ger l , the fol low ing a r e e quivalent: (a) gldim End Λ (Λ ⊕ X ) ≤ l + 2 , (b) gldim End Λ (Λ ⊕ DT r X ) ≤ l + 2 , (c) gldim End Λ (Λ ⊕ T rD X ) ≤ l + 2 . Remark. A straigh tforw ard consequence o f the ab ov e coro lla ry is that if Λ is a selfinjectiv e artin algebra, then Λ ⊕ X is an Auslander gener- ator (that is, a generator- cog enerator for mo d Λ such that the globa l dimension of its endomorphism ring giv es the represen t a tion dimension of Λ) if and only if Λ ⊕ DT r X is an Auslande r generator. Let A and C b e in mo d Λ. Kno wing that for sub-bifunctors F = F M of Ext 1 Λ ( − , − ) an y Λ-mo dule has an F -pro jectiv e and an F - injectiv e resolution, one can define t he righ t deriv ed func tors Ext i F ( C , − ) and Ext i F ( − , A ) of Hom Λ ( C , − ) and Hom Λ ( − , A ) res p ectiv ely , in the same w ay as in the case P ( F ) = P (Λ ) . Moreov er, it can b e pro v ed that Ext i F ( C , − )( A ) is then isomorphic to Ext i F ( − , A )( C ) and that for i = 1 w e ha v e that Ext 1 F ( C , A ) = F ( C , A ). Although Ext 1 F ( C , A ) is a subgroup of Ext 1 Λ ( C , A ), v ery little is kno wn ab out how the Ext i Λ ( C , A )-groups and the relative Ext i F ( C , A )- groups a re related for i > 1. In the next prop osition we consider a case where these t w o c oincide. This w ill help us in the next section to compare Iy ama’s condition o n maximal orthog o nal modules to our result. F or X and Y in mo d Λ w e write X ⊥ k Y if Ext i Λ ( X , Y ) = (0), for 0 < i ≤ k . Abusing the no tation, w e will write X ⊥ k Y ev en for k = 0 but this will mean no condition on X and Y . Prop osition 1.3. L et M 1 and M 2 b e in mo d Λ such that M 2 is a gener ator for mo d Λ an d M 1 is a c o ge n er ator for mo d Λ . The f o l lowing ar e e quivalent for a p ositive inte ger k : (a) M 2 ⊥ k M 1 , (b) F or any C in mo d Λ , Ext i F M 2 ( C , M 1 ) ≃ Ext i Λ ( C , M 1 ) , for 0 < i ≤ k 6 LADA (c) F or any D in mo d Λ , Ext i F M 1 ( M 2 , D ) ≃ Ext i Λ ( M 2 , D ) , for 0 < i ≤ k . Pr o of. (a) ⇒ (b) Let C b e in mo d Λ and let · · · → P l f l − → · · · → P 1 f 1 − → P 0 f 0 − → C → 0 b e a minimal F M 2 -pro jectiv e resolution of M 1 . Set K − 1 = M 1 and K i = Ker f i , fo r i = 0 , 1 , 2 , . . . . W e first show that fo r all i = − 1 , 0 , 1 , 2 , . . . w e ha v e Ext 1 F M 2 ( K i , M 1 ) ≃ Ext 1 Λ ( K i , M 1 ) . Applying the functor Hom Λ ( − , M 1 ) to t he short exact s equence 0 → K i +1 → P i +1 → K i → 0 w e get the long exact sequence Hom Λ ( P i +1 , M 1 ) → Hom Λ ( K i +1 , M 1 ) → Ext 1 Λ ( K i , M 1 ) → Ext 1 Λ ( P i +1 , M 1 ) . But since M 2 ⊥ k M 1 and P i +1 is in add M 2 , w e hav e Ext 1 Λ ( P i +1 , M 1 ) = (0). So Ext 1 Λ ( K i , M 1 ) is the cokerne l of the map Hom Λ ( K i +1 , M 1 ) → Hom Λ ( P i +1 , M 1 ). The short ex act sequence 0 → K i +1 → P i +1 → K i → 0 is F M 2 -exact so w e also ha v e the fo llo wing long e xact sequence Hom Λ ( P i +1 , M 1 ) → Hom Λ ( K i +1 , M 1 ) → Ext 1 F M 2 ( K i , M 1 ) → Ext 1 F M 2 ( P i +1 , M 1 ) and since Ext 1 F M 2 ( P i +1 , M 1 ) = (0), Ext 1 F M 2 ( K i , M 1 ) is the cokernel of the map Hom Λ ( K i +1 , M 1 ) → Hom Λ ( P i +1 , M 1 ). Hence Ext 1 F M 2 ( K i , M 1 ) ≃ Ext 1 Λ ( K i , M 1 ) , for all i = − 1 , 0 , 1 , 2 , . . . . In particular we ha v e that Ext 1 F M 2 ( C , M 1 ) ≃ Ext 1 Λ ( C , M 1 ) . Next w e s