TILTING THEORY AND FUNCTOR CATEGORIES III. THE MAPS CATEGORY. 1 1 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES 0 2 Abstract. In this paper we continue the project of generalizing tilting the- n oryto the category of contravariant functors Mod(C), from askeletallysmall a preadditive category C tothe category of abeliangroups, initiated in[17]. In J [18] we introduced the notion of a a generalized tilting category T, and ex- 1 tendedHappel’stheoremtoMod(C). Weprovedthatthereisanequivalenceof 2 triangulated categories Db(Mod(C))∼=Db(Mod(T)). Inthecaseofdualizing varieties,weprovedaversionofHappel’stheoremforthecategoriesoffinitely ] presented functors. We alsoproved inthis paper, that there exists arelation T between covariantlyfinitecoresolvingcategories, andgeneralizedtiltingcate- R gories. Extendingtheoremsforartinalgebrasprovedin[4],[5]. Inthisarticle weconsiderthecategoryofmaps,andrelatetiltingcategoriesinthecategory . h offunctors,withrelativetiltinginthecategoryofmaps. Ofspecialinterestis t thecategorymod(modΛ)withΛanartinalgebra. a m [ 1 1. Introduction and basic results v 1 4 This is the last article in a series of three in which, having in mind applications 2 to the categoryof functors from subcategoriesof modules overa finite dimensional 4 algebra to the category of abelian groups, we generalize tilting theory, from rings . 1 to functor categories. 0 In the first paper [17] we generalized classical tilting to the category of con- 1 travariant functors from a preadditive skeletally small category C, to the category 1 of abelian groups and generalized Bongartz’s proof [10] of Brenner-Butler’s the- : v orem [11]. We then applied the theory so far developed, to the study of locally i X finite infinite quivers with no relations, and computed the Auslander-Reiten com- ponents of infinite Dynkin diagrams. Finally, we applied our results to calculate r a the Auslander-Reitencomponents ofthe categoryofKoszulfunctors (see[19],[20], [21]) on a regular component of a finite dimensional algebra over a field. These results generalize the theorems on the preprojective algebra obtained in [15]. Following [12], in [18] we generalized the proof of Happel’s theorem given by Cline,ParshallandScott: givenageneralizedtiltingsubcategoryT ofMod(C),the derived categories of bounded complexes Db(Mod(C)) and Db(Mod(T)) are equiv- alent, and we discussed a partial converse [14]. We also saw that for a dualizing Date:January21,2010. 1991 Mathematics Subject Classification. 2000]Primary 05C38, 15A15; Secondary 05A15, 15A18. Key words and phrases. ClassicalTilting,FunctorCategories. The second author thanks CONACYT for giving him financial support during his graduate studies. Thispaperisinfinalformandnoversionofitwillbesubmittedforpublicationelsewhere. 1 2 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES variety C and a tilting subcategory T ⊂ mod(C) with pseudokerneles, the cate- gories of finitely presented functors mod(C) and mod(T) have equivalent derived bounded categories, Db(mod(C)) ∼= Db(mod(T)). Following closely the results for artinalgebrasobtained in [3], [4], [5], by Auslander, Buchweits and Reiten, we end thepaperprovingthatforaKrull-SchmidtdualizingvarietyC,thereareanalogous relations between covariantly finite subcategories and generalized tilting subcate- gories of mod(C). This paper is dedicated to study tilting subcategories of mod(C). In order to have a better understanding of these categories, we use the relation between the categoriesmod(C)andthecategoryofmaps,maps(C),givenbyAuslanderin[1]. Of specialinterestisthecasewhenCisthecategoryoffinitelygeneratedleftΛ-modules overanartinalgebraΛ,sinceinthiscasethe categorymaps(C)is equivalenttothe categoryoffinitelygeneratedΓmodules,mod(Γ),overtheartinalgebraoftriangular Λ 0 matrices Γ = . In this situation, tilting subcategories on mod(mod(Λ)) (cid:18) Λ Λ (cid:19) will correspond to relative tilting subcategories of mod(Γ), which in principle, are easier to compute. The paper consists of three sections: In the first section we establish the notation and recall some basic concepts. In the second one, for a variety of annuli with pseudokerneles C, we prove that generalizedtilting subcategoriesofmod(C)areincorrespondencewithrelativetilt- ing subcategories of maps(C) [9]. In the third section, we explore the connections Λ 0 between mod Γ, with Γ = and the category mod(mod(Λ)). We com- (cid:18) Λ Λ (cid:19) pare the Auslander-Reiten sequences in mod(Γ) with Auslander-Reiten sequences in mod(mod(Λ)). We end the paper proving that, some important subcategories of mod(C) related with tilting, like: contravariantly, covariantly, functorially finite [see 18], correspond to subcategories of maps(C) with similar properties. 1.1. Functor Categories. In this subsection we will denote by C an arbitrary skeletallysmallpreadditivecategory,andMod(C)willbethecategoryofcontravari- antfunctors from C to the categoryof abelian groups. The subcategoryof Mod(C) consistingofallfinitelygeneratedprojectiveobjects,p(C),isaskeletallysmalladdi- tivecategoryinwhichidempotentssplit,thefunctorP :C →p(C),P(C)=C(−,C), is fully faithful and induces by restriction res : Mod(p(C)) → Mod(C), an equiva- lenceofcategories. Forthisreason,wemayassumethatourcategoriesareskeletally small, additive categories,such that idempotents split. Such categorieswere called annuli varieties in [2], for short, varieties. To fix the notation, we recall known results on functors and categories that we use through the paper, referring for the proofs to the papers by Auslander and Reiten [1], [4], [5]. Given a category C we will write for short, C(−,?) instead of HomC(−,?) and when it is clear from the context we use just (−,?). Definition 1.1. Given a variety C, we say C has pseudokernels; if given a map f :C →C , thereexistsamapg :C →C suchthatthesequenceof representable 1 0 2 1 (−,g) (−,f) functors C(−,C )−−−→C(−,C )−−−→C(−,C ) is exact. 2 1 0 A functor M is finitely presented; if there exists an exact sequence C(−,C )→C(−,C )→M →0 1 0 THE MAPS CATEGORY 3 We denote by mod(C) the full subcategory of Mod(C) consisting of finitely pre- sented functors. It was proved in [1] mod(C) is abelian, if and only if, C has pseudokernels. 1.2. Krull-Schmidt Categories. We start giving some definitions from [6]. Definition 1.2. Let R be a commutative artin ring. An R-variety C, is a va- riety such that C(C ,C ) is an R-module, and composition is R-bilinear. Under 1 2 these conditions Mod(C) is an R-variety, which we identify with the category of contravariant functors (Cop,Mod(R)). An R-variety C is Hom-finite, if for each pair of objects C ,C in C, the R- 1 2 module C(C ,C ) is finitely generated. We denote by (Cop,mod(R)), the full sub- 1 2 category of (Cop,Mod(R)) consisting of the C-modules such that; for every C in C the R-module M(C) is finitely generated. The category (Cop,mod(R)) is abelian and the inclusion (Cop,mod(R))→(Cop,Mod(R)) is exact. The category mod(C) is a full subcategory of (Cop,mod(R)). The functors D : (Cop,mod(R)) → (C,mod(R)), and D : (C,mod(R)) → (Cop,mod(R)), are defined asfollows: foranyC inC,D(M)(C)=Hom (M(C),I(R/r)), withr theJacobson R radical of R, and I(R/r) is the injective envelope of R/r. The functor D defines a duality between (C,mod(R)) and (Cop,mod(R)). If C is an Hom-finite R-category andM isinmod(C),thenM(C)isafinitelygeneratedR-moduleanditistherefore in mod(R). Definition 1.3. An Hom-finite R-variety C is dualizing, if the functor D :(Cop,mod(R))→(C,mod(R)) induces a duality between the categories mod(C) and mod(Cop). ItisclearfromthedefinitionthatfordualizingcategoriesC thecategorymod(C) has enough injectives. To finish, we recall the following definition: Definition 1.4. An additive category C is Krull-Schmidt, if every object in C decomposes in a finite sum of objects whose endomorphism ring is local. In [18 Theo. 2] we see that for a dualizing Krull-Schmidt variety the finitely presented functors have projective covers. Theorem 1.5. Let C a dualizing Krull-Schmidt R-variety. Then mod(C) is a dualizing Krull-Schmidt variety. 1.3. Contravariantly finite categories. [4]Let X be a subcategoryofmod(C), which is closed under summands and isomorphisms. A morphism f : X → M in mod(C), with X in X, is a right X-approximation of M, if (−,X)X −(−−−,h−)−X→ (−,M)X →0is anexactsequence,where (−,?)X denotes the restrictionof(−,?) to the category X. Dually, a morphism g : M → X, with X in X, is a left X-approximation of M, if (X,−)X −(−g−,−−)−X→(M,−)X →0 is exact. A subcategory X of mod(C) is called contravariantly (covariantly) finite in mod(C), if every object M in mod(C) has a right (left) X-approximation; and functorially finite, if it is both contravariantly and covariantly finite. AsubcategoryX ofmod(C)isresolving (coresolving), ifitsatisfiesthefollowing three conditions: (a) it is closed under extensions, (b) it is closed under kernels of 4 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES epimorphisms (cokernels of monomorphisms), and (c) it contains the projective (injective) objects. 1.4. Relative Homological Algebra and Frobenius Categories. In this sub- section we recall some results on relative homological algebra introduced by Aus- lander and Solberg in [9],[see also 14, 23]. Let C be an additive category which is embedded as a full subcategory of an abelian category A, and suppose that C is closed under extensions in A. Let S be a collection of exact sequences in A f g 0→X −→Y −→Z →0 f iscalledanadmissible monomorphism,andgiscalledanadmissible epimorphism. Apair(C,S)iscalledanexact category providedthat: (a)Anysplitexactsequence whose terms are in C is in S. (b) The composition of admissible monomorphisms (resp., epimorphisms) is an admissible monomorphism (resp., epimorphism). (c) It is closed under pullbacks (pushouts) of admissible epimorphisms (admissible monomorphisms). Let(C,S)beanexactsubcategoryofanabeliancategoryA. Sincethecollection S is closed under pushouts, pullbacks and Baer sums, it gives rise to a subfunctor F of the additive bifunctor Ext1(−,−) : C ×Cop → Ab [9]. Given such a functor C F, we say that an exact sequence η :0→A→B →C →0 in C is F-exact, if η is inF(C,A), wewillwritesometimesExt1(−,?)insteadofF(−,?). AnobjectP in F C is F-projective,if foreachF-exactsequence 0→A→B →C →0,the sequence 0 → (P,N) → (P,E) → (P,M) → 0 is exact. Analogously we have the definition of an F-injective object. If for any object C in C there is an F-exact sequence 0 → A → P → C → 0, with P an F-projective, then we say (C,S) has enough F- projectives. Dually, if for any object C in C there is an F-exact sequence 0 → C → I → A → 0, with I an F-injective, then (C,S) has enough F− injectives. An exact category (C,S) is called Frobenius, if the category (C,S) has enough F-projectives and enough F-injectives and they coincide. Let F be a subfunctor of Ext1(−,−). Suppose F has enough projectives. Then C for any C in C there is an exact sequence in C of the form ···Pn −d→n Pn−1 −d−n−−→1 ···→P1 −d→1 P0 −d→0 C →0 where P is F-projective for i ≥ 0 and 0 → Imd → P → Imd → 0 is F-exact i i+1 i i for all i≥0. Such sequence is called an F-exact projective resolution. Analogously we have the definition of an F-exact injective resolution. When (C,S) has enough F-injectives (enough F- projectives), using F-exact injectiveresolutions(respectively,F-exactprojectiveresolutions),wecanprovethat foranyobjectC inC,(A inC ), there existsa rightderivedfunctor ofHomC(C,−) ( HomC(−,A) ). We denote by ExtiF(C,−) the right derived functors of HomC(C,−) and by ExtiF(−,A) the right derived functors of HomC(−,A). 2. The maps category, maps(C) In this section C is an annuli variety with pseudokerneles. We will study tilting subcategories of mod(C) via the equivalence of categories between the maps cate- gory, module the homotopy relation, and the category of functors, mod(C), given THE MAPS CATEGORY 5 by Auslander in [1]. We will provide maps(C) with a structure of exact category such that, tilting subcategories of mod(C) will correspond to relative tilting sub- categoriesofmaps(C). Webeginthe sectionrecallingconceptsandresultsfrom[1], [14] and [23]. Theobjectsinmaps(C)aremorphisms(f ,A ,A ):A −f→1 A ,andthemapsare 1 1 0 1 0 pairs(h ,h ):(f ,A ,A )→(g ,B ,B ),suchthatthefollowingsquarecommutes 1 0 1 1 0 1 1 0 A f1// A 1 0 h1 (cid:15)(cid:15) h0 (cid:15)(cid:15) B g1// B 1 0 Wesaythattwomaps(h ,h ),(h′,h′):(f ,A ,A )→(g ,B ,B )arehomotopic, 1 0 1 0 1 1 0 1 1 0 if there exist a morphisms s : A → B such that h − h′ = g s. Denote by 0 1 0 0 1 maps(C) the category of maps modulo the homotopy relation. It was proved in [1] that the categories maps(C) and mod(C) are equivalent. The equivalence is given by a functor Φ:maps(C)→mod(C) induced by the functor Φ:map(C)→mod(C) given by Φ(A −f→1 A )=Coker((−,A )−(−−−,f−1→) (−,A )). 1 0 1 0 The category maps(C) is not in general an exact category, we will use instead the exact category P0(A) of projective resolutions, which module the homotopy relation, is equivalent to maps(C). SinceweareassumingC haspseudokerneles,thecategoryA=mod(C)isabelian. We can consider the categories of complexes C(A), and its subcategory C−(A), of bounded above complexes, both are abelian. Moreover, if we consider the class of j π exact sequences S: 0 → L. −→ M. −→ N. → 0, such that, for every k, the exact sequences 0→L −j→k M −π→k N →0 split, then (S,C(A)), (S,C−(A)) are exact k k k categories with enough projectives, in fact they are both Frobenius. In the first case the projective are summands of complexes of the form: 0 1 0 1     0 0 0 0 ···Bk+2 Bk+1 −−−−−→ Bk+1 Bk −−−−−→ Bk Bk−1··· a a a In the second case of the form: 0 1 0 1     0 0 0 0 0 1 ···B B −−−−−→ B B −−−−−→ B B −h−−−−→i B →0 k+3 k+2 k+2 k+1 k+1 k k a a a If we denote by C−(A) the stable category, it is well known [23], [14], that the homotopy category K−(A) and C−(A) are equivalent. Now, denote by P0(A) the full subcategory of C−(A) consisting of projective resolutions, this is, complexes of projectives P.: ···Pk →Pk−1 →···→P1 →P0 →0 such that Hi(P.)=0 for i6=0. Then we have the following: Proposition 2.1. The category P0(A) is closed under extensions and kernels of epimorphisms. 6 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES Proof. If 0 → P. → E. → Q. → 0 is an exact sequence in P0(A), then 0 → P → j E → Q → 0 is a splitting exact sequence in A with P , Q projectives, hence j j j j E is also projective. By the long homology sequence we have the exact sequence: j ···→Hi(P.)→Hi(E.)→Hi(Q.)→Hi−1(P.) →···, with Hi(P.)=Hi(Q.)=0, for i6=0. This implies E.∈P0(A). Now, let 0 → T. → Q. → P. → 0 be an exact sequence with Q., P. in P0(A). This implies that for each k, 0 → T → Q → P → 0 is an exact and splittable k k k sequence,hence eachT is projectiveand, bythe longhomologysequence,we have k the following exact sequence ···→H1(T.)→H1(Q.)→H1(P.)→H0(T.)→H0(Q.)→H0(P.)→0 with Hi+1(P.)=Hi(Q.)=0 for i≥1. This implies Hi(T.)=0, for i6=0. (cid:3) If SP0(A) denotes the collection of exact sequences with objects in P0(A), then (P0(A),SP0(A)) is an exact subcategory of (C−(A),S). The category P0(A) has enough projectives, they are the complexes of the form: 0 1 0 1     0 0 0 0 (2.1) ···→P P −−−−−→ P P −−−−−→ P P →0 3 2 2 1 1 0 a a a Denote by R0(A) the category P0(A) module the homotopy relation. This is: R0(A) is a full subcategory of C−(A) = K−(A). It is easy to check that R0(A) is the category with objects in P0(A) and maps the maps of complexes, module the maps that factor through a complex of the form: 0 1 0 1     0 0 0 0 0 1 ···→P P −−−−−→ P P −−−−−→ P P −h−−−−→i P →0 3 2 2 1 1 0 0 a a a We have the following: Proposition 2.2. There is a functor Ψ : P0(A) → maps(C) which induces an equivalence of categories Ψ :R0(A)→maps(C) given by: Ψ(P.)=Ψ(···→(−,A )−(−−−,f−2→) (−,A )−(−−−,f−1→) (−,A )→0)=A −f→1 A 2 1 0 1 0 Proof. Since C has pseudokerneles, any map A −f→1 A induces an exact sequence 1 0 (−,An)−(−−−,f−n→) (−,An−1)→···→(−,A2)−(−−−,f−2→) (−,A1)−(−−−,f−1→) (−,A0) and Ψ is clearly dense. Let (−,ϕ):P.→Q. be a map of complexes in P0(A): ··· // (−,A2)(−,f2// )(−,A1)(−,f1// )(−,A0) // 0 (2.2) (−,ϕ2) (cid:15)(cid:15) (−,ϕ1) (cid:15)(cid:15) (−,ϕ0) (cid:15)(cid:15) ··· // (−,B2)(−,g2// )(−,B1)(−,g1// )(−,B0) // 0 THE MAPS CATEGORY 7 (−,ϕ) If Ψ(P. −−−→Q.) is homotopic to zero, then we have a map s :A →A such 0 0 1 that g s =ϕ : 0 0 0 A f1// A 1 0 ϕ1 (cid:15)(cid:15) s(cid:127)(cid:127)(cid:127)0(cid:127)(cid:127)(cid:127)ϕ(cid:127)0 (cid:15)(cid:15) B g1// B 1 0 and s lifts to a homotopy s : P. → Q.. Conversely, any homotopy s : P. → Q. 0 induces an homotopy in maps(C). Then Ψ is faithful. If Ψ(P.) = (f ,A ,A ), Ψ(Q.) = (g ,B ,B ) and (h ,h ) : Ψ(P.) → Ψ(Q.) is a 1 1 0 1 1 0 0 1 map in maps(C), then (h ,h ) lifts to a map (−,h) = (−,h ) : P. → Q., and Ψ is 0 1 i full. (cid:3) Corollary 2.3. There is an equivalence of categories Θ : R0(A) → mod(C) given by Θ =ΦΨ, with Θ =ΦΨ. Proposition 2.4. Let P. be an object in P0(A), denote by rpdimP. the relative projective dimension of P. , and by pdimΘ(P.) the projective dimension of Θ(P.). Then we have rpdimP. = pdimΘ(P.). Moreover, if Ωi(P.) is the relative syzygy of P., then for all i≥0, we have Ωi(Θ(P.)) =Θ(Ωi(P.)). Proof. Let P. be the complex resolution 0→(-,An)−(−−−,f−n→) (-,An−1)→···→(-,A2)−(−−−,f−2→) (-,A1)−(−−−,f−1→) (-,A0)→0 and M =Coker(−,f ), then pdimM ≤n. 1 Now, consider the following commutative diagram 0 // An − 1 10 (cid:15)(cid:15) An (cid:0)0(cid:1) // An(cid:0) (cid:1) An−1 a A A f3 // A n 3 2 1 1 1 − 1 (cid:15)(cid:15) fn (cid:15)(cid:15) f3 01 (cid:15)(cid:15) f2 An (cid:0)0(cid:1) // An (cid:0)An−(cid:1)1 A3 (cid:0)A(cid:1)2 (cid:0)00(cid:1) // A2 (cid:0)A(cid:1)1 a a a (cid:15)(cid:15)(−fn1) (cid:15)(cid:15)(−f31) (cid:15)(cid:15)(f2−1) An fn // An−1 A2 f2 // A1 1 1 1 1 − 1 (cid:15)(cid:15) fn 01 (cid:15)(cid:15) fn−1 (cid:15)(cid:15) f2 01 (cid:15)(cid:15) f1 An (cid:0)0(cid:1) // An A(cid:0)n−(cid:1)1(cid:0)00//(cid:1)An−1 (cid:0)An−(cid:1)2 A2 (cid:0)A(cid:1)1 (cid:0)00(cid:1) // A1 (cid:0)A(cid:1)0 a a a a (cid:15)(cid:15)(−fn1) (cid:15)(cid:15)(fn−1−1) (cid:15)(cid:15)(f2−1) (cid:15)(cid:15)(−f11) An fn // An−1 fn−1 // An−2 A1 f1 // A0 SetQ =0→(−,A )→0,andforn−1≥i≥1considerthefollowingcomplex n n Q : i 0→(−,An)→(−,An) (−,An−1)→···→ a →(−,A ) (−,A )→(−,A ) (−,A )→0 i+2 i+1 i+1 i a a 8 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES Then we have a relative projective resolution 0→Qn →Qn−1 →···→Q1 →Q0 →P.→0 with relative syzygy the complex: Ωi(P.):0→(−,An)→(−,An−1)→(−,An−2)···(−,Ai+2)→(−,Ai+1)→0 for n−1≥i≥0. Therefore: we have an exact sequence 0→Θ(Ω(P.)) →Θ(Q )→Θ(P.)→0 0 inmod(C). Since Θ(Q )=(−,A ) andΘ(P.)=M, we haveΩ(Θ(P.))=Θ(Ω(P.)), i i and we can prove by induction that Ωi(Θ(P.)) = Θ(ΩiP.), for all i ≥ 0. It follows rpdimP.≥pdimΘ(P.). Conversely, applying Θ to a relative projective resolution 0→Qn →Qn−1 →···→Q1 →Q0 →P.→0, we obtain a projective resolution of Θ(P.) 0→Θ(Qn)→Θ(Qn−1)→···→Θ(Q1)→Θ(Q0)→Θ(P.)→0. It follows rpdimP.≤pdimΘ(P.). (cid:3) As a corollary we have: Corollary 2.5. Let C a dualizing Krull-Schmidt variety. If P. and Q are are complexes in P0(mod(C)) without summands of the form (2.1), then there is an isomorphism Extk (P.,Q.)=Extk (Θ(P.),Θ(Q.)) C−(mod(C)) mod(C) Proof. By Proposition 2.4, we see that Θ(ΩiP.) = Ωi(Θ(P.)), i ≥ 0. It is enough to prove the corollary for k = 1. Assume that (*) 0 → Q. −(−−−,j−i→) E. −(−−−,p−i→) P. → 0 is a exact sequence in Extk (P.,Q.), with Q. = ··· → (−,B ) −(−−−,g−2→) C−(mod(C) 2 (−,B ) −(−−−,g−1→) (−,B ) → 0, P. = ··· → (−,A ) −(−−−,f−2→) (−,A ) −(−−−,f−1→) (−,A ) → 1 0 2 1 0 0,E.=···→(−,E )−(−−−,h−2→) (−,E )−(−−−,h−1→) (−,E )→0. Sincetheexactsequence 2 1 0 0 → (−,B ) −(−−−,j−i→) (−,E ) −(−−−,p−i→) (−,A ) → 0 splits E = A B , i ≥ 0. Then i i i i i i we have an exact sequence in mod(C) ` ρ σ (2.3) 0→Θ(Q.)−→Θ(E.)−→Θ(P.)→0 If (2.3) splits, then there exist a map δ : Θ(E.) → Θ(Q.) such that δρ = 1 , Θ(Q.) We have a lifting of δ, (−,l ) : E. → Q. such that the following diagram is i i∈Z commutative ··· // (−,B1) (−,g1)// (−,B0) π // Θ(Q.) // 0 (−,l1ji) (−,l0j0) (cid:15)(cid:15) (cid:15)(cid:15) ··· // (−,B1) (−,g1)// (−,B0) π // Θ(Q.) // 0 The complex Q. has not summand of the form (2.1), hence, Q. is a minimal pro- jective resolution of Θ(Q.). THE MAPS CATEGORY 9 Since π : (−,B ) → Θ(Q.) is a projective cover, the map (−,l j ) : (−,B ) → 0 0 0 0 (−,B ) is an isomorphism, and it follows by induction that all maps (−,l j ) are 0 i i isomorphisms, which implies that the map {(−,ji)}i∈Z : Q. → E. is a splitting homomorphism of complexes. Given an exact sequence (**) 0 → G → H → F → 0, in mod(C), we take minimal projective resolutions P. and Q. of F and G, respectively, by the Horse- shoe’s Lemma, we have a projective resolution E. for H, with E = Q ⊕P , and i i i 0 → Θ(Q.) → Θ(E.) → Θ(P.) → 0 is a exact sequence in mod(C) isomorphic to (**). (cid:3) 2.1. Relative Tilting in maps(C). Let C a dualizing Krull-Schmidt variety. In order to define an exact structure on maps(C) we proceed as follows: we identify first C with the category p(C) of projective objects of A = mod(C), in this way maps(C)isequivalenttomaps(p(C))whichisembeddedintheabeliancategoryB = maps(A). We can define an exact structure (maps(C),S) giving a subfunctor Fof Ext1(−,?). Let Ψ :P0(A)→maps(C) be the functor givenabove andα: maps(C) B → maps(p(C)) the natural equivalence. Since Ψ is dense any object in maps(C) is of the form Ψ(P.) and we define Ext1(αΨ(P.) , αΨ(Q.)) as αΨ(Ext1 (P.,Q.)). F C−(A) We obtain the exact structure on maps(C) using the identification α. Once we have the exact structure on maps(C) the definition of a relative tilting subcategory TC of maps(C) is very natural, it will be equivalent to the following: Definition 2.6. A relative tilting category in the category of maps, maps(C), is a subcategory TC such that : (i) Given T :T1 →T0 in TC , and P.∈P0(C) such that Ψ(P.)=T, there exist an integer n such that rpdimP.≤n. (ii) Given T : T1 → T0, T′ : T1′ → T0′ in TC and Ψ(P.) = T, Ψ(Q.) = T′, P.,Q.∈P0(mod(C)). Then Extk (P.,Q.)=0 for all k≥1. C−(mod(C)) (iii) Given an object C in C, denote by (−,C)◦ the complex 0 → (−,C) → 0 concentrated in degree zero. Then there exists an exact sequence 0→(−,C)◦ →P0 →P1 →···→Pn →0 with Pi ∈P0(mod(C)) and Ψ(Pi)∈TC. By definition, the following is clear Theorem 2.7. Let Φ : maps(C) → mod(C) be functor above, TC is a relative tiltingsubcategoryofmaps(C)if andonlyif Φ(TC)is atiltingsubcategoryof mod(C) 3. The Algebra of Triangular Matrices Let Λ be an artin algebra. We want to explore the connections between mod Γ, Λ 0 with Γ= andthe categorymod(modΛ). In particularwe wantto com- (cid:18) Λ Λ (cid:19) pare the Auslander-Reiten quivers and subcategories which are tilting, contravari- antly, covariantly and functorially finite. We identify mod Γ with the category of Λ-maps, maps(Λ) [see 7 Prop. 2.2]. We refer to the book by Fossum, Griffits and Reiten [13] or to [16] for properties of modules over triangular matrix rings. 10 R.MART´INEZ-VILLAANDM.ORTIZ-MORALES 3.1. Almost Split Sequences. In this subsection we want to study the rela- tion between the almost split sequences in mod Γ and almost split sequences in mod(modΛ).We will see that except for a few specialobjects in mod Γ, the almost split sequences will belong to the class S of the exact structure, so in particular will be relative almost split sequences. ForanyindecomposablenonprojectiveΓ-moduleM =(M ,M ,f)wecancom- 1 2 pute DtrM as follows: To construct a minimal projective resolution of M ([13], [16]), let P −p→1 P → 1 0 M → 0 be a minimal projective presentation. Taking the cokernel, we have an 1 exact sequence M −→f M −f→2 M →0, and a commutative diagram 1 2 3 0 // P1 // P1⊕Q1 // Q1 // 0 p1 (cid:15)(cid:15) (cid:15)(cid:15) q1 (cid:15)(cid:15) 0 // P0 // P0⊕Q0 // Q0 // 0 p0 (cid:15)(cid:15) (cid:15)(cid:15) q0 (cid:15)(cid:15) M f // M f2 // M // 0 1 2 3 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 0 0 with Q the projective cover of M .The presentation can be written as: 0 3 P P M 1 → 0 → 1 →0 (cid:18) P1⊕Q1 (cid:19) (cid:18) P0⊕Q0 (cid:19) (cid:18) M2 (cid:19) and trM will look as as follows: P∗⊕Q∗ P∗⊕Q∗ M 0 0 → 1 1 →tr 1 →0 (cid:18) Q∗0 (cid:19) (cid:18) Q∗1 (cid:19) (cid:18) M2 (cid:19) which corresponds to the commutative exact diagram: 0 // Q∗ // Q∗⊕P∗ // P∗ // 0 0 0 0 0 q∗ (cid:15)(cid:15) (cid:15)(cid:15) p∗ (cid:15)(cid:15) 1 1 0 // Q∗ // Q∗⊕P∗ // P∗ // 0 1 1 1 1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) trM ⊕Q∗ // trM ⊕P∗ // trM // 0 3 2 1 (cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15) 0 0 0 with Q∗, P∗, projectives coming from the fact that the presentationsof M and 2 M in the first diagram are not necessary minimal. 3 Then τM is obtained as τ(M ,M ,f) = τM ⊕D(P∗) → τM ⊕D(Q∗), with 1 2 2 3 kernel 0→τM →τM ⊕D(P∗)→τM ⊕D(Q∗). 2 3 We consider now the special cases of indecomposable Γ-modules of the form: M −1−M→M,(M,0,0),(0,M,0),withM anonprojectiveindecomposableΛ-module. j π Proposition 3.1. Let 0 → τM −→ E −→ M → 0 be an almost split sequence of Λ-modules. (a) Then the exact sequences of Γ-modules: