Almost split sequences in tri-exact categories
aa r X i v : . [ m a t h . R T ] J un ALMOST SPLIT SEQUENCES IN TRI-EXACT CATEGORIES
SHIPING LIU AND HONGWEI NIU
Abstract.
We shall study the existence of almost split sequences in tri-exactcategories, that is, extension-closed subcategories of triangulated categories.Our results unify and extend the existence theorems for almost split sequencesin abelian categories and exact categories (that is, extension-closed subcate-gories of abelian categories), and those for almost split triangles in triangulatedcategories in [4, 17, 20, 24, 26, 29]. As applications, we shall obtain some newresults on the existence of almost split sequences in the derived categories ofall modules over an algebra with a unity or a locally finite dimensional algebragiven by a quiver with relations.
Introduction
Since its introduction in the last seventies; see [5, 6], the Auslander-Reiten the-ory of almost split sequences has been playing a fundamental role in the modernrepresentation theory of algebras; see, for example, [3, 7]. Later, Happel introducedthe analogous theory of almost split triangles in triangulated categories; see [15, 16],making the Auslander-Reiten theory applicable in other areas of mathematics suchas algebraic topology and algebraic geometry; see [28, 18, 19]. Since then, thistheory has been further developed separately for exact categories and triangulatedcategories; see [4, 17, 20, 22, 24, 26, 29]. Our purpose is to unify and extend theseresults by working with tri-exact categories. Observe that the existence of almostsplit sequences in a Krull-Schmidt category will help us to classify the indecom-posable objects and describe certain morphisms in terms of the Auslander-Reitenquiver; see [25]. We shall outline the content of the paper section by section.In Section 1, in addition to laying down the foundation, we shall also studymodules over an R -algebra, which are reflexive with respect to the minimal injectiveco-generator for Mod R , where R is a commutative ring. These modules will playthe same role as those of finite length over an artin algebra. In case the algebrais reflexive and noetherian, we shall establish a duality between the R -noetherianmodules and the R -artinian modules; see (1.4), which generalizes the well-knownMatlis duality; see [4, 13].In Section 2, we shall study mainly the stable categories of a tri-exact category.The stable categories were first considered by Auslander and Reiten for modulesover an artin algebra in order to establish the existence of almost split sequences;see [6]. Later, Lenzing and Zuazua defined the stable categories of an abeliancategory without projective or injection objects; see [24], which carry over easily Mathematics Subject Classification.
Key words and phrases.
Modules; algebras; almost split sequences; almost split triangles;abelian categories; derived categories; triangulated categories.The first named author is supported in part by the Natural Science and Engineering ResearchCouncil of Canada. to an exact category; see [26]. We shall extend them to tri-exact categories andshow that every exact category is equivalent to a tri-exact category with equivalentstable categories. This ensures that the study of almost split sequences in exactcategories and abelian categories is covered under our tri-exact setting. We shouldpoint out that a triangulated category coincides with its stable categories.In Section 3, we shall study the existence of an individual almost split sequencein a tri-exact category. Historically, one derives an almost split sequence froman Auslander-Reiten formula in an abelian category; see [4] and [24, (1.1)], andan almost split triangle from a Serre formula in a triangulated category; see [20,(2.2)] and [29, (I.2.3)], but the converses do not hold in general. These formulaeinvolve taking the “dual” of some stable Hom-spaces against injective modules overvarious rings; see [4, 7, 20, 24]. Recently, some necessary and sufficient conditionswere found for the existence of an almost split sequence in an exact R -category,where the “dual” is taken against an injective co-generator for Mod R ; see [26,(2.2)]. By taking the “dual” against injective modules over rings mapping to thestable endomorphisms of two prescribed objects, we shall obtain some necessaryand sufficient conditions for the existence of an almost split sequence in a tri-exactcategory; see (3.7), which essentially cover all the previously mentioned results.In Section 4, we shall be concerned with the global existence of almost split se-quences in a tri-exact category. It is known that an Ext-finite abelian R -categorywith R being artinian has almost split sequences if and only if it admits an Auslander-Reiten duality; see [7, 14, 24] and a Hom-finite triangulated category over a fieldhas almost split triangles on the right (or left) if and only if it admits a right (orleft) Serre functor; see [29, (I.2.3)]. We shall deal this problem for Hom-reflexiveKrull-Schmidt tri-exact categories. This class of categories includes the categoryof noetherian modules and that of artinian modules over a noetheiran R -algebrawith R being notherian complete local, which are not Hom-finite if the algebra isnot artinian; see [4]. We shall show that such a tri-exact R -category has almostsplit sequences on the right (or left) if and only if it admits a full right (or left)Auslander-Reiten functor; see (4.8). In the right (or left) triangulated case, the ex-istence of almost split sequences on the right (or left) is equivalent to the existenceof a right (or left) Auslander-Reiten functor, or equivalently, a right (or left) Serrefunctor with a proper image; see (4.10).In Section 5, we shall study the existence of almost split triangles in the de-rived categories of an abelian category with enough projective objects and enoughinjective objects. This has been done for the bounded derived category of finitedimensional modules over a finite dimensional algebra; see [16, 15]. In the mostgeneral case, we shall show that an almost split triangle in the bounded derived cate-gory starts with a bounded complex of injective objects and ends with a boundedcomplex of projective objects; see (5.2) and (5.3). In case the abelain categoryadmits a Nakayama functor with respect to a subcategory of projective objects;see (5.4), we shall establish an existence theorem of an almost split triangle in thebounded derived category; see (5.8). In case the subcategory of projective objectsis Hom-reflexive, we shall describe all possible almost split triangles in the boundedderived category; see (5.12). These results will be applicable to the derived cate-gories of module over an general algebra, a reflexive notherian algebra, or a locallyfinite dimensional algebra given by a quiver with relations. LMOST SPLIT SEQUENCES 3 Preliminaries
The main objective of this section is to fix the notation and the terminology, whichwill be used throughout this paper, and collect some preliminary results. However,we shall also obtain some new results on modules over an algebra. Throughout thispaper, morphisms in any category are composed from the right to the left.
1) Modules.
All rings and algebras except for those given by a quiver withrelations have an identity. Let Σ be a ring or an algebra. We shall denote byMod Σ the category of all left Σ -modules, and by mod Σ the full subcategory ofMod Σ of modules of finite length. For convenience, we shall identify the categoryof all right Σ -modules with Mod Σ op , where Σ op is the opposite ring or the oppositealgebra of Σ . A map f : M → N in Mod Σ is called socle essential provided thatIm( f ) ∩ Soc( N ) is non-zero whenever Soc( N ) is non-zero.Let M be a left or right Σ -module. Then M ∗ = Hom Σ ( M, Σ ) is a right or left Σ -module, respectively. Given u ∈ M , we have ˆ u ∈ M ∗∗ = Hom Σ ( M ∗ , Σ ), sending f ∈ M ∗ to f ( u ). The map ρ M : M → M ∗∗ , sending u to ˆ u, is clearly a natural Σ -linear map. It is well known; see, for example, [30, (3.15)] that M is finitelygenerated projective if and only if it has a finite projective basis { u i ; f i } ≤ i ≤ n ,where u i ∈ M and f i ∈ M ∗ , such that u = P ni =1 f i ( u ) u i (or u = P ni =1 u i f i ( u ))for all u ∈ M . We shall denote by proj Σ the full subcategory of Mod Σ of finitelygenerated projective modules. The following statement is probably well-known.1.1. Lemma.
Let Σ be a ring or an algebra. Then
Hom Σ ( − , Σ ) : proj Σ → proj Σ op is a duality.Proof. Let P ∈ proj Σ with a finite projective basis { u i ; f i } ≤ i ≤ n . Given f ∈ P ∗ and u ∈ P, we obtain( P ni =1 f i ˆ u i ( f ))( u ) = P ni =1 ( f i f ( u i )) ( u ) = P ni =1 f i ( u ) f ( u i ) = f ( P ni =1 f i ( u ) u i ) . Since u = P ni =1 f i ( u ) u i , we conclude that f = P ni =1 f i ˆ u i ( f ). That is, { f i ; ˆ u i } ≤ i ≤ n is a projective basis of P ∗ . In particular, P ∗ ∈ proj Σ op . If u ∈ P is non-zero, since u = P ni =1 f i ( u ) u i , we see that ˆ u is non-zero. That is, ρ P is a monomorphism. Asshown above, P ∗∗ has a projective basis { ˆ u i ; ˆ f i } ≤ i ≤ n . Given ϕ ∈ P ∗∗ , we obtain ϕ = P ni =1 ˆ f i ( ϕ )ˆ u i = P ni =1 ϕ ( f i )ˆ u i = ρ P ( P ni =1 ϕ ( f i ) u i ) . Thus, ρ P is an isomorphism. The proof of the lemma is completed.Throughout this paper, R will stand for a commutative ring and I R for a minimalinjective co-generator for Mod R ; see [1, (18.19)]. We shall use frequently the functor D = Hom R ( − , I R ) : Mod R → Mod R . The following statement is probably known.1.2. Proposition.
Let U be a module over a commutative ring R . (1) If U is of finite length n , then DU is also of length n . (2) If U is finitely co-generated, then DU is finitely generated. (3) If DU is artinian or noetherian, then U is noetherian or artinian respectively.Proof. Assume first that U is simple. In particular, U = Ru for some u ∈ U .Consider some non-zero linear functions f, g ∈ DU . Since I R is a minimal injectiveco-generator, soc( I R ) contains exactly one copy of U ; see [1, (18.19)]. Therefore, g ( U ) = f ( U ), and hence, g ( u ) = rf ( u ) for some r ∈ R . This yields g = rf . Thus, DU is also simple. By induction, we can establish Statement (1). SHIPING LIU AND HONGWEI NIU
Assume next that U is finitely co-generated, that is, U has an essential socle S = S ⊕ · · · ⊕ S t , where the S i are simple; see [1, (10.4)]. Consider the canonicalprojections p i : S → S i and the canonical injections q i : S i → S , and fix somemonomorphisms f i : S i → I R , for i = 1 , . . . , t . Letting q : S → U be the inclusion,we obtain R -linear maps g i : U → I R such that g i q = f i p i , for i = 1 , . . . , t . Givenany R -linear map g : U → I R , as seen above, gqq i = r i f i for some r i ∈ R . Thisyields gq = P ti =1 gqq i p i = P ti =1 r i f i p i = ( P ti =1 r i g i ) q. Since q is an essentialmonomorphism, g = P ti =1 r i g i . Statement (2) is established.Finally, given a submodule V of U , we denote by V ⊥ the submodule of DU of R -linear maps vanishing on V and by ⊥ ( V ⊥ ) the submodule of U of elementsannihilated by the R -linear maps in V ⊥ . Then, V ⊆ ⊥ ( V ⊥ ). We claim that V = ⊥ ( V ⊥ ). Otherwise, we can find an R -linear map h : U → I R such that h ( ⊥ ( V ⊥ )) = 0 but h ( V ) = 0, contrary to the definition. Using this claim, we mayeasily establish Statement (3). The proof of the proposition is completed.Let A be an R -algebra. A left or right A -module M is called R -noetherian or R -artinian if R M is noetherian or artinian; and A is called a noetherian or reflexive R -algebra if A A is R -noetherian or R -reflexive, respectively. Note that our definition ofa noetherian R -algebra is different from the classical one, where R is assumed to benoetherian. Consider the exact functors D = Hom R ( − , I R ) : Mod A → Mod A op and D = Hom R ( − , I R ) : Mod A op → Mod A . Given a left or right A -module M , we obtaina canonical A -linear monomorphism σ M : M → D M so that σ M ( x )( f ) = f ( x ), for x ∈ M and f ∈ DM . We shall say that M is R -reflexive if σ M is bijective.1.3. Lemma.