ho w t ha t for a ll i = − 1 , 0 , 1 , 2 , . . . and 2 ≤ j ≤ k Ext j − 1 Λ ( K i +1 , M 1 ) ≃ Ext j Λ ( K i , M 1 ) . T o do this, w e apply the f unctor Hom Λ ( − , M 1 ) to the short exact se- quence 0 → K i +1 → P i +1 → K i → 0 and we get the long exact se quences Ext j − 1 Λ ( P i +1 , M 1 ) → Ext j − 1 Λ ( K i +1 , M 1 ) → Ext j Λ ( K i , M 1 ) → Ext j Λ ( P i +1 , M 1 ) . But since M 2 ⊥ k M 1 and P i +1 is in add M 2 , w e hav e t ha t Ext j − 1 Λ ( P i +1 , M 1 ) = 0 and Ext j Λ ( P i +1 , M 1 ) = 0 whic h implies that Ext j − 1 Λ ( K i +1 , M 1 ) ≃ Ext j Λ ( K i , M 1 ) . No w using the ab o v e w e can see that for all 2 ≤ i ≤ k we ha ve Ext i F M 2 ( C , M 1 ) ≃ Ext i − 1 F M 2 ( K 0 , M 1 ) ≃ · · · ≃ Ext 1 F M 2 ( K i − 2 , M 1 ) ≃ RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 7 Ext 1 Λ ( K i − 2 , M 1 ) ≃ Ext 2 Λ ( K i − 3 , M 1 ) ≃ · · · ≃ Ext i Λ ( C , M 1 ) whic h completes the pro of. (b) ⇒ (a) Set C = M 2 . The n we hav e that Ext i F M 2 ( M 2 , M 1 ) = (0) , since M 2 is F M 2 -pro jectiv e, hence Ext i Λ ( C , M 1 ) = (0), f o r 0 < i ≤ k . The pro of of (a) ⇔ (c) is symm etric.  2. Cotil ting and maximal or thogonal modules In this section w e state and prov e the main theorem and giv e the connections with Iy ama’s result. But let us start b y recalling Iy ama’s definition for maximal l - o rthogonal Λ-mo dules. Set M ⊥ k = { Y ∈ mo d Λ | M ⊥ k Y } and ⊥ k M = { X ∈ mo d Λ | X ⊥ k M } . Definition 2.1. A Λ-mo dule M is called maximal l -ortho gonal if M ⊥ l = add M = ⊥ l M . The following conjecture was stated in [11]. Conjecture 2.2 (O. Iy a ma ) . Let M 1 and M 2 b e maximal l -ortho g onal in mo d Λ. Then their endomorphism ring s, End Λ ( M 1 ) a nd End Λ ( M 2 ), are derive d equiv alen t. Before we con tinue, we giv e a characterization of maximal orthogonal mo dules that can b e found in a more general setting in [11, Prop osition 2.2.2]. F or con v enience, w e restate it and pro v e it here in the languag e of relativ e homology . W e call a Λ-mo dule k-selfortho gonal if M ⊥ k M holds. Prop osition 2.3. L et M b e in mod Λ and l a p ositive inte ger. The fol lowing ar e e quivalent for any inte ger k , s uch that 0 ≤ k ≤ l . (a) M i s maxima l l -ortho gonal in mo d Λ , (b) M is a gener ator-c o gener ator for mo d Λ , M is l -self o rtho gonal and p d F M X ≤ l − k for any X in ⊥ k M , (c) M is a gener ator-c o gener ator for mo d Λ , M is l -selfortho gonal and id F M Y ≤ l − k for any Y in M ⊥ k . Pr o of. (a) ⇒ (b) Let · · · → M l − k f l − k − − → M l − k − 1 → · · · → M 1 f 1 − → M 0 f 0 − → C → 0 b e a minimal F M -pro jectiv e resolution of X . Set K − 1 = X and K i = Ker f i , for i = 0 , 1 , 2 , . . . . W e wan t to sho w that K l − k − 1 is in add M . In order to do this, w e show tha t Ext j Λ ( M , K l − k − 1 ) = (0) , for a ll j = 1 , 2 , . . . , l . F or a ll i , applying the functor Hom Λ ( M , − ) to the short exact sequence 0 → K i +1 → M i +1 → K i → 0 w e get the long exact sequence Hom Λ ( M , M i +1 ) → Hom Λ ( M , K i ) → Ext 1 Λ ( M , K i +1 ) → Ext 1 Λ ( M , M i +1 ) . 8 LADA Since M is l -selforthogo nal with l ≥ 1 and M i +1 is in a dd M , w e ha ve that Ext 1 Λ ( M , M i +1 ) = (0). Moreov er, since the short exact sequence 0 → K i +1 → M i +1 → K i → 0 is F M -exact, the map Hom Λ ( M , M i +1 ) → Hom Λ ( M , K i ) is an epimorphism. Hence Ext 1 Λ ( M , K i +1 ) = (0 ). F rom the same short exact se quence we also g et the long ex act seq uences Ext j − 1 Λ ( M , M i +1 ) → Ext j − 1 Λ ( M , K i ) → Ext j Λ ( M , K i +1 ) → Ext j Λ ( M , M i +1 ) . F or 2 ≤ j ≤ l , since M is l -selforthogonal and M i +1 is in add M , w e ha v e that Ext j − 1 Λ ( M , M i +1 ) = (0) and Ex t j Λ ( M , M i +1 ) = (0), whic h implies that Ext j − 1 Λ ( M , K i ) ≃ Ext j Λ ( M , K i +1 ) , 2 ≤ j ≤ l No w, using the ab ov e, w e can compute the groups Ext j Λ ( M , K l − k − 1 ) as follo ws: for 2 ≤ j ≤ l − k w e ha v e Ext j Λ ( M , K l − k − 1 ) ≃ Ext 1 Λ ( M , K l − k − j ) = (0) and fo r j = l − k + s, 1 ≤ s ≤ k we ha ve Ext l − k + s Λ ( M , K l − k − 1 ) ≃ Ext s Λ ( M , X ) = (0) . So w e ha v e Ext j Λ ( M , K l − k − 1 ) ≃ Ext 1 Λ ( M , K l − k − j ) = (0) , 2 ≤ j ≤ l and since M is maximal l - orthogonal this implie s that K l − k − 1 is in add M . Hence pd F M X ≤ l − k . (b) ⇒ (a) Let X be in mo d Λ such that Ext i Λ ( X , M ) = ( 0 ), for 0 < i ≤ l . W e will sho w that X is t hen in add M . Note that k ≤ l , so Ext i Λ ( X , M ) = (0), for 0 < i ≤ k o r equiv a len tly X is in ⊥ k M , hence b y assumption p d F M X ≤ l − k . If k = 0, this will imply tha t X is F M -pro jectiv e, hence X is in add M . Assume tha t k > 0 and let 0 → M l − k f l − k − − → M l − k − 1 → · · · → M 1 f 1 − → M 0 f 0 − → X → 0 b e a minimal F M -pro jectiv e resolution of X . Set K − 1 = X , K i = Ker f i , for i = 0 , 1 , . . . , l − k − 2 and K l − k − 1 = M l − k . The n, for an y i , applying the functor Hom Λ ( X , − ) to the short exact sequenc e 0 → K i → M i → K i − 1 → 0 w e get long ex act seq uences Ext j Λ ( X , M i ) → Ext j Λ ( X , K i − 1 ) → Ext j + 1 Λ ( X , K i ) → Ext j + 1 Λ ( X , M i ) . Since M is l -selforthogonal and M i is in add M for all i , we hav e tha t the first and last term of the ab o v e sequence v anish for all j = 1 , 2 , . . . , l − 1. So w e ha v e Ext j Λ ( X , K i − 1 ) ≃ Ext j + 1 Λ ( X , K i ) for all i and fo r j = 1 , 2 , . . . , l − 1. But then w e ha v e Ext 1 Λ ( X , K 0 ) ≃ Ext 2 Λ ( X , K 1 ) ≃ · · · ≃ Ext l − k Λ ( X , K l − k − 1 ) = Ext l − k Λ ( X , M l − k ) = (0) . RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 9 This implies that the short exact sequence 0 → K 0 → M 0 → X → 0 splits a nd henc e X is in add M . So M is maximal l - orthogonal. The pro of of (a) ⇔ (c) is symm etric.  Setting k = 0 in the ab o v e pro p osition, a nd using Lemma 1.1, w e ha v e the following nice c haracterization of maximal l -orthogonal mo d- ules. Corollary 2.4. L et M b e in mo d Λ and l a p ositive inte ge r. The fol lowing ar e e quivalent: (a) M is maximal l -ortho g onal, (b) M is a gener ator-c o gener ator for mo d Λ , M is l -selfortho gonal and gldim End Λ ( M ) ≤ l + 2 . Before w e state and prov e our main theorem w e recall some defini- tions. A Λ- mo dule M is called c otilting , if it has the fo llo wing prop- erties: ( 1) Ext i Λ ( M , M ) = (0), for i > 0, (2) id Λ M < ∞ and ( 3 ) I (Λ ) ⊆ \ add M . Similarly , if F = F M is a sub-bifunctor of Ext 1 Λ ( − , − ), a Λ-mo dule M is called F -c otilting if : (1) Ext i F ( M , M ) = (0), for i > 0, (2) id F M < ∞ a nd (3) I ( F ) ⊆ \ add F M , where \ add F M denotes the full sub category of mo d Λ consisting of all mo dules that hav e a finite resolution in add M whic h is in addition F -exact. The not io ns o f tilting and F-tilting mo dules are defined dually . F or an ar t in algebra Λ it is kno wn that when gldim Λ < ∞ , a Λ- mo dule M is tilting if and only if M it is cotilting. The pro of is based on the one t o one corresp ondences b et w een equiv a lence classes of tilting or cotilting Λ-mo dules and certain sub categories of mo d Λ ( see for ex- ample [12, Theorem 2.1]). A relativ e v ersion of these corresp ondences is prov ed in [2] and using these one can prov e the following: Lemma 2.5. L et F b e a sub-bifunctor of Ext 1 Λ ( − , − ) with enough pr o- je ctives and M a Λ -mo dule. Assume that gldim F Λ < ∞ . Then M is F -tilting if and only if M is F -c otilting. W e are now in p osition to giv e the main theorem of this paper. Theorem 2.6. L et Λ b e an artin algebr a with M 1 and M 2 two gener ator- c o gener ators f o r mo d Λ . Supp ose that ther e exists some p osi tive inte ger l such that gldim End Λ ( M i ) ≤ l + 2 for i = 1 , 2 . The fol lo wing ar e e quivalent: (a) Ext i F M 1 ( M 2 , M 2 ) = (0 ) and Ext i F M 2 ( M 1 , M 1 ) = (0 ) for 0 < i ≤ l , (b) M 2 is an F M 1 -c otilting mo dule, (c) M 1 is an F M 2 -c otilting mo dule, (d) Hom Λ ( M 2 , M 1 ) is a c o tilting End Λ ( M 2 ) op - End Λ ( M 1 ) -bimo dule. Pr o of. (a) ⇒ (b) First, since gldim End Λ ( M 1 ) ≤ l + 2, apply ing L emma 1.1, w e see that id F M 1 M 2 ≤ l . Also, b y assumption, w e hav e Ext i F M 1 ( M 2 , M 2 ) = 10 LADA (0) for 0 < i ≤ l and since id F M 1 M 2 ≤ l w e get Ext i F M 1 ( M 2 , M 2 ) = (0) for i > 0. It remains to show tha t I ( F M 1 ) is contained in \ add F M 1 M 2 . T o do this, w e consider a minimal F M 2 -pro jectiv e resolution of M 1 · · · → P l → P l − 1 f l − 1 − − → · · · → P 1 → P 0 → M 1 → 0 and w e set K l − 1 = Ker f l − 1 . Since gldim End Λ ( M 2 ) ≤ l + 2, applying Lemma 1.1 w e see that K l − 1 is F M 2 -pro jectiv e, so it is in add M 2 . Applying the functor Hom Λ ( − , M 1 ) to this sequence we get a long exact sequence 0 → Hom Λ ( M 1 , M 1 ) → Hom Λ ( P 0 , M 1 ) → Hom Λ ( P 1 , M 1 ) → · · · → Hom Λ ( P l − 1 , M 1 ) → Hom Λ ( K l − 1 , M 1 ) . But, b y assumption, Ext i F M 2 ( M 1 , M 1 ) = (0) f o r 0 < i ≤ l , so the a b o v e sequence is exact and the map Hom Λ ( P l − 1 , M 1 ) → Hom Λ ( K l − 1 , M 1 ) is an epimorphism, whic h implies that the sequence 0 → K l − 1 → P l − 1 f l − 1 − − → · · · → P 1 → P 0 → M 1 → 0 is F M 1 -exact. Th us M 2 is an F M 1 -cotilting module. (b) ⇒ (a) Since M 2 is an F M 1 -cotilting mo dule, Ext i F M 1 ( M 2 , M 2 ) = (0) for i > 0. Mor eov er, I ( F M 1 ) is contained in \ add F M 1 M 2 , so there exists an F M 1 -exact sequence ( η ) : 0 → P n → P n − 1 f n − 1 − − → · · · → P 1 f 1 − → P 0 f 0 − → M 1 → 0 with P i in add M 2 , for all i . W e will show that ( η ) is also F M 2 -exact. W e set K i = Ker f i for i = 1 , . . . , n − 2, K 0 = M 1 and K n − 1 = P n . Then, for an y i , applying the functor Hom Λ ( M 2 , − ) to the short exact sequence ( η i ) : 0 → K i → P i → K i − 1 w e get long ex act seq uences Ext j F M 1 ( M 2 , P i ) → Ext j F M 1 ( M 2 , K i − 1 ) → Ext j + 1 F M 1 ( M 2 , K i ) → Ext j + 1 F M 1 ( M 2 , P i ) for j > 0. But since P i is in add M 2 , for a ll i, the first and the last term of these sequences are zero, hence the t w o middle t erms a re isomorphic. So w e ha v e Ext 1 F M 1 ( M 2 , K i ) ≃ Ext 2 F M 1 ( M 2 , K i +1 ) ≃ · · · ≃ Ext n − i F M 1 ( M 2 , K n − 1 ) = Ext n − i F M 1 ( M 2 , P n ) = (0) . for i = 0 , 1 , . . . , n − 1. But by applying the functor Hom Λ ( M 2 , − ) to the short exact sequence s ( η i ), we also get the long e xact sequences Hom Λ ( M 2 , P i ) → Hom Λ ( M 2 , K i − 1 ) → Ext 1 F M 1 ( M 2 , K i ) and since Ext 1 F M 1 ( M 2 , K i ) = (0) for a ll i , w e hav e that the map Hom Λ ( M 2 , P i ) → Hom Λ ( M 2 , K i − 1 ) is an e pimorphism for all i , whic h implies tha t ( η ) is RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 11 F M 2 -exact. W e no w apply the functor Hom Λ ( − , M 1 ) to ( η ) and w e get the complex 0 → Hom Λ ( M 1 , M 1 ) → Hom Λ ( P 0 , M 1 ) → Hom Λ ( P 1 , M 1 ) → · · · → Hom Λ ( P n , M 1 ) → 0 Since ( η ) is F M 2 -exact, the i − th -homology of the abov e c omplex is Ext i F M 2 ( M 1 , M 1 ). But sinc e ( η ) is F M 1 -exact, the abov e complex is acyclic. Hence Ext i F M 2 ( M 1 , M 1 ) = (0), for i > 0, whic h completes the pro of. (a) ⇔ (c) The proo f is symmetric to the pro o f of (a) ⇔ (b) (b) ⇒ (d) Since gldim End Λ ( M 1 ) ≤ l + 2, by Lemma 1.1, we g et gldim F M 1 Λ ≤ l and then, b y Lemma 2.5, w e hav e that M 2 is an F M 1 -tilting Λ -mo dule. But then, since I ( F M 1 ) = add M 1 , the mo d- ule Hom Λ ( M 2 , M 1 ) is a cotilting End Λ ( M 2 ) op -mo dule, as shown in [2]. This is equiv alen t to Hom Λ ( M 2 , M 1 ) b eing a cotilting End Λ ( M 2 ) op - End Λ ( M 1 )-bimo dule. (d) ⇒ (a) W e first show that Ext i F M 1 ( M 2 , M 2 ) = (0) for 0 < i ≤ l . Recall that b y Lemma 1.1 w e ha ve that gldim F M 1 Λ ≤ l and let 0 → M 2 → I 0 → I 1 → · · · → I l → 0 b e a m inimal F M 1 -injectiv e resolution of M 2 . Then b y applying the functor Hom Λ ( − , M 1 ) w e get a minimal pro jectiv e resolution of the End Λ ( M 1 )- mo dule Hom Λ ( M 2 , M 1 ) 0 → Hom Λ ( I l , M 1 ) → Hom Λ ( I l − 1 , M 1 ) → · · · → Hom Λ ( I 1 , M 1 ) → Hom Λ ( I 0 , M 1 ) → Hom Λ ( M 2 , M 1 ) → 0 Applying the f unctor Hom End Λ ( M 1 ) ( − , Hom Λ ( M 2 , M 1 )) to the last se- quence w e get the follo wing comm utativ e exact diagram where the notation has b een simplifie d 0 / / (( M 2 , M 1 ) , ( M 2 , M 1 )) ≀   / / (( I 0 , M 1 ) , ( M 2 , M 1 )) ≀   / / · · · / / (( I l , M 1 ) , ( M 2 , M 1 )) ≀   / / 0 0 / / ( M 2 , M 2 ) / / ( M 2 , I 0 ) / / · · · / / ( M 2 , I l ) / / 0 F rom the abov e diagram w e see that Ext i F M 1 ( M 2 , M 2 ) ≃ Ext i End Λ ( M 1 ) (Hom Λ ( M 2 , M 1 ) , Hom Λ ( M 2 , M 1 )) = (0) , for i > 0, since Hom Λ ( M 2 , M 1 ) is a cotilting End Λ ( M 1 )-mo dule. Symmetrically , starting with a minimal F M 2 -pro jectiv e resolution of M 1 and a pplying the functor Hom Λ ( M 2 , − ) we can sho w that Ext i F M 2 ( M 1 , M 1 ) ≃ Ext i End Λ ( M 2 ) op (Hom Λ ( M 2 , M 1 ) , Hom Λ ( M 2 , M 1 )) = (0) 12 LADA for i > 0 . Here w e are using that Hom Λ ( M 2 , M 1 ) is a cotilting End Λ ( M 2 ) op - mo dule.  The following easy consequence of the ab ov e theorem generalizes the Theorem 5.3 .2 in [11]. Corollary 2.7. L et M 1 and M 2 b e ma x i mal l - ortho gonal m o dules in mo d Λ such that Ext i F M 1 ( M 2 , M 2 ) = (0) an d Ext i F M 2 ( M 1 , M 1 ) = (0) fo r 0 < i ≤ l . Th e n their endomorphism rings, End Λ ( M 1 ) an d End Λ ( M 2 ) , ar e derive d e quivalent. Pr o of. Le t M 1 and M 2 b e maximal l -orthog onal modules in mod Λ satisfying the assu mption o f the Corollary . By Prop osition 2.4 and Prop osition 1.1 we hav e that M 1 and M 2 are generator-cogenerators for mo d Λ with gldim End Λ ( M i ) ≤ l + 2, i = 1 , 2. Then using Theorem 2.6 and w e get that Hom Λ ( M 2 , M 1 ) is a cotilting End Λ ( M 2 ) op -End Λ ( M 1 )- bimo dule and hence End Λ ( M 1 ) a nd End Λ ( M 2 ) a re deriv ed equiv alent b y a resu lt o f Happ el [9].  Although it is obvious fro m our Theorem 2.6 that Iyama’s orthog- onalit y condition on tw o maximal l -ortho g onal m o dules M 1 and M 2 implies the v anishing of Ext i F M 1 ( M 2 , M 2 ) a nd Ext i F M 2 ( M 1 , M 1 ) fo r 0 < i ≤ l , it is in teresting to give a direct pro of of this fact. This is done in the next prop osition. Prop osition 2.8. L et M 1 and M 2 b e maximal l -ortho gona l in mo d Λ . Assume that ther e exists a p ositive inte ger k , such that k ≤ l ≤ 2 k + 1 and M 2 ⊥ k M 1 . Then (a) Ext i F M 2 ( M 1 , M 1 ) = (0) , 0 < i ≤ l , (b) Ext i F M 1 ( M 2 , M 2 ) = (0) , 0 < i ≤ l . Pr o of. (a) F or 0 < i ≤ k , since M 2 ⊥ k M 1 , b y Proposition 1.3 we ha ve that Ext i F M 2 ( M 1 , M 1 ) = Ext i Λ ( M 1 , M 1 ) . But M 1 is l -o r thogonal, so Ext i Λ ( M 1 , M 1 ) = (0), for 0 < i ≤ k . Hence Ext i F M 2 ( M 1 , M 1 ) = (0) , for 0 < i ≤ k . F or i > k + 1, since l ≤ 2 k + 1, we hav e that i > l − k . But since M 2 is maximal l -orthogonal, by Lemma 2.3 w e ha v e that p d F M 2 M 1 ≤ l − k . So Ext i F M 2 ( M 1 , M 1 ) = (0), f o r i > k + 1. F or i = k + 1, if l < 2 k + 1, we hav e again that i > l − k , so using the same argumen t as b efore we g et that Ext i F M 2 ( M 1 , M 1 ) = (0) . It remains to sho w that Ext k +1 F M 2 ( M 1 , M 1 ) = (0 ), in the case where l = 2 k + 1. In order to do this, cons ider a minimal F M 2 -pro jectiv e resolution of M 1 · · · → P k f k − → P k − 1 → · · · → P 1 f 1 − → P 0 → M 1 → 0 RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 13 and set K i = Ker f i , fo r i = 0 , 1 , . . . . Then w e ha v e Ext k +1 F M 2 ( M 1 , M 1 ) ≃ Ext k F M 2 ( K 0 , M 1 ) ≃ · · · ≃ Ext 1 F M 2 ( K k − 1 , M 1 ) ≃ Ext 1 Λ ( K k − 1 , M 1 ) ≃ · · · ≃ Ext k Λ ( K 0 , M 1 ) . T o sho w t hese isomorphisms w e use the same argumen t s as in the pro of of 1.3 a nd w e omit here the details. Apply ing the functor Hom Λ ( − , M 1 ) to the short exact sequence 0 → K 0 → P 0 → M 1 → 0 w e get the long exact sequence Ext k Λ ( P 0 , M 1 ) → Ext k Λ ( K 0 , M 1 ) → Ext k +1 Λ ( M 1 , M 1 ) . Since M 1 is l -orthogonal and k + 1 < l , we ha v e that Ext k +1 Λ ( M 1 , M 1 ) = (0) and since M 2 ⊥ k M 1 and P 0 is in add M 2 , w e ha v e that Ext k Λ ( P 0 , M 1 ) = (0). So Ext k Λ ( K 0 , M 1 ) = (0 ) and hence Ext k +1 F M 2 ( M 1 , M 1 ) = (0 ), whic h completes the pro of. The pro of of (b) is symme tric.  The conv erse of Prop osition 2.8 is not in general true, as the following example shows . Example 2.9. Let Q be the quiv er 1 α / / 2 β         3 γ [ [ 6 6 6 6 6 6 and K Q the path algebra of Q o v er some field K . Let also I be the ideal of K Q generated by all paths o f length 5 and c onsider the fa ctor algebra Λ = K Q/I . Set M 1 = P 1 ⊕ P 2 ⊕ P 3 ⊕ S 1 ⊕ P 3 / r 2 P 3 and M 2 = P 1 ⊕ P 2 ⊕ P 3 ⊕ S 1 ⊕ P 1 / r 2 P 1 , where P i denotes the indecomp osable pro j ectiv e Λ-mo dule corresp ond- ing to v ertex i , for i = 1 , 2 , 3 and S 1 denotes the s imple mo dule in v ertex 1. It is easy to ve rify tha t M 1 and M 2 are maximal 2-orthogo nal mo dules in mo d Λ. Moreo ver, the mo dules M 1 and M 2 are connected via the follo wing exact sequence: ( η ) : 0 → P 1 / r 2 P 1 → S 1 ⊕ P 1 → S 1 ⊕ P 3 → P 3 / r 2 P 3 → 0 . Observ e that the ab ov e s equence is both an F M 2 -exact and an F M 1 - exact sequence. Henc e ( η ) can be view ed both as an F M 2 -pro jectiv e resolution of P 3 / r 2 P 3 and as an F M 1 -injectiv e res olution of P 1 / r 2 P 1 . But then, if we apply the functor Hom Λ ( − , M 1 ) to ( η ), the resulting 14 LADA complex is acyclic, since ( η ) is F M 1 -exact, so Ext i F M 2 ( P 3 / r 2 P 3 , M 1 ) = (0), for i > 0 and hence Ext i F M 2 ( M 1 , M 1 ) = (0 ), for i > 0. Also, if we apply the functor Hom Λ ( M 2 , − ) to ( η ), the resulting complex is acyclic, since ( η ) is F M 2 -exact, so Ext i F M 1 ( M 2 , P 1 / r 2 P 1 ) = (0), for i > 0 and hence Ext i F M 1 ( M 2 , M 2 ) = (0), for i > 0. Th us, w e hav e show n that M 1 and M 2 satisfy the conclusions (a) and (b) of Prop osition 2.8. Next w e consider the short exact s equence 0 → P 3 / r 2 P 3 → P 1 / So c P 1 → P 1 / r 2 P 1 → 0 . The sequence is non split, so Ext 1 Λ ( P 1 / r 2 P 1 , P 3 / r 2 P 3 ) 6 = (0), hence Ext 1 Λ ( M 2 , M 1 ) 6 = (0) whic h implies that Iyama’s orthogonalit y con- dition do es not ho ld for M 1 and M 2 . In fact, in the ab ov e example, w e can compute all maximal 2 -orthogonal Λ-mo dules and w e can then see that they are all connected w ith se- quences whic h ha ve the pro p erties o f ( η ), meaning that w e can get one from the other b y exc hanging one indecomp osable summand using ap- pro ximations. T h us, for the ab ov e example, Iy ama’s conjecture is true. W e complete this section with a proposition sho wing the connection b et w een these exch ange sequenc es and the v a nishing of the relativ e Ext F . Prop osition 2.10. L et M i = N ⊕ X i b e in mo d Λ such that N is a gener ator-c o gener ator for mod Λ and X i ar e ind e c omp osable not c on- taine d in add N , for i = 1 , 2 . Assume that ther e exists a p ositive inte ger l such that gldim End Λ ( M i ) ≤ l + 2 , for i = 1 , 2 . Th e n the fol lowing ar e e quivalent: (a) Ext i F M 1 ( M 2 , M 2 ) = (0 ) and Ext i F M 2 ( M 1 , M 1 ) = (0 ) for 0 < i ≤ l , (b) ther e exists an exact se quenc e ( η ) : 0 → X 2 → N 0 f 0 − → N 1 → · · · → N m f m − → X 1 → 0 wher e e ach map Ker f j → M j is a minimal left add N -appr oximation , e ach map M j → Im f j is a m inimal right add N -appr oxima tion and ( η ) is in addition F M 1 -exact and F M 2 -exact. Pr o of. (a) ⇒ (b) Using Theorem 2.6 w e hav e that Hom Λ ( M 2 , M 1 ) is a cotilting End Λ ( M 1 )-mo dule. Als o w e kno w that Ho m Λ ( M 1 , M 1 ) is triv- ially a cotilting End Λ ( M 1 )-mo dule. So the End Λ ( M 1 )-mo dules Hom Λ ( X 2 , M 1 ) and Hom Λ ( X 1 , M 1 ) are complemen ts o f the almost complete cotilting End Λ ( M 1 )-mo dule Hom Λ ( N , M 1 ). Then, b y [7], there exists an exact sequence 0 → Hom Λ ( X 1 , M 1 ) f ⋆ m − → Hom Λ ( N m , M 1 ) → · · · f ⋆ 1 − → Hom Λ ( N 1 , M 1 ) f ⋆ 0 − → Hom Λ ( N 0 , M 1 ) → Hom Λ ( X 2 , M 1 ) → 0 RELA TIVE HOMOLO GY AN D MAXIMA L l -O R THOGONAL MODULES 15 where eac h map Im f ⋆ j → Hom Λ ( N j , M 1 ) is a minimal left add Hom Λ ( N , M 1 )- appro ximation and eac h Hom Λ ( N j , M 1 ) → Cok er f ⋆ j is a minimal right add Hom Λ ( N , M 1 )-approx imation. Since M 1 is a cog enerator for mo d Λ, w e hav e that fo r all j , f ⋆ j = Hom Λ ( f j , M 1 ) for some f j : M j → M j + 1 and the seque nce ( η ) : 0 → X 2 → N 0 f 0 − → N 1 f 1 − → · · · → N m f m − → X 1 → 0 is F M 1 -exact. Moreo v er eac h map Ker f j → M j is a minimal left add N - appro ximation, eac h map M j → Im f j is a minimal r ig h t add N - appro ximation. It r emains to sho w t hat ( η ) is in also F M 2 -exact. T o do this, w e apply the functor Hom Λ ( M 2 , − ) to ( η ) and w e get the complex 0 → Hom Λ ( M 2 , X 2 ) → Hom Λ ( M 2 , N 0 ) → Hom Λ ( M 2 , N 1 ) → · · · → Hom Λ ( M 2 , N m ) → Hom Λ ( M 2 , X 1 ) → 0 . But ( η ) can b e view ed as an F M 1 -injectiv e resolution of X 2 and then the j -th- homology of the ab ov e complex is Ext j F M 1 ( M 2 , X 2 ) whic h is, b y assu mption, zero. So the complex is acyclic whic h implies that ( η ) is F M 2 -exact. (b) ⇒ (a) Assume that there exists a sequence ( η ) as in (b). T hen ( η ) can b e view ed b oth as an F M 1 -injectiv e resolution of X 2 and as an F M 2 - pro jectiv e resolution of X 1 . If w e apply the functor Hom Λ ( M 2 , − ) to ( η ), the resulting complex will b e acyclic since ( η ) is F M 2 -exact, hence Ext i F M 1 ( M 2 , X 2 ) = (0), for i > 0, whic h implies that Ext i Λ ( M 2 , M 2 ) = (0), for i > 0 , since N is F M 1 -injectiv e. Similarly , if w e apply the functor Hom Λ ( − , M 1 ) to ( η ), the resulting complex will b e a cyclic since ( η ) is F M 1 -exact, hence Ext i F M 2 ( X 1 , M 1 ) = (0), fo r i > 0 , whic h implies that Ext i Λ ( M 2 , M 2 ) = (0) , for i > 0, sinc e N is F M 2 -injectiv e.  Note that the in teger m that app ears in the sequence ( η ) of the a b o v e prop osition can b e at most l − 1 , by Prop osition 1 .1. Reference s [1] Au slander, M.; Solb erg, Ø.; Relative homology and represen tation theory I, R elative ho molo gy and h omolo gic al ly finite sub c ate gories , Comm. Alg., 21 (9), 299 5-303 1 (1993). [2] Au slander, M.; Solb erg, Ø .; Relative homology and repres e n tation theo ry I I, R elative c otilting t he ory , Comm. Alg.,21 (9), 3033 -3079 (19 93). [3] Au slander, M.; Solb erg, Ø.; Relative homology and represen tation theory II I, Cotilting mo dules and We dderburn c orr esp ondenc e , Comm. in Alg., 21(9), 3081- 3097. 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Lecture Note Ser., v ol. 119, Ca m- bridge Univ. Press, Cambridge, 1988. [10] Iyama, O; Higher dimensional Auslander -Reiten theory on maximal orthogona l sub c ategories, Adv ances in Mathematics 210 (20 07), 22 -50. [11] Iyama, O; Auslander corres po ndence, Adv ances in Mathematics 210(20 07)51- 82. [12] So lber g, Ø; Infinite dimensio nal tilting mo dules o v er finite dimensional alge- bras, Handbo o k of Tilting, L o ndon Mathematical So ciety Lecture Note Series (No. 332 ), Cambridge Univ. Pr ess, 2007 . [13] V an den Ber gh, M.; Non-commutativ e cr epant resolutions ; The leg acy of Niels Henrik Ab el, Springer, Berlin, 2 004, pp. 749-7 70. Mag dalini Lad a, Institutt for ma tema tiske f ag, N TNU, N–7491 Trond- heim, Nor w a y E-mail addr ess : magda lin@ma th.ntnu.no