Let A be an R -algebra. The full subcategory RMod A of Mod A of R -reflexive modules is abelian, contains all modules of finite R -length, and admitsa duality D : RMod A → RMod A op .Proof. Considering the canonical monomorphisms and applying the Snake Lemma,we see that RMod A is closed under taking submodules and quotient modules, thatis, it is abelian. If M ∈ Mod A is of R -length n , by Proposition 1.2(1), so is D M ,and consequently, σ M : M → D M is an isomorphism. Finally, by the definitionof reflexive modules, D : RMod A → RMod A op and D : RMod A op → RMod A aremutual quasi-inverse. The proof of the lemma is completed.Consider now the endofunctors ν A = D Hom A ( − , A ) and ν - A = Hom A ( D ( − ) , A )of Mod A . Put inj A = ν A (proj A ) which, by Lemma 1.1, contains only injectivemodules. Let mod + A stand for the full subcategory of Mod A of finitely generatedmodules, and mod − A for that of modules finitely co-generated by inj A .1.4. Theorem.
Let A be a reflexive noetherian R -algebra. (1) The functors ν A : proj A → inj A and ν - A : inj A → proj A are mutual quasi-inverses, where proj A and inj A have as objects all R -noetherian projectivemodules and all R -artinian injective modules, respectively. (2) There exists a duality D = Hom R ( − , I R ) : mod + A op → mod − A , where mod + A and mod − A are abelian subcategories of RMod A , whose objects are all R -noetherian modules and all R -artinian modules, respectively.Proof. Since A A is R -reflexive and R -noetherian, we deduce from Lemma 1.3 thatmod + A is an abelian subcategory of RMod A , whose objects are clearly the R -noetherian A -modules. In particular, the objects of proj A are the R -noetherianprojective A -modules. On the other hand, since A op is also a reflexive noetherian LMOST SPLIT SEQUENCES 5 R -algebra, mod + A op is an abelian subcategory of RMod A op . Consider the equiva-lence Hom A ( − , A ) : proj A → proj A op and the duality D : RMod A op → RMod A in Lemmas 1.1 and 1.3, we see that inj A is a subcategory of RMod A , whereas thefunctors ν A : proj A → inj A and ν - A : inj A → proj A are mutual quasi-inverses.Being finitely co-generated by inj A , by Lemma 1.3, mod − A is a subcategory ofRMod A . Given M ∈ RMod A op , in view of the duality D : RMod A op → RMod A ,we see that M ∈ mod − A if and only if DM ∈ mod + A op . Thus, we obtain aduality D : mod + A op → mod − A . In particular, mod − A is abelian. Since mod + A op contains only R -noetherian modules, by Proposition 1.2(3), mod − A contains only R -artinian modules. On the other hand, if M ∈ Mod A is R -artinian, then R M is finitely co-generated, and by Proposition 1.2(2), DM is finitely generated over R . In particular, DM ∈ mod + A op , and hence, D M ∈ mod − A . Since mod − A isabelian and σ M : M → D M is a monomorphism, M ∈ mod − A . Finally, let I bean R -artinian injective A -module. Then, I is an injective object in mod − A . Hence, I ∼ = DP , where P is a projective object in mod + A op . It is then easy to see that P ∈ proj A op , that is, I ∈ inj A. The proof of the theorem is completed.
Remark.
A noetherian algebra over a commutative noetherian complete local ringis reflexive; see [4, Section 5]. Thus, Theorem 1.4 generalizes the well-known Matlisduality; see [4, Section 5] and [13, (3.2.13)].We conclude this subsection with algebras given by a quiver with relations. Let Q be a locally finite quiver with vertex set Q . An infinite path in Q is called left infinite if it has no starting point and right infinite if it has no ending point.Given a field k , an ideal J in the path algebra kQ is called weakly admissible if itlies in the ideal generated by the paths of length two; and locally admissible if, forany x ∈ Q , there exists n x ∈ Z for which R contains all paths of length ≥ n x ,starting or ending with x . Consider Λ = kQ/J , where J is weakly admissible, witha complete set of pairwise orthogonal primitive idempotents { e x | x ∈ Q } . Onecalls Λ locally finite dimensional if e x Λ e y is finite dimensional for all x, y ∈ Q ;and strongly locally finite dimensional if J is locally admissible; see [12, Section1(4)]. Assume that Λ is locally finite dimensional. We shall denote by Mod Λ thecategory of all left Λ -modules M such that M = ⊕ x ∈ Q e x M , and by mod b Λ the fullsubcategory of Mod Λ of finite dimensional modules. Given x ∈ Q , one obtainsa projective module P x = Λ e x and an injective module I x = D ( Λ op e x ); see [12,Section 3]. Let proj Λ and inj Λ be the strictly additive subcategories of Mod Λ generated by the P x with x ∈ Q and by the I x with x ∈ Q , respectively. Observethat Λ is strongly locally finite dimensional if and only if all P x and I x with x ∈ Q are finite dimensional.
2) Additive categories.
Throughout this paper, all functors between additivecategories are additive. Let A be an additive category. A strictly additive sub-category of A is a full subcategory which is closed under finite direct sums, directsummands and isomorphisms. An object in A is called strongly indecomposable if ithas a local endomorphism ring. One says that A is Krull-Schmidt if every non-zeroobject is a finite direct sum of strongly indecomposable objects; and in this case, A / I is Krull-Schmidt, for every ideal I in A .A morphism f : X → Y in A is called left minimal if every morphism h : Y → Y such that f = hf is an automorphism; left almost split if f is not a section such thatevery non-section morphism g : X → M factors through it; and minimal left almost SHIPING LIU AND HONGWEI NIU split if it is left minimal and left almost split. In the dual situations, one says that f is right minimal , right almost split , and minimal right almost split , respectively.
3) Stable categories of exact categories.
Let C be an exact category, thatis an extension-closed subcategory of an abelian category A ; see [26, Section 2].Given X, Y ∈ C , one writes Ext C ( X, Y ) = Ext A ( X, Y ). A morphism f : X → Y in C is called injectively trivial in C if the push-out mapExt C ( Z, f ) : Ext C ( Z, X ) → Ext C ( Z, Y ) : δ f · δ vanishes for all Z ∈ C ; and projectively trivial in C if the pull-up mapExt C ( f, Z ) : Ext C ( Y, Z ) → Ext C ( X, Z ) : ζ ζ · f vanishes for all Z ∈ C . Denoting by I C the ideal of injectively trivial morphismsand by P C that of projectively trivial morphisms, one obtains the injectively stablecategory C = C / I C and the projectively stable category C = C / P C of C ; see[26, 24]. In the sequel, we shall write as usual Hom C ( X, Y ) = Hom C ( X, Y ) andHom C ( X, Y ) = Hom C ( X, Y ), for all
X, Y ∈ C .
4) Triangulated categories.
Let A be a triangulated category, whose trans-lation functor will always be written as [1] . An exact triangle X f / / Y g / / Z δ / / X [1]in A is called almost split if f is minimal left almost split and g is minimal rightalmost split; see [16, (4.1)]. In this case, X and Z will be called the starting term and the ending term , respectively.1.5. Definition.
A full subcategory C of a triangulated category A is called exten-sion-closed provided, for any exact triangle X / / Y / / Z / / X [1] in A , that Y ∈ C whenever X, Z ∈ C . In this case, we shall simply call C a tri-exact category. Observe that an extension-closed subcategory of a triangulated category is strictlyadditive, but it is not necessarily closed under the translation functor [1].1.6.
Definition.
An extension-closed subcategory C of a triangulated category A will be called(1) left triangulated provided, for any exact triangle X / / Y / / Z / / X [1] in A , that X ∈ C whenever Y, Z ∈ C .(2) right triangulated provided, for any exact triangle X / / Y / / Z / / X [1]in A , that Z ∈ C whenever X, Y ∈ C .Let C be a full subcategory of A . For n ∈ Z , we denote by C [ n ] the full subcate-gory of A generated by the objects X [ n ] with X ∈ C . Clearly, X ∈ C [ n ] if and onlyif X [ − n ] ∈ C . Turning exact triangles in A , we obtain the following observation.1.7. Lemma.
An extension-closed subcategory C of a triangulated category is lefttriangulated if and only if C [ − ⊆ C ; and right triangulated if and only if C [1] ⊆ C . Observe that a left or right triangulated subcategory of a triangulated categoryis a left or right triangulated category as defined in [2, (1.1)] and will be simplycalled a left triangulated category with a left translation [ −
1] or a right triangulatedcategory with a right translation [1], respectively.
LMOST SPLIT SEQUENCES 7
5) Derived categories.
Let A be an additive category. We shall denote by C ( A ) the complex category of A . The full subcategories of C ( A ) of bounded-abovecomplexes, of bounded-below complexes, and of bounded complexes will be writtenas C − ( A ), C − ( A ) and C b ( A ), respectively. Given ∗ ∈ { + , − , b, ∅} , let K ∗ ( A )stand for the homotopy category , that is the quotient of C ∗ ( A ) modulo the null-homotopic morphisms, which is triangulated with a canonical projection functor P ∗ : C ∗ ( A ) → K ∗ ( A ); and D ∗ ( A ) for the derived category , that is the localizationof K ∗ ( A ) with respect to the quasi-isomorphisms, which is also triangulated witha canonical localization functor L ∗ : K ∗ ( A ) → D ∗ ( A ). Given a morphism f . in C ∗ ( A ) , we shall write ¯ f . = P ∗ ( f . ) ∈ K ∗ ( A ) and ˜ f . = L ∗ ( ¯ f . ) ∈ D ∗ ( A ) . Givena complex M . ∈ C b ( A ) , its width w ( M . ) is an integer defined by w ( M . ) = 0 if M i = 0 for all i ∈ Z ; and otherwise, w ( M . ) = t − s + 1, where s ≤ t such that M s and M t are non-zero, but M i = 0 for all i / ∈ [ s, t ]. Furthermore, given a complex( X . , d . ) over A and some integer n , one defines two brutal truncations κ ≥ n ( X . ) : · · · / / / / X n d n / / X n +1 d n +1 / / X n +2 / / · · · and κ ≤ n ( X . ) : · · · / / X n − d n − / / X n − d n − / / X n / / / / · · · , where X n is the component of degree n in both complexes, with two canonicalmorphisms µ . n : κ ≥ n ( X . ) → X . such that µ pn = 1 Xp for p ≥ n and µ pn = 0 for p < n ;and π . n : X . → κ ≤ n ( X . ) such that π pn = 1 Xp for p ≤ n and π pn = 0 for p > n. Thefollowing statement is well-known; see [27, (III.4.4.2)] and [17, (1.3)].1.8.
Lemma.
Let A be an additive category. If X . ∈ C ( A ) and n ∈ Z , then K ( A ) has an exact triangle κ ≥ n ( X . ) / / X . / / κ ≤ n − ( X . ) / / κ ≥ n ( X . )[1] . Consider now the derived category of an abelian category A . Fix an integer n . We shall denote by D ≤ n ( A ) and D ≥ n ( A ) the full subcategories of D ( A ) ofcomplexes M . with H i ( M . ) = 0 for all i > n and of complexes M . with H i ( M . ) = 0for all i < n respectively, where H i ( M . ) is the i -th cohomology of M . . By Lemma1.6, D ≤ n ( A ) is right triangulated and D ≥ n ( A ) is left triangulated in D ( A ). Let( X . , d . ) be a complex over A . Writing d n = q n p n , where p n : X n → C n is thecokernel of d n − , and d n − = i n j n − , where i n : K n → X n is the kernel of d n , weobtain two smart truncations τ ≥ n ( X . ) : · · · / / / / C n q n / / X n +1 d n +1 / / X n +2 / / · · · , where C n is of degree n , with a canonical projection p . n : X . → τ ≥ n ( X . ) so that p nn = p n and p sn = 1 X s for all s > n ; and τ ≤ n ( X . ) : · · · / / X n − d n − / / X n − j n − / / K n / / / / · · · where K n is of degree n , with a canonical injection i . n : τ ≤ n ( X . ) → X . so that i nn = i n and i tn = 1 X t for all t < n. Lemma.
Let B be an abelian subcategory of an abelian category A . Consider X . ∈ C ∗ ( A ) with ∗ ∈ {∅ , − , + , b } and Y . ∈ C ( B ) . If X . ∼ = Y . in D ( A ) , then thereexists Z . ∈ C ∗ ( B ) such that X . ∼ = Z . in D ∗ ( A ) .Proof. We shall only consider the case where X . ∈ C b ( A ) and Y . ∈ C ( B ) such that X . ∼ = Y . in D ( A ). Let s, t with s ≤ t be such that X p = 0 for p [ s, t ]. Then,H p ( Y . ) = 0 for all p [ s, t ], and hence, the canonical injection i . t : τ ≤ t ( Y . ) → Y . and the canonical projection p . s : τ ≤ t ( Y . ) → τ ≥ s ( τ ≤ t ( Y . )) are quasi-isomorphisms; SHIPING LIU AND HONGWEI NIU see [27, (III.3.4.1), (III.3.4.2)]. As a consequence, τ ≥ s ( τ ≤ t ( Y . )) ∈ C b ( B ) such that X . ∼ = τ ≥ s ( τ ≤ t ( Y . )) in D b ( A ). The proof of the lemma is completed.In the same fashion, one can show that D ∗ ( A ) with ∗ ∈ {− , + , b } fully embedsin D ( A ); see [27, (III.3.4.3), (III.3.4.4), (III.3.4.5)]. In the sequel, we shall alwaysregard D ∗ ( A ) as a full triangulated subcategory of D ( A ).2. Tri-exact structure and stable categories
The objective of this section is to study the tri-exact structure in a non-axiomaticfashion and introduce the stable categories of a tri-exact category. They are analo-gous to the exact structure and the stable categories of an exact category as de-scribed in [24, 26]. More importantly, we shall show that every exact category isequivalent to a tri-exact category with equivalent stable categories.Throughout this section C will denote a tri-exact category, say an extension-closed subcategory of a triangulated category A . Given X, Y ∈ C , we shall writeExt C ( X, Y ) = Hom A ( X, Y [1]), whose elements will be called extensions of Y by X . Given an extension δ ∈ Ext C ( X, Y ) and two morphisms f ∈ Hom C ( M, X ) and g ∈ Hom C ( Y, N ), we shall define δ · f = δ ◦ f ∈ Hom A ( M, Y [1]) = Ext C ( M, Y )and g · δ = g [1] ◦ δ ∈ Hom A ( X, N [1]) = Ext C ( X, N ) . This yields the following trivial observation.2.1.
Lemma.
Let C be a tri-exact category. Given an extension δ and two morphisms f, g in C , the following equations hold whenever the composites make sense :( g · δ ) · f = g · ( δ · f ); ( δ · f ) · g = δ · ( f g ); g · ( f · δ ) = ( gf ) · δ. The tri-exact structure of C consists of the tri-exact sequences as defined below.2.2. Definition.
Let C be an extension-closed subcategory of a triangulated cate-gory A . A sequence of morphisms X / / Y / / Z in C is called a tri-exact se-quence if it embeds in an exact triangle X / / Y / / Z δ / / X [1]in A . In this case, we say that the tri-exact sequence is defined by δ ∈ Ext C ( Z, X ),and call X the starting term and Z the ending term . Remark.
It is well-known; see [16, (1.2)] that a tri-exact sequence is a pseudo-exactsequence as defined in [25, Section 1]. The converse, however, is not true.The following statement describes some basic properties of the tri-exact structureof a tri-exact category.2.3.
Lemma.
Let C be a tri-exact category, and let X f / / Y g / / Z be a tri-exactsequence defined by an extension δ ∈ Ext C ( Z, X ) . (1) A morphism u : X → M factors through f if and only if u · δ = 0 . (2) A morphism v : N → Z factors through g if and only if δ · v = 0 . (3) The morphism f is a section if and only if g is a retraction if and only if δ = 0 . (4) If X or Z is strongly indecomposable, then g is right minimal or f is leftminimal, respectively. LMOST SPLIT SEQUENCES 9
Proof.
Assume that C is an extension-closed subcategory of a triangulated category A . Statements (3) and (4) follow from some well-known properties of A ; see [16,(1.4)] and [20, (2.4),(2.5)]. Given a morphism u : X → M in C , we obtain acommutative diagram with rows being exact triangles X f / / u (cid:15) (cid:15) Y g / / (cid:15) (cid:15) Z δ / / X [1] u [1] (cid:15) (cid:15) M f ′ / / L g ′ / / Z u · δ / / M [1]in A , where L ∈ C . If u · δ = 0 , then f ′ is a section, and hence, u factors through f . If u factors through f then, by rotating the top exact triangle to the left, we seethat u ◦ ( − δ [ − u · η = u [1] ◦ δ = 0. This establish Statement (1).Dually, we can prove Statement (2). The proof of the lemma is completed.We are ready to introduce the stable categories of C . Fix an object M in C .Given a morphism f : X → Y in C , we obtain two Z -linear mapsExt C ( M, f ) : Ext C ( M, X ) → Ext C ( M, Y ) : δ f · δ and Ext C ( f, M ) : Ext C ( Y, M ) → Ext C ( X, M ) : ζ ζ · f. This consideration yields a covariant functor Ext C ( M, − ) : C →
Mod Z and a con-travariant functor Ext C ( − , M ) : C →
Mod Z . A morphism f : X → Y in C is called injectively trivial if Ext C ( M, f ) = 0 forall M ∈ C ; and projectively trivial if Ext C ( f, M ) = 0 for all M ∈ C ; compare [24,Section 2]. Moreover, an object X ∈ C is called Ext-injective if 1 X is injectivelytrivial, or equivalently, Ext C ( M, X ) = 0 for all M ∈ C ; and Ext-projective if 1 X is projectively trivial, or equivalently, Ext C ( X, N ) = 0 for all N ∈ C . Clearly, theinjectively trivial morphisms and the projectively trivial morphisms form two idealswritten as P C and I C in C , respectively. The following observation is important.2.4. Lemma.
Let C be an extension-closed subcategory of a triangulated category. (1) If X ∈ C ∩ C [1] , then P C ( M, X ) = 0 for all M ∈ C . (2) If X ∈ C ∩ C [ − , then I C ( X, N ) = 0 for all N ∈ C .Proof. We shall only prove Statement (1). Let f : M → X be projectively trivial,where X ∈ C ∩ C [1]. Since X [ − ∈ C , we see that 1 X ∈ Ext C ( X, X [ − C ( f, X [ − X ) = 0 , that is, f = 0. The proof of the lemma is completed.We are ready to define the stable categories of a tri-exact category.2.5. Definition.
Let C be a tri-exact category. We shall call C = C / I C the injec-tively stable category , and C = C / P C the projectively stable category , of C . Remark.
In view of Lemmas 1.7 and 2.4, we see that C = C in case C is a lefttriangulated category, and C = C in case C is a right triangulated category.We shall put Hom C ( X, Y ) = Hom C ( X, Y ) and Hom C ( X, Y ) = Hom C ( X, Y ), forall
X, Y ∈ C . Consider a morphism f : X → Y in C . We shall write ¯ f and f for itsimages in Hom C ( X, Y ) and Hom C ( X, Y ), respectively. In this way, we may define¯ f · ζ = f · ζ and δ · f = δ · f , for all ζ ∈ Ext ( M, X ) and δ ∈ Ext ( M, Y ).2.6.
Lemma.
Let C be a Krull-Schmidt tri-exact category. (1) If X ∈ C is indecomposable, then there exists an indecomposable object M in C such that Hom C ( X, − ) ∼ = Hom C ( M, − ) and Ext C ( X, − ) ∼ = Ext C ( M, − ) . (2) If X ∈ C is indecomposable, then there exists an indecomposable object N in C such that Hom C ( − , X ) ∼ = Hom C ( − , N ) and Ext C ( − , X ) ∼ = Ext C ( − , N ) .Proof. We shall only prove the first statement. Let X ∈ C be indecomposable.Since C is Krull-Schmidt, End( X ) is semiperfect; see [26, (1.1)]. Thus, End( X )has a complete orthogonal set { e , . . . , e n } of primitive idempotents such that e i End( X ) e i is local, for i = 1 , . . . , n ; see [1, (27.6)]. Since End( X ) is local, wemay assume that e = 1 X . Let q : M → X and p : X → M be morphisms suchthat pq = 1 M and qp = e . Observing that End( M ) ∼ = e End( X ) e , we see that M is indecomposable in C . Given Y ∈ C , since 1 X − e is projectively trivial,we obtain two Z -linear isomorphisms Ext C ( p, Y ) : Ext C ( M, Y ) → Ext C ( X, Y ) andHom C ( p, Y ) : Hom C ( M, Y ) → Hom C ( X, Y ), which are evidently natural in Y . Theproof of the lemma is completed.Next, we shall relate exact categories to tri-exact categories. Fix an abeliancategory A and consider its derived category D ( A ). Given an object X in A , weobtain a stalk complex X [ n ] whose component of degree − n is X . Given a morphism f : X → Y in A , we obtain a morphism f [ n ] : X [ n ] → Y [ n ] whose component ofdegree − n is f . Consider the canonical embedding functor D : A → D ( A ) : X X [0]; f ˜ f [0] , where ˜ f [0] is the image of f [0] under L ◦ P : C ( A ) → D ( A ); see [27, (III.3.4.7)].Let C be an extension-closed subcategory of A . We shall denote by ˆ C the fullsubcategory of D ( A ) of complexes X . with H ( X . ) ∈ C and H i ( X . ) = 0 for i = 0.2.7. Lemma.
Let C be an extension-closed subcategory of an abelian category A . (1) The category ˆ C is an extension-closed subcategory of D ( A ) . (2) Given X . ∈ ˆ C , there exists a natural isomorphism θ X . : X . → H ( X . )[0] in ˆ C . Proof.
Given X . ∈ ˆ C , consider the smart truncations τ ≤ ( X . ) and τ ≥ ( τ ≤ ( X . )).Since H i ( X . ) = 0 for all i = 0, the canonical injection i . : τ ≤ ( X . ) → X . andthe canonical projection p . : τ ≤ ( X . ) → τ ≥ ( τ ≤ ( X . )) are quasi-isomorphisms; see[27, (III.3.4)]. Observing that τ ≥ ( τ ≤ ( X . )) = H ( X . )[0], we obtain a isomorphism θ X . : X . → H ( X . )[0] in ˆ C , which is evidently natural in X . .Let X . / / Y . / / Z . / / X . [1] , where X . , Z . ∈ ˆ C , be an exact triangle in D ( A ). By the long exact sequence of cohomology, H i ( Y . ) = 0 for all i = 0 and0 / / H ( X . ) / / H ( Y . ) / / H ( Z . ) / / A . SinceH ( X . ) , H ( Z . ) ∈ C , we see that H ( Y . ) ∈ C . The proof of the lemma is completed.By Lemma 2.7, restricting the canonical embedding D : A → D ( A ) yields anequivalence D C : C → ˆ C . We shall say that a functor F : C → Mod Z is essentiallyequivalent to a functor ˆ F : ˆ C → Mod Z provided that F ∼ = ˆ F ◦ D C . Lemma.
Let C be an extension-closed subcategory of an abelian category A ,and consider an object X in C . (1) The functors
Ext C ( X, − ) and Ext C ( − , X ) are essentially equivalent to thefunctors Ext C ( X [0] , − ) and Ext C ( − , X [0]) , respectively. (2) The functors
Hom C ( X, − ) and Hom C ( − , X ) are essentially equivalent to thefunctors Hom ˆ C ( X [0] , − ) and Hom ˆ C ( − ,X [0]) , respectively. LMOST SPLIT SEQUENCES 11
Proof.
Given an extension δ ∈ Ext C ( M, N ) represented by a short exact sequence0 / / N f / / L g / / M / / C , it is well-known that D ( A ) has an induced exact triangle N [0] ˜ f [0] / / L [0] ˜ g [0] / / M [0] ˜ δ / / N [1] . This yields an isomorphism E M,N : Ext C ( M, N ) → Ext C ( M [0] , N [0]) : δ ˜ δ, such, for all f ∈ Hom C ( X, M ), δ ∈ Ext C ( Z, X ) and g ∈ Hom C ( N, Z ), that E N,M ( g · δ · f ) = ˜ f [0] · ˜ δ · ˜ g [0]; see [27, (IV.2.1.1)]. Thus, Statement (1) holds.To show Statement (2), we claim that a morphism f : X → Y in C is injectivelytrivial if and only if ˜ f [0] is injectively trivial in ˆ C . Assume first that ˜ f [0] is injec-tively trivial in ˆ C . Given any δ ∈ Ext C ( Z, X ), we obtain E Z,Y ( f · δ ) = ˜ f [0] · ˜ δ = 0,and hence, f · δ = 0. That is, f is injectively trivial in C . Conversely, assume that f is injectively trivial in C . Let ζ . ∈ Ext C ( Z . , X [0]). Setting Z = H ( Z . ), by Lemma2.7(2), we have an isomorphism θ Z . : Z . → Z [0] in ˆ C . Thus, ζ . · θ − Z . = ˜ δ for some δ ∈ Ext C ( Z, X ). Observing that ˜ f [0] · ˜ δ = E Z,Y ( f · δ ) = E Z,Y (0) = 0 , we obtain˜ f [0] · ζ . = 0. That is, ˜ f [0] is injectively trivial in ˆ C . This establishes our claim. Asa consequence, D C : C → ˆ C induces an equivalence between the injectively stablecategories. In particular, Hom C ( X, − ) is essentially equivalent to Hom ˆ C ( X [0] , − ).In a dual fashion, we may establish the second part of Statement (2). The proof ofthe lemma is completed.The following statement is needed in the next section.2.9. Lemma.
Let C be an extension-closed subcategory of an abelian category A ,and let F, G : C → Mod Z be functors essentially equivalent to ˆ F, ˆ G : ˆ C → Mod Z respectively. Then there exists a ( mono, iso ) morphism η : F → G if and only ifthere exists a ( mono, iso ) morphism ˆ η : ˆ F → ˆ G .Proof. Let ζ : F → ˆ F ◦ D C and ξ : G → ˆ G ◦ D C be isomorphisms. Firstly, assumethat ˆ η : ˆ F → ˆ G is a morphism. Given X ∈ C , we set η X = ξ − X ◦ ˆ η X [0] ◦ ζ X , which isa monomorphism or isomorphism in case ˆ η X [0] is a monomorphism or isomorphism,respectively. This yields a desired (mono, iso)morphism η : F → G .Conversely, assume that η : F → G is a morphism. Given X . ∈ ˆ C , by Lemma2.7(2), there exists a natural isomorphism θ X . : X . → X [0], where X = H ( X . ) . We define ˆ η X . to be the composite of the following morphismsˆ F ( X . ) ˆ F ( θ X . ) / / ˆ F ( X [0]) ζ − X / / F ( X ) η X / / G ( X ) ξ X / / ˆ G ( X [0]) ˆ G ( θ − X . ) / / ˆ G ( X . ) , which is a monomorphism or isomorphism if η X is a monomorphism or isomorphismrespectively. This yields a desired (mono, iso)morphism ˆ η : ˆ F → ˆ G . The proof ofthe lemma is completed. 3. Almost split sequences
The objective of this section is to study the existence of an individual almost splitsequence in a tri-exact category. Using similar but more general techniques, we shallunify and extend the results under various classical settings; see [2, 20, 26, 24, 29].
In particular, Auslander’s existence theorem for an almost split sequence in thecategory of all modules over a ring; see [4] and Krause’s existence theorem for analmost triangle in a triangulated category fit well into our setting; see [20].Throughout this section, C stands for a tri-exact category, say an extension-closed subcategory of a triangulated category A . The following notion plays afundamental role in our investigation.3.1. Definition.
Let C be a tri-exact category. A tri-exact sequence X f / / Y g / / Z defined by an extension δ ∈ Ext C ( Z, X ) is called almost split if f is minimal leftalmost split and g is minimal right almost split. In this case, δ is called almost-zero . Remark.
An almost split sequence with a non-zero middle term is an Auslander-Reiten sequence as defined in [25, (1.3)]. The converse is probably not true.Modifying slightly the proof of the proposition stated in [15, (3.5)], we obtainthe uniqueness of an almost split sequence in a tri-exact category as follows.3.2.
Proposition.
Let C be a tri-exact category. If X / / Y / / Z is an almostsplit sequence in C , then it is unique up to isomorphism for X and for Z . The following statement says in particular that the study of almost split se-quences under various classical settings can be unified under our tri-exact setting.3.3.
Proposition.
Let C be an extension-closed subcategory of an abelian category A . Then every almost split sequence / / X / / Y / / Z / / in C inducesan almost split sequence X [0] / / Y [0] / / Z [0] in ˆ C ; and every almost split se-quence in ˆ C can be obtained in this way.Proof. Since D C : C → ˆ C is an equivalence, the first part of the proposition followsimmediately. Assume that ˆ C has an almost split sequence X . f . / / Y . g . / / Z . defined by an extension η . ∈ Ext C ( Z . , X . [1]). Applying the long exact sequence ofcohomology, we obtain a short exact sequence δ : 0 / / H ( X . ) f / / H ( Y . ) g / / H ( Z . ) / / C , where f = H ( f . ) and g = H ( g . ). In view of Lemma 2.7(2), there exists acommutative diagram with vertical isomorphisms X . f . / / θ X . (cid:15) (cid:15) Y . g . / / θ Y . (cid:15) (cid:15) Z . ζ . (cid:15) (cid:15) η . / / X . [1] θ X . [1] (cid:15) (cid:15) H ( X . )[0] ˜ f [0] / / H ( Y . )[0] ˜ g [0] / / H ( Z . )[0] ˜ δ / / H ( X . )[1]in D ( A ), where the rows are exact triangles. In particular, ˜ f [0] is minimal leftalmost split and ˜ g [0] is minimal right almost split in ˆ C . Since D C : C → ˆ C is anequivalence, δ is almost split in C . The proof of the proposition is completed.The following characterization of an almost split sequence in a tri-exact categoryis adapted from those in the classical settings; see [6, 20]. LMOST SPLIT SEQUENCES 13
Theorem.
Let C be a tri-exact category. If X f / / Y g / / Z is a tri-exactsequence in C , then the following statements are equivalent. (1) The sequence is an almost split sequence in C . (2) The morphism f is left almost split and g is right almost split. (3) The morphism f is left almost split and Z is strongly indecomposable. (4) The morphism g is right almost split and X is strongly indecomposable. (5) The morphism f is minimal left almost split or g is minimal right almost split.Proof. Let η ∈ Ext C ( Z, X ), defining the tri-exact sequence stated in the theorem.If Statement (2) holds, then X and Z are strongly indecomposable; see [6, (2.3)],and by Lemma 2.3(4), g is right minimal and f is left minimal, that is, Statement(1) holds. Moreover, by Lemma 2.3(4), either of Statements (3) and (4) impliesStatement (5). Assume now that Statement (5) holds. To prove Statement (2), weassume that C is an extension-closed subcategory of a triangulated category A . If g is minimal right almost split, using the same argument given in [20, (2.6)], we mayshow that f is left almost split. If f is minimal left almost split, one can duallyshow that g is right almost split. The proof of the theorem is completed.The rest of this section is devoted to the study of the existence of an almostsplit sequence in a tri-exact category. We start with some properties of almost-zeroextensions; compare [18, (2.2)], [24, (3.1)] and [29, Page 306].3.5. Lemma.
Let C be a tri-exact category with δ ∈ Ext C ( Z, X ) being almost-zero. (1) A factorization δ = η · f exists whenever a non-zero extension η ∈ Ext C ( L, X ) or a non-zero morphism f ∈ Hom C ( Z, L ) is given. (2) A factorization δ = ¯ g · ζ exists whenever a non-zero extension ζ ∈ Ext C ( Z, L ) or a non-zero morphism ¯ g ∈ Hom C ( L, X ) is given.Proof. Given a non-zero extension η ∈ Ext C ( L, X ), using the same proof of the firststatement of the sublemma stated in [29, Page 306], we obtain δ = η · f for somemorphism f ∈ Hom C ( Z, L ). Given a non-zero extension ζ ∈ Ext C ( Z, L ), by a dualargument, we can show that δ = ¯ g · ζ for some ¯ g ∈ Hom C ( L, X ).Now, let f : Z → L be a non projectively trivial morphism in C . Then, thereexists some ζ ∈ Ext C ( L, M ) such that 0 = ζ · f ∈ Ext C ( Z, M ). By the first part ofStatement (2), there exists some g : M → X in C such that δ = g · ( ζ · f ) = ( g · ζ ) · f .This establishes the second part of Statement (1). Dually, one can verify the secondpart of Statement (2). The proof of the lemma is completed.Given X, Z ∈ C , by Lemma 2.1, Ext C ( Z, X ) is an End( X )-End( Z )-bimodule sothat f · δ · g = f · δ · g , for f ∈ End( X ) , δ ∈ Ext C ( Z, X ) and g ∈ End( Z ) . Given M ∈ C strongly indecomposable, we always write S M = End( M ) / rad(End( M )),which is a simple left End( M )-module if M is not Ext-injective; and a simple rightEnd( M )-module if M is not Ext-projective.3.6. Theorem.
Let C be a tri-exact category with X, Z ∈ C strongly indecomposable.Consider non-zero ring homomorphisms Γ → End( X ) and Σ → End( Z ) . Let Γ I be an injective cogenerator of the left Γ -module S X and I Σ an injective cogeneratorof the right Σ -module S Z . If δ ∈ Ext C ( Z, X ) is non-zero, then δ being almost zerois equivalent to each of the following statements. (1) There exists a monomorphism
Ψ : Ext C ( Z, − ) → Hom Γ (Hom C ( − , X ) , Γ I ) suchthat Ψ X ( δ ) lies in the socle of the left End( X ) -module Hom Γ (End( X ) , Γ I ) . (2) There exists a monomorphism
Φ : Ext C ( − , X ) → Hom Σ (Hom C ( Z, − ) , I Σ ) suchthat Φ Z ( δ ) lies in the socle of the right End( Z ) -module Hom Σ (End( Z ) , I Σ ) .Proof. Let δ ∈ Ext C ( Z, X ) be non-zero. We shall only consider Statement (1).Let I be an injective cogenerator of the left Γ -module S X . Assume first that δ is almost-zero. Consider the End( X )-submodule S of Ext C ( Z, X ) generated by δ ,which is simple by Lemma 3.5(2). Since End( X ) is local, S ∼ = S X as left End( X )-modules, and consequently, S ∼ = S X as Γ -modules. Thus, we may find a Γ -linearmap ψ : Ext C ( Z, X ) → I with ψ ( δ ) = 0 . Given L ∈ C , by Lemma 3.5(2), we obtaina non-degenerate Z -bilinear form < − , − > L : Hom C ( L, X ) × Ext C ( Z, L ) → I : (¯ g, ζ ) ψ (¯ g · ζ ) . Since ψ is left Γ -linear, by Lemma 2.1, we obtain a Z -linear monomorphismΨ L : Ext C ( Z, L ) → Hom Γ (Hom C ( L, X ) , I ) : ζ < − , ζ > L , which is natural in L . This yields a monomorphism Ψ as stated in Statement (1).Since Ψ X is a left End( X )-linear monomorphism, Ψ X ( S ) is a simple submodule ofthe left End( X )-module Hom Γ (End( X ) , I ) . This establishes Statement (1).Conversely, assume that Ψ : Ext C ( Z, − ) → Hom Γ ( Hom C ( − , X ) , I ) is a monomor-phism such that Ψ X ( δ ) is in the left End( X )-socle of Hom Γ (End( X ) , I ) . Then,Ψ X ( δ ) vanishes on rad(End( X )) . Consider the non-splitting tri-exact sequence( ∗ ) X f / / Y g / / Z in C defined by δ . Let u : X → L be a non-section morphism in C . In view of thecommutative diagramExt C ( Z, X ) Ψ X / / Ext C ( Z, ¯ u ) (cid:15) (cid:15) Hom Γ ( End( X ) , I ) Hom Γ (Hom C (¯ u,X ) ,I ) (cid:15) (cid:15) Ext C ( Z, L ) Ψ L / / Hom Γ (Hom C ( L, X ) , I ) , we obtain Ψ L (¯ u · δ ) = Ψ X ( δ ) ◦ Hom C (¯ u, X ) . For any morphism v : L → X in C , since vu ∈ rad(End( X )) , we see that Ψ L (¯ u · δ )(¯ v ) = Ψ X ( δ )(¯ v ¯ u ) = 0 . Thus, Ψ L (¯ u · δ ) = 0,and hence, ¯ u · δ = 0. By Lemma 2.3(1), u factors through f . That is, f is leftalmost split. Since Z is strongly indecomposable, by Theorem 3.4(3), the tri-exactsequence ( ∗ ) is almost split. The proof of the theorem is completed. Remark.
In view of Lemmas 2.8 and 2.9 and Proposition 3.3, it is easy to see thatTheorem 3.6 covers the result stated in [26, (2.2)].We are ready to obtain our main existence theorem of an almost splits sequence.3.7.
Theorem.
Let C be a tri-exact category with X, Z ∈ C strongly indecomposable.Consider non-zero ring homomorphisms Γ → End( X ) and Σ → End( Z ) . Let Γ I be an injective cogenerator of the left Γ -module S X and I Σ an injective cogeneratorof the right Σ -module S Z . The following statements are equivalent. (1) There exists an almost split sequence X / / Y / / Z in C . (2) There exists a monomorphism
Ψ : Ext C ( Z, − ) → Hom Γ (Hom C ( − , X ) , Γ I ) suchthat the left End( X ) -linear map Ψ X is socle essential. (3) There exists a monomorphism
Φ : Ext C ( − , X ) → Hom Σ (Hom C ( Z, − ) , I Σ ) suchthat the right End( Z ) -linear map Φ Z is socle essential. LMOST SPLIT SEQUENCES 15
Proof.
By Theorem 3.6, it suffices to prove that Statement (2) implies Statement(1). Let I be an injective co-generator of the left Γ -module S X with a monomor-phism Ψ : Ext C ( Z, − ) → Hom Γ (Hom C ( − , X ) , I ) such that the left End( X )-linearmap Ψ X : Ext C ( Z, X ) → Hom Γ (End( X ) , I ) is socle essential. Consider the canon-ical projection p : End( X ) → S X and fix a non-zero Γ -linear map q : S X → I .Then, qp is a non-zero element in Hom Γ (End( X ) , I ) annihilated by rad(End( X )).Since End( X ) is local, qp belongs to the left End( X )-socle of Hom Γ (End( X ) , I ).Since Ψ X is socle essential, there exists δ ∈ Ext C ( Z, X ) such that Ψ X ( δ ) lies in theleft End( X )-socle of Hom Γ (End( X ) , Γ I ). By Theorem 3.6(1), δ defines an almostsplit sequence as stated in Statement (1). The proof of the theorem is completed. Remark.
Observe, for any ring Σ , that a Σ -linear monomorphism f : M → N is socle essential if M has a nonzero socle or N has an essential socle. Thus, wesee from Lemmas 2.8 and 2.9 and Proposition 3.3 that Theorem 3.7 includes theresults stated in [24, (4.1)] and [26, (2.3)].By abuse of terminology, a functor F is called a subfunctor of another functor G if there exists a monomorphism F → G . We shall drop the additional hypotheseson Ψ Z and Φ X stated in Theorem 3.7 in some special cases as below.3.8. Theorem.
Let C be a tri-exact category with X, Z ∈ C strongly indecomposable.Consider non-zero surjective ring homomorphisms Γ → End( X ) and Σ → End( Z ) .Let Γ I be an injective envelope of the left Γ -module S X and I Σ an injective envelopeof the right Σ -module S Z . The following statements are equivalent. (1) There exists an almost split sequence X / / Y / / Z in C . (2) Ext C ( Z, − ) is a non-zero subfunctor Hom Γ (Hom C ( − , X ) , Γ I ) . (3) Ext C ( − , X ) is a non-zero subfunctor of Hom Σ (Hom C ( Z, − ) , I Σ ) . Proof.
By Theorem 3.7, it suffices to prove that Statement (2) implies Statement(1). Let Ψ : Ext C ( Z, − ) → Hom Γ (Hom C ( − , X ) , I ) be a non-zero monomorphism,where I = Γ I . In particular, there exists some L ∈ C such that Ext C ( Z, L ) = 0.Thus, Ψ L ( δ )( ¯ f ) = 0 for some δ ∈ Ext C ( Z, L ) and f ∈ Hom C ( L, X ). SinceExt C ( Z, L ) Ψ L / / Ext C ( Z, ¯ f ) (cid:15) (cid:15) Hom Γ ( Hom C ( L, X ) , I ) Hom Γ (Hom C ( ¯ f,X ) ,I ) (cid:15) (cid:15) Ext C ( Z, X ) Ψ X / / Hom Γ (End( X ) , I )is a commutative diagram, we obtain Ψ X ( ¯ f · δ )(1 X ) = Ψ L ( δ )( ¯ f ) = 0 . Thus, Ψ X isa non-zero left End( X )-linear monomorphism, which is also left Γ -linear.Since the ring homomorphism ρ : Γ → End( X ) is surjective, S X is a sim-ple left Γ -module and ρ ∗ = Hom Γ ( ρ, I ) : Hom Γ ( End( X ) , I ) → Hom Γ ( Γ , I ) is aleft Γ -linear monomorphism. Let ν : Hom Γ ( Γ , I ) → I be the canonical Γ -linearisomorphism. Then, ψ = ν ◦ ρ ∗ ◦ Ψ X : Ext C ( Z, X ) → I is a non-zero Γ -linearmonomorphism. Since S X is the essential Γ -socle of I , it is contained in Im( ψ ).In particular, Ext C ( Z, X ) has a simple left Γ -submodule S such that ψ ( S ) = S X .Since ρ is surjective, S is a simple left End( X )-submodule of Ext C ( Z, X ) , and hence,the left End( X )-linear monomorphism Ψ X is socle essential. By Theorem 3.7(2), C has a desired almost split sequence. The proof of the theorem is completed. Remark. (1) Let Z ∈ Mod Λ be finitely presented, strongly indecomposable andnot projective, where Λ is a ring. Since End Λ op (Tr Z ) op ∼ = End Λ ( Z ); see [4, Sec- tion I.3], we have a surjective ring homomorphism from Σ = End Λ op (Tr Z ) op ontoEnd Λ ( Z ). Let I be the injective envelope of the right Σ -module S Z . Then, X = Hom Σ (Tr Z, I ) is a strongly indecomposable module in Mod Λ such thatExt Λ ( − , X ) ∼ = Hom Σ (Hom Λ ( Z, − ) , I ); see [4, (I.11.3), (I.3.4)]. By Theorem 3.8,there exists an almost split sequence 0 / / X / / Y / / Z / / Λ . Thisis Auslander’s theorem stated in [4, (II.5.1)].(2) Let A be a locally finitely presented Grothendieck ableian category witha finitely presented strongly indecomposable object Z . Consider the canonicalprojection Σ = End( Z ) → End( Z ) . Given any injective module I ∈ Mod Σ op ,Krause obtained a monomorphism Ext A ( − , τ I ( Z )) → Hom Σ (Hom A ( Z, − ) , I ) , forsome τ I ( Z ) ∈ A ; see [21, (1.2)]. However, it is not known whether or not τ I ( Z ) isstrongly indecomposable even if I is the injective envelope of S Z . Thus, we cannotapply Theorem 3.8 to obtain an almost split sequence in A . We can weaken the assumption that both X and Z are strongly indecomposableas stated in Theorem 3.8 in some special cases as below.3.9. Theorem.
Let C be an extension-closed subcategory of a triangulated category,and let Z ∈ C be strongly indecomposable with a non-zero monomorphism Φ : Hom C ( − , X ) → Hom
End( Z ) (Hom C ( Z, − ) , I ) , where X ∈ C ∩ C [1] and I is an injective envelope of the right End( Z ) -module S Z .If Φ X is bijective, then C has an almost split sequence X [ − / / Y / / Z. Proof.
Assume that Φ X is bijective. Observing that X [ − ∈ C is such thatExt C ( − , X [ − C ( − , X ). By Theorem 3.8(3), it amounts to show thatEnd( X ) is local. Put Σ = End( Z ). By Lemma 2.4(1), Hom C ( Z, X ) = Hom C ( Z, X ),which is a right Σ -module. Now, Φ X : End( X ) → Hom Σ (Hom C ( Z, X ) , I ) is aright End( X )-linear isomorphism, while Φ Z : Hom C ( Z, X ) → Hom Σ ( Σ , I ) is aright Σ -linear monomorphism. Considering the canonical Σ -linear isomorphism ρ : Hom Σ ( Σ , I ) → I, we obtain a commutative diagram of surjective Z -linear mapsEnd Σ ( I ) Hom Σ ( ρ, I ) / / θ (cid:15) (cid:15) Hom Σ (Hom Σ ( Σ , I ) , I ) Hom Σ (Φ Z , I ) (cid:15) (cid:15) End( X ) Φ X / / Hom Σ (Hom C ( Z, X ) , I ) . Since End Σ ( I ) is local; see [1, (25.4)], it suffices to show that θ is a ring homo-morphism. Fix arbitrarily a morphism u : Z → X in C . Given f ∈ End Σ ( I ), inview of the above commutative diagram, we see that Φ X ( θ ( f )) = f ◦ Φ Z ◦ ρ, andconsequently, we obtain an equation(1) Φ X ( θ ( f ))( u ) = f (Φ Z ( u )(1 Σ )) . On the other hand, considering the commutative diagramEnd( X ) Hom C ( u, X ) (cid:15) (cid:15) Φ X / / Hom Σ (Hom C ( Z, X ) , I ) Hom Σ (Hom C ( Z,u ) ,I ) (cid:15) (cid:15) Hom C ( Z, X ) Φ Z / / Hom Σ ( Σ , I ) , we obtain an equation(2) Φ Z ( u )(1 Σ ) = Φ X (1 X )( u ) . LMOST SPLIT SEQUENCES 17
Let f i ∈ End Γ ( I ), and write g i = θ ( f i ) ∈ End( X ), for i = 1 ,
2. We deduce fromthe equation (1) thatΦ X ( θ ( f f ))( u ) = ( f f )(Φ Z ( u )(1 Z )) = f [ f (Φ Z ( u )(1 Z ))] = f (Φ X ( g )( u )) . Since Φ X is right End( X )-linear, combining the equations (1) and (2) yieldsΦ X ( g g )( u ) = Φ X ( g )( g u ) = f (Φ Z ( g u )(1 Σ )) = f (Φ X (1 X )( g u )) = f (Φ X ( g )( u )) . Thus, Φ X ( θ ( f f )) = Φ X ( g g ) , and consequently, θ ( f f ) = g g = θ ( f ) θ ( f ) . Since θ is surjective, θ (1 I ) = 1 X . The proof of the theorem is completed.
Remark. If C is a left triangulated subcategory of a triangulated category, then C ⊆ C [1]. In particular, Theorem 3.9 covers the essential part of Krause’s resultstated in [20, (2.2)], where the isomorphism End( X ) ∼ = End End( Z ) ( I ) is only verifiedto be an abelian group isomorphism.In a dual fashion, we may establish the following statement.3.10. Theorem.
Let C be an extension-closed subcategory of a triangulated category,and let X ∈ C be strongly indecomposable with a non-zero monomorphism Ψ : Hom C ( Z, − ) → Hom
End( X ) (Hom C ( − , X ) , I ) where Z ∈ C ∩ C [ − and I is an injective envelope of the left End( X ) -module S X .If Ψ Z is bijective, then C has an almost split sequence X / / Y / / Z [1] . Auslander-Reiten functors
The objective of this section is to study the existence of almost split sequences ina Hom-reflexive tri-exact R -category, where R is a commutative ring. Our mainresults will relate the global existence of almost split sequences in the categoryto the existence of an Auslander-Reiten functor, which is a generalization of anAuslander-Reiten duality considered in [24]; and in the left or right triangulatedcase, to the existence of a Serre functor with a proper image, which differs slightlyfrom the classical notion of a Serre functor defined in [29, (I.1)]; see also [10].Throughout this section, C will stand for a tri-exact R -category, say an extension-closed subcategory of a triangulated R -category A . We shall say that C has almostsplit sequences on the right (respectively, left ) if every strongly indecomposable notExt-projective (respectively, not Ext-injective) object is the ending (respectively,starting) term of an almost split sequence; and that C has almost split sequences if ithas almost split sequences on the right and on the left. Recall that the exact functor D = Hom R ( − , I R ) : Mod R → Mod R restricts to a duality D : RMod R → RMod R ,where I R is the minimal injective co-generator for Mod R , whereas RMod R is thecategory of reflexive R -modules. We shall say that C is Hom-reflexive (respectively,
Hom-finite ) if Hom C ( X, Y ) is reflexive (respectively, of finite length) over R , forall X, Y ∈ C ; and
Ext-reflexive if Ext C ( X, Y ) is reflexive over R , for all X, Y ∈ C .Hom-finite R -categories are Hom-reflexive; see (1.3), and the converse is not true.The following statement is important for our purpose; compare [26, (2.4)].4.1. Lemma.
Let R be a commutative ring, and let M, N ∈ Mod R defining a non-degenerate R -bilinear form < − , − > : M × N → I R . If M or N is reflexive, then both M and N are reflexive with R -linear isomorphisms φ M : M → DN : u and ψ N : N → DM : v < − , v> . Proof.
By the hypothesis, φ M and ψ N are monomorphisms. Consider the canonicalmonomorphisms σ M : M → D M and σ N : N → D N . It is easy to verify that φ M = D ( ψ N ) ◦ σ M and ψ N = D ( φ M ) ◦ σ N , where D ( φ M ) and D ( ψ N ) are surjective.If M is reflexive, so are DM and N ; see (1.3). In particular, σ M and σ N aresurjective, so are φ M and ψ N . The proof of the lemma is completed.We shall first strengthen the results on the existence of an individual almostsplit sequence under the Hom-reflexive setting. The following preparatory result iswell-known under some classical settings; see, for example, [14, 24, 29].4.2. Lemma.
Let C be a Hom-reflexive tri-exact R -category. Consider an almost-zero extension δ ∈ Ext C ( Z, X ) and a linear form θ ∈ D Ext C ( Z, X ) such that θ ( δ ) = 0 . Given any object L ∈ C , there exist natural R -linear isomorphismsΩ L,X : Hom C ( L, X ) → D Ext C ( Z, L ) : ¯ g θ ◦ Ext C ( Z, ¯ g ) and Θ Z,L : Hom C ( Z, L ) → D Ext C ( L, X ) : f θ ◦ Ext C ( f , X ) . Proof.
Given L ∈ C , by Lemma 3.5, we obtain two non-degenerate R -bilinear forms < − , − > L : Hom C ( L, X ) × Ext C ( Z, L ) → I R : ( ¯ g, ζ ) θ ( ¯ g · ζ )and L < − , − > : Ext C ( L, X ) × Hom C ( Z, L ) → I R : ( ζ, f ) → θ ( ζ · f ) . Since C is Hom-reflexive, by Lemma 1.3, Hom C ( L, X ) and Hom C ( Z, L ) are re-flexive R -modules. By Lemma 4.1, we obtain two isomorphisms Ω L,X and Θ Z,L as stated in the lemma, which are clearly natural in L ; see (2.1). The proof of thelemma is completed.The following result improves Theorem 3.7 under the Hom-reflexive setting.4.3. Theorem.
Let C be a Hom-reflexive tri-exact R -category with X, Z ∈ C stronglyindecomposable. The following statements are equivalent. (1)
There exists an almost split sequence X / / Y / / Z in C . (2) There exists a non-zero isomorphism Ω X : Hom C ( − , X ) → D Ext C ( Z, − ) . (3) There exists a non-zero isomorphism Θ Z : Hom C ( Z, − ) → D Ext C ( − , X ) . Proof.
Given an almost-zero extension δ ∈ Ext C ( Z, X ), we choose θ ∈ D Ext C ( Z, X )such that θ ( δ ) = 0. By Lemma 4.2, we see that Hom C ( Z, − ) ∼ = D Ext C ( − , X ) andHom C ( − , X ) ∼ = D Ext C ( Z, − ). Thus, Statement (1) implies Statements (2) and (3).Let now Ω X : Hom C ( − , X ) → D Ext C ( Z, − ) be a nonzero isomorphism. Inparticular, Z is not Ext-projective, and hence, we have a nonzero canonical algebrahomomorphism R → End( Z ). Moreover, since C is Hom-reflexive, we obtain anisomorphism Ψ X : Ext C ( Z, − ) → D Hom C ( − , X ) . By Theorem 3.7(2), we obtainan almost split sequence as stated in Statement (1). Similarly, we may show thatStatement (3) implies Statement (1). The proof of the theorem is completed.
Remark.
In case R is artinian, Theorem 4.3 is known for an Ext-finite abelian R -category; see [14, 24] and for a Hom-finite exact R -category; see [26].We shall weaken the condition that both X and Z are strongly indecomposablestated in Theorem 4.3 in some special cases as below.4.4. Theorem.
Let C be a Hom-reflexive extension-closed subcategory of a trian-gulated R -category. LMOST SPLIT SEQUENCES 19 (1) If X ∈ C is strongly indecomposable and Z ∈ C ∩ C [1] , then C has an almostsplit sequence X / / Y / / Z if and only if Hom C ( − , X ) ∼ = D Ext C ( Z, − ) = 0 . (2) If Z ∈ C is strongly indecomposable and X ∈ C ∩ C [ − , then C has an almostsplit sequence X / / Y / / Z if and only if Hom C ( Z, − ) ∼ = D Ext C ( − , X ) = 0 . Proof.
We shall only prove the sufficiency of Statement (1). Let X ∈ C be stronglyindecomposable and Hom C ( − , X ) ∼ = D Ext C ( Z, − ) = 0, where Z = M [1] for some M ∈ C . It suffices to show that End( M ) is local. Since Ext C ( Z, − ) ∼ = Hom C ( M, − ) , we obtain an isomorphism Ψ X : Hom C ( − , X ) → D Hom C ( M, − ). In particular,Ψ X,X : End( X ) → D Hom C ( M, X ) is a right End( X )-linear isomorphism andΨ M,X : Hom C ( M, X ) → D End( M ) is a right End( M )-linear isomorphism. Since M ∈ C [ − C ( M, X ) = Hom C ( M, X ), which is aleft End( X )-module. Since End( X ) is reflexive, we obtain a commutative diagramEnd( M ) θ (cid:15) (cid:15) σ / / D End( M ) D (Ψ M,X ) (cid:15) (cid:15) End( X ) Ψ X,X / / D Hom C ( M, X )of R -linear isomorphisms, where σ is the canonical isomorphism. It remains toshow that θ is an algebra homomorphism. Indeed, we fix arbitrarily a morphism u ∈ Hom C ( M, X ). Given any f ∈ End( M ), in view of the above commutativediagram, we obtain an equation(1) Ψ X,X ( θ ( f ))( u ) = Ψ M,X ( u )( f ) . On the other hand, consider the commutative diagramEnd( X ) Ψ X,X / / Hom C (¯ u,X ) (cid:15) (cid:15) D Hom C ( M, X ) D Hom C ( M,u ) (cid:15) (cid:15) Hom C ( M, X ) Ψ M,X / / D End( M ) . Given any v ∈ End( X ), we obtain an equation(2) Ψ M,X (¯ v ¯ u )( f ) = Ψ X,X (¯ v )( uf ) . Let now f, g ∈ End( M ). Since Ψ X,X is also right End( X )-linear, we deduce fromthe equations (1) and (2) thatΨ X,X ( θ ( f ) θ ( g ))( u ) = Ψ X,X ( θ ( f ))( θ ( g ) u ) = Ψ M,X ( θ ( g )¯ u )( f ) = Ψ X,X ( θ ( g ))( uf ) . Since Ψ
M,X is right End( M )-linear, we deduce from the equation (1) thatΨ X,X ( θ ( g ))( uf ) = Ψ M,X ( uf )( g ) = Ψ M,X ( u )( f g ) = Ψ X,X ( θ ( f g ))( u ) . This yields Ψ
X,X ( θ ( f ) θ ( g )) = Ψ X,X ( θ ( f g )), and hence, θ ( f g ) = θ ( f ) θ ( g ). Theproof of the theorem is completed.4.5. Corollary.
Let C be a Hom-reflexive extension-closed subcategory of a trian-gulated R -category. (1) If C is left triangulated with X ∈ C strongly indecomposable, then X is the start-ing term of an almost split sequence if and only if the functor D Hom C ( − , X ) is representable by a nonzero object in C [ − . (2) If C is right triangulated with Z ∈ C strongly indecomposable, then Z is theending term of an almost split sequence if and only if the functor D Hom C ( Z, − ) is representable by a nonzero object in C [1] . In order to study the global existence of almost split sequences in C , we need togeneralize the classical notion of an Auslander-Reiten duality; see [7, 24].4.6. Definition.
Let C be a tri-exact R -category.(1) A right Auslander-Reiten functor for C is a functor τ : C → C with binatural R -linear isomorphisms Θ X,Y : Hom C ( X, Y ) → D Ext C ( X, τ Y ) , where X, Y ∈ C . (2) A left Auslander-Reiten functor for C is a functor τ − : C → C with binatural R -linear isomorphisms Ω X,Y : Hom C ( X, Y ) → D Ext C ( τ − Y, X ) , where X, Y ∈ C . The following statement collects some properties of an Auslander-Reiten functor.4.7.
Proposition.
Let C be a Hom-reflexive tri-exact R -category. Then, a right ( or left ) Auslander-Reiten functor for C is faithful. If C is, in addition, right ( orleft ) triangulated, then a right ( or left ) Auslander-Reiten functor is fully faithful.Proof.
We shall only consider a right Auslander-Reiten functor τ : C → C withbinatural R -linear isomorphisms Θ X,Y : Hom C ( X, Y ) → D Ext C ( Y, τ X ). Given amorphism u : X → Y , considering the commutative diagramEnd( Y ) Hom C ( u,Y ) (cid:15) (cid:15) Θ Y,Y / / D Ext C ( Y, τ Y ) D Ext C ( Y,τ ( u )) (cid:15) (cid:15) Hom C ( X, Y ) Θ X,Y / / D Ext C ( Y, τ X ) , we obtain Θ X,Y ( u ) = Θ Y,Y (1 Y ) ◦ Ext C ( Y, τ ( u )) . If τ ( u ) = 0, then Θ X,Y ( u ) = 0,and hence, u = 0. That is, τ is faithful. Next, assume that C is a Hom-reflexiveright triangulated subcategory of a triangulated R -category. By Lemmas 1.7 and2.4(2), C [1] ⊆ C = C . This yields a functor F = [1] ◦ τ : C → C with isomorphismsΦ
X,Y : Hom C ( X, Y ) → D Hom C ( Y, F X ) , which are binatural in X and Y . Given f ∈ Hom C ( X, Y ) and g ∈ Hom C ( Y, F X ), considering the commutative diagramHom C ( Y, F X ) Φ Y,FX (cid:15) (cid:15)
Hom C ( Y, Y ) Hom(
Y,g ) o o Φ Y,Y (cid:15) (cid:15)
Hom( f,Y ) / / Hom C ( X, Y ) Φ X,Y (cid:15) (cid:15) D Hom C ( F X, F Y ) D Hom C ( Y, F Y ) D Hom( g,F Y ) o o D Hom(
Y,F ( f )) / / D Hom C ( Y, F X ) , we obtain the following equations( ∗ ) Φ Y,F X ( g )( F ( f )) = Φ Y,Y (1 Y )( F ( f ) g ) = Φ X,Y ( f )( g ) . Since
F X ∈ C [1], by Lemma 2.4(1), Hom C ( Y, F X ) = Hom C ( Y, F X ). Thus, weobtain an isomorphism Φ
Y,F X : Hom C ( Y, F X ) → D Hom C ( F X, F Y ) , which makesHom C ( X, Y ) Φ X,Y / / F (cid:15) (cid:15) D Hom C ( Y, F X )Hom C ( F X, F Y ) σ / / D Hom C ( F X, F Y ) D (Φ Y,FX ) O O commute, where σ is the canonical isomorphism. Indeed, for any f ∈ Hom C ( X, Y )and g ∈ Hom C ( Y, F X ), in view of the equations in ( ∗ ), we see that LMOST SPLIT SEQUENCES 21 D (Φ Y,F X )( σ ( F ( f )))( g ) = σ ( F ( f ))(Φ Y,F X ( g )) = Φ Y,F X ( g )( F ( f )) = Φ X,Y ( f )( g ) . As a consequence, F : Hom C ( X, Y ) → Hom C ( F X, F Y ) is an isomorphism. Thatis, F is fully faithful, and so is τ . The proof of the proposition is completed.We are now able to relate the existence of almost split sequences to the existenceof an Auslander-Reiten functor.4.8. Theorem.
Let C be a Hom-reflexive Krull-Schmidt tri-exact R -category. (1) There exist almost split sequences on the right ( respectively, left ) in C if andonly if it admits a full right ( respectively, left ) Auslander-Reiten functor. (2)
There exist almost split sequences in C if and only if it admits a right Auslander-Reiten equivalence, or equivalently, a left Auslander-Reiten equivalence ; and inthis case, C is Ext-reflexive.Proof. We shall prove the theorem only for right Auslander-Reiten functors. Con-sider a full right Auslander-Reiten functor τ : C → C . Let Z ∈ C be inde-composable but not Ext-projective. By definition, there exist R -linear isomor-phisms Θ Z,Y : Hom C ( Z, Y ) → D Ext C ( Y, τ Z ), which is natural in Y . Therefore,Hom C ( Z, − ) ∼ = D Ext C ( − , τ Z ) . By Proposition 4.7, End( τ Z ) ∼ = End( Z ), which is lo-cal. By Lemma 2.6(1), Ext C ( − , τ Z ) ∼ = Ext C ( − , X ) for some indecomposable X ∈ C .Then, Hom C ( Z, − ) ∼ = D Ext C ( − , X ) . By Theorem 4.3, there exists an almost splitsequence X / / L / / Z in C .Assume that C has almost split sequences on the right. For each indecomposableand not Ext-projective object M ∈ C , we fix an indecomposable object τ M ∈ C , analmost-zero extension δ M ∈ Ext C ( M, τ M ), and a linear form θ M ∈ D Ext C ( M, τ M )such that θ M ( δ M ) = 0. Let X, Y, Z ∈ C be indecomposable and not Ext-projective.Considering θ X and θ Y , by Lemma 4.2, we obtain two R -linear isomorphisms Θ X,Y : Hom C ( X, Y ) → D Ext C ( Y, τ X ) : f θ X ◦ Ext C ( f , X )and Ω τX,τY : Hom C ( τ X, τ Y ) → D Ext C ( Y, τ X ) : ¯ g θ Y ◦ Ext C ( Y, ¯ g ) . This yields an R -linear isomorphism τ X,Y = Ω − τX,τY Θ X,Y : Hom C ( X, Y ) → Hom C ( τ X, τ Y ) . Let f ∈ Hom C ( X, Y ) and ζ ∈ Ext C ( Y, τ X ). Since Θ X,Y ( f ) = Ω τX,τY ( τ X,Y ( f ))by definition, we obtain an equation( ∗ ) θ X ( ζ · f ) = θ Y ( τ X,Y ( f ) · ζ ) . Let g ∈ Hom C ( Y, Z ) and δ ∈ Ext C ( Z, τ X ). By the above equation, we obtain θ X ( δ · ( gf )) = θ X (( δ · g ) · f ) = θ Y ( τ X,Y ( f ) · δ · g ) = θ Z (( τ Y,Z ( g ) τ X,Y ( f )) · δ )That is, Θ X,Z ( gf )( δ ) = Ω X,Z ( τ Y,Z ( g ) τ X,Y ( f ))( δ ) , from which we conclude that τ X,Z ( gf ) = τ Y,Z ( g ) τ X,Y ( f ) . In particular, since τ X,X is bijective, τ X,X (1 X ) = ¯1 τX .Since C is Krull-Schmidt, by Lemma 2.6(1), we may extend τ to a fully faithfulfunctor τ : C → C . It remains to show that the isomorphism Θ X,Y is binatural. Itis natural in Y by Lemma 4.2. We claim, for h ∈ Hom C ( X, Z ), that the diagram
Hom C ( Z, Y ) Θ Z , Y / / Hom C ( h ,Y ) (cid:15) (cid:15) D Ext C ( Y, τ Z ) D Ext C ( Y,τ
X,Z ( h )) (cid:15) (cid:15) Hom C ( X, Y ) Θ X , Y / / D Ext C ( Y, τ X )is commutative. Indeed, for any u ∈ Hom C ( Z, Y ) and δ ∈ Ext C ( Y, τ Z ), we obtain D Ext C ( Y, τ
X,Z ( h ))( Θ Z,Y ( u ))( δ ) = Θ Z,Y ( u )( τ X,Z ( h ) · δ ) = θ Z ( τ X,Z ( h ) · δ · u ) . On the other hand, applying the definition of Θ X,Y and the equation ( ∗ ), we obtain Θ X,Y ( uh )( δ ) = θ X (( δ · u ) · h ) = θ Z ( τ X,Z ( h ) · δ · u ) . This shows that Θ X,Y is natural in X . This establishes Statement (1).Next, assume that the right Auslander-Reiten functor τ : C → C is an equiv-lence. Since τ is dense, C has almost split sequences on the left. Let X, Y ∈ C be indecomposable. If Y is Ext-injective, then Ext C ( X, Y ) = 0. Otherwise, weobtain Y ∼ = τ Z , where Z ∈ C is indecomposable and not Ext-projective. In thiscase, D Ext C ( X, Y ) ∼ = D Ext C ( X, τ Z ) ∼ = Hom C ( Z, X ). Since Hom C ( Z, X ) is reflex-ive, by Lemma 1.3, so are Hom C ( Z, X ) and Ext C ( X, Y ). Being Krull-Schmidt, C isExt-reflexive. The proof of the theorem is completed. Remark.
Theorem 4.8(2) generalizes Lenzing and Zuazua’s result stated in [24,(1.1)] for Ext-finite abelian categories over a commutative artinian ring.
Example.
Let Q be a quiver of type A ∞ with a unique source vertex. Thecategory of finitely presented representations of Q over a field k is a Hom-finiteabelian k -category, which admits a right Auslander-Reiten functor, but no leftAuslander-Reiten functor; see [9, (1.15), (3.7)].Finally, we shall specialize to left or right triangulated R -categories. For thispurpose, we shall modify the classical notion of a Serre functor; see [29, (I.1)].4.9. Definition.
Let C be a tri-exact R -category.(1) A left Serre functor for C is a functor S : C → C with binatural R -linearisomorphisms Φ X,Y : Hom C ( X, Y ) → D Hom C ( S Y, X ) , with X, Y ∈ C . (2) A right Serre functor for C is a functor S : C → C with binatural R -linearisomorphisms Ψ X,Y : Hom C ( X, Y ) → D Hom C ( Y, S X ), with X, Y ∈ C . Remark.
In case C is a triangulated R -category, by Lemma 2.4, our left or rightSerre functors coincide with those given by Reiten and van den Bergh in [29, (I.1)].4.10. Theorem.
Let C be a Hom-reflexive Krull-Schmidt tri-exact R -category. (1) If C is right triangulated, then it has almost split sequences on the right if andonly if it admits a right Auslander-Reiten functor, or equivalently, a right Serrefunctor whose image lies in C [1] . (2) If C is left triangulated, then it has almost split sequences on the left if and onlyif it admits a left Auslander-Reiten functor, or equivalently, a left Serre functorwhose image lies in C [ − . (3) If C is triangulated, then it has almost split sequences if and only if it admits aright or left Serre equivalence, or equivalently, a right or left Serre equivalence. LMOST SPLIT SEQUENCES 23
Proof.
Since Statement (3) is an immediate consequence of Statements (1) and(2), we shall only prove Statement (1). Assume that C is a right triangulatedsubcategory of a triangulated R -category. The first equivalence stated in Statement(1) follows from Proposition 4.7 and Theorem 4.8(2). For the second equivalence,observe that C [1] ⊆ C = C ; see (1.7) and (2.4). Let τ : C → C be a right Auslander-Reiten functor with binatural isomorphisms Θ X,Y : Hom C ( X, Y ) → D Ext C ( Y, τ X ).Since Ext C ( Y, τ X ) = Hom C ( Y, ( τ X )[1]), we see that S = [1] ◦ τ : C → C is aright Serre functor, whose image lies in C [1]. Conversely, let S : C → C be a rightSerre functor with binatural isomorphisms Ψ
X,Y : Hom C ( X, Y ) → D Hom C ( Y, S X ).Suppose that S ( C ) ⊆ C [1]. Then, Hom C ( Y, S X ) = Ext C ( Y, ( S X )[ − X, Y ∈ C .Thus, τ = [ − ◦ S : C → C is a right Auslander-Reiten functor. The proof of thetheorem is completed.
Remark.
Theorem 4.10 generalizes Reiten and van den Bergh’s result stated in[29, (I.2.4)] for a Hom-finite triangulated category over a field.
Example.
Let Λ = kQ/J be a strongly locally finite dimensional algebra over afield k , where Q is a locally finite quiver without infinite paths and J is locallyadmissible. Since all modules in mod b Λ have finite projective and finite injective di-mension, we will see from Theorem 5.12 that D b (mod b Λ ) has almost split sequences.Further, for each n ∈ Z , the right triangulated category D ≤ n (mod b Λ ) has almostsequences on the left, and the left triangulated category D ≥ n (mod b Λ ) has almostsequences on the right. More examples can be found at the end of this paper.5. Almost split triangles in derived categories
The main objective of this section is to study almost split triangles in the derivedcategories of an abelian category with enough projective objects and enough in-jective objects. Our results are applicable to the derived categories of modulescategories over an algebra with a unity or a locally finite dimension algebra givenby a quiver with relations. In particular, they include Happel’s result obtained in[17] for the bounded derived category of finite dimensional modules over a finite di-mensional modules. We shall start with an arbitrary abelian category A and quotethe following well-known statement; see, for example, [31, (10.4.7)].5.1. Lemma.
Let X . , Y . be complexes over an abelian category A . If X . is bounded-above of projective objects or Y . is bounded-below of injective objects, then thereexists an isomorphism L X . ,Y . : Hom K ( A ) ( X . , Y . ) → Hom D ( A ) ( X . , Y . ) , which isinduced from the localization functor L : K ( A ) → D ( A ) . Let P and I be strictly additive subcategories of A of projective objects and ofinjective objects, respectively. By Lemma 5.1, we can view K b ( P ) and K b ( I ) as fullsubcategories of D b ( A ). A projective resolution over P of a complex Z . ∈ C − ( A ) isa quasi-isomorphism s . : P . → Z . with P . ∈ C − ( P ) , which is finite if P . ∈ C b ( P ) . Dually, an injective co-resolution over I of a complex X . ∈ C + ( A ) is a quasi-isomorphism t . : X . → I . with I . ∈ C + ( I ) , which is finite if I . ∈ C b ( I ) . Theorem.
Let A be an abelian category such that D ∗ ( A ) with ∗ ∈ {∅ , + , − , b } has an almost split triangle X . / / Y . / / Z . / / X . [1] , where X . is a bounded-below complex and Z . is a bounded-above complex. (1) If Z . admits a projective resolution over a strictly additive subcategory P ofprojective objects of A , then it admits a finite projective resolution over P . (2) If X . admits an injective co-resolution over a strictly additive subcategory I ofinjective objects of A , then it admits a finite injective co-resolution over I .Proof. We shall view D ∗ ( A ) as a full triangulated subcategory of D ( A ); see [27,Chapter III]. Let P be a strictly additive subcategory of projective objects A with s . : P . → Z . a quasi-isomorphism with P . ∈ C − ( P ). Write W . = X . [1], a complexin D + ( A ) ∩ D ∗ ( A ). Let n be an integer such that W i = 0 for all i < n . Write δ . : Z . → X . [1] for the third morphism in the almost split triangle stated in thetheorem. By Lemma 5.1, δ . ˜ s . = ˜ t . for some complex morphism t . : P . → W . .Consider the brutal truncation κ ≥ n ( P . ) and the associated canonical morphism µ . : κ ≥ n ( P . ) → P . . Being bounded, κ ≥ n ( P . ) lies in D ∗ ( A ). We claim that ¯ µ . is aretraction in K ( A ). Otherwise, by Lemma 5.1, ˜ s . ˜ µ . is not a retraction in D ∗ ( A ),and hence, ˜ t . ˜ µ . = δ . (˜ s . ˜ µ . ) = 0. By Lemma 5.1, ¯ t . ¯ µ . = 0. In particular, there existmorphisms h i : P i → W i − with i ≥ n such that t i = t i µ i = h i +1 d iP + d i +1 W h i , forall i ≥ n. Setting h i = 0 : P i → W i − for i < n , we obtain t i = h i +1 d iP + d i +1 W h i , for all i ∈ Z . That is, ¯ t . = 0 , and hence, δ . = 0, a contradiction. This establishesour claim. In particular, H i ( P . ) = 0 for all i < n .Let u . : P . → κ ≥ n ( P . ) be a complex morphism such that ¯ µ . ¯ u . = ¯1 P . . Then,there exist f i +1 : P i +1 → P i such that 1 P i − u i = f i +1 d iP + d i − P f i , for i ∈ Z . Inparticular, 1 P n − = f n d n − P + d n − P f n − and d n − P = d n − P f n d n − P . Write d n − P = jv, where v : P n − → C is the cokernel of d n − P . Since Im( d n − P ) = Ker( d n − P ), by theSnake Lemma, j : C → P n is a monomorphism. Since ju = juf n ju , we obtain1 Q = ( uf n ) j , and hence, C ∈ P . Thus, the smart truncation τ ≥ n ( P . ) lies in C b ( P ).Since H i ( P . ) = 0 for all i < n , the canonical projection p . : P . → τ ≥ n ( P . ) is a quasi-isomorphism; see [27, (III.3.4.2)]. Therefore, Z . ∼ = τ ≥ n ( P . ) in D ∗ ( A ). By Lemma5.1, τ ≥ n ( P . ) is a finite projective resolution of Z . over P . Dually, we may establishStatement (2). The proof of the theorem is completed.If A has enough projective (respectively, injective) objects, then every bounded-above (respectively, bounded-below) complex over A admits a projective resolution(respectively, injective co-resolution); see [11, (7.5)].5.3. Corollary.
Let A be an abelian category such that D b ( A ) has an almost splittriangle X . / / Y . / / Z . / / X . [1] . (1) If A has enough projective objects, then Z . has a finite projective resolution. (2) If A has enough injective objects, then X . has a finite injective co-resolution. Example.
Given any ring Σ , Corollary 5.3 applies in D b (Mod Σ ); and if Σ isnoetherian, then Corollary 5.3(1) applies in D b (mod + Σ ).Next, we shall obtain some sufficient conditions for the existence of an almostsplit triangle in the derived categories of A . For this purpose, we need to assumethat A is an abelian R -category and consider D = Hom R ( − , I R ) : Mod R → Mod R, where I R is a minimal injective co-generator for Mod R .5.4. Definition.
Let A be an abelian R -category. Given P a strictly additive sub-category of projective objects of A , a functor ν : P → A is called a Nakayama func-tor if there exist binatural isomorphisms β P,X : Hom A ( X, νP ) → D Hom A ( P, X ),for all P ∈ P and X ∈ A . LMOST SPLIT SEQUENCES 25
Remark.
Given a Nakayama functor ν : P → A , we see easily that νP is aninjective object of A , for every P ∈ P . Hence ν P , the image of P under ν , is astrictly additive subcategory of injective objects of A .As an example, we have the following probably known statement.5.5. Lemma.
Let A be an R -algebra. Then ν A = D Hom A ( − , A ) : proj A → Mod A is a Nakayama functor for Mod A .Proof. Given P ∈ proj A and X ∈ Mod A , it is well known; see [1, (20.10)] thatthere exists a binatural R -linear isomoprhism η P,X : Hom A ( P, A ) ⊗ A X → Hom A ( P, X ) : f ⊗ x [ u f ( u ) x ] . Considering the R - A -bimodule Hom A ( P, A ) and the adjoint isomorphism, we obtainthe following binatural isomorphismsHom R (Hom A ( P, X ) , I R ) ∼ −→ Hom R (Hom A ( P, A ) ⊗ A X, I R ) ∼ −→ Hom A ( X, Hom R (Hom A ( P, A ) , I R )) . The proof of the lemma is completed.The following statement collects some properties of a Nakayama functor.5.6.
Lemma.
Let A be an abelian category with P a strictly additive subcategory ofprojective objects of A . Then every Nakayama functor ν : P → A is faithful, and itis fully faithful in case P is Hom-reflexive over R .Proof. Let ν : P → A be a Nakayama functor with binatural R -linear isomorphisms β P,X : Hom A ( X, νP ) → D Hom A ( P, X ) , where P ∈ P and X ∈ A . Fix two objects L, P ∈ P . Given f : L → P and g : P → νL , considering the commutative diagramHom A ( νL, νP ) β P,νL (cid:15) (cid:15)
Hom A ( νL, νL ) ( gf,νL ) / / β L,νL (cid:15) (cid:15) ( νL,νf ) o o Hom A ( L, νL ) β L,L (cid:15) (cid:15)
Hom A ( P, νL ) β L,P (cid:15) (cid:15) ( f,νL ) o o D Hom A ( P, νL ) D Hom A ( L, νL ) D ( L,gf ) / / D ( f,νL ) o o D Hom A ( L, L ) D Hom A ( L, P ) , D ( L,f ) o o we obtain the following equations( ∗ ) β P,νL ( νf )( g ) = β L,νL (1 νL )( gf ) = β L,L ( gf )(1 L ) = β L,P ( g )( f ) . We claim these equations imply the commutativity of the diagramHom A ( L, P ) σ / / ν (cid:15) (cid:15) D Hom A ( L, P ) D ( β L,P ) (cid:15) (cid:15) Hom A ( νL, νP ) β P,νL / / D Hom A ( P, νL ) , where σ is the canonical injection. Indeed, using the equations in ( ∗ ), we see that D ( β L,P )( σ ( f ))( g ) = σ ( f )( β L,P ( g )) = β L,P ( g )( f ) = β P,νL ( νf )( g ) . As a consequence, ν : Hom A ( L, P ) → Hom A ( νL, νP ) is a monomorphism, and it isan isomorphism if Hom A ( L, P ) is reflexive. The proof of the lemma is completed.
Remark. If P is Hom-reflexive over R , then every Nakayama functor ν : P → A co-restricts an equivalence ν : P → ν P . In this case, we shall always denote by ν - : ν P → P a quasi-inverse of ν : P → ν P . Example.
Let Λ = kQ/I be a locally finite dimensional algebra over a field k ,where Q is locally finite and J is weakly admissible. Then proj Λ is Hom-finite over k ; see [12, (3.2)], and we have a Nakayama functor ν Λ : proj Λ → Mod Λ , sending P x to I x ; see [12, (3.2), (3.6)]. This yields an equivalence ν Λ : proj Λ → inj Λ with aquasi-inverse ν - Λ : inj Λ → proj Λ , sending I x to P x .Let F : A → B be a functor between additive categories. Applying F component-wise, one may extend F to a functor C ( A ) → C ( B ), sending null-homotopic mor-phisms to null-homotopic ones and cones to cones. The latter functor induces atriangle-exact functor K ( A ) → K ( B ); see [27, (V.1.1.1)]. For the simplicity of no-tation, these functors will be written as F : C ( A ) → C ( B ) and F : K ( A ) → K ( B ).5.7. Proposition.
Let A be an abelian R -category admitting a Nakayama functor ν : P → A , where P is a strictly additive subcategory of projective objects of A . (1) The triangle-exact functor ν : K ( P ) → K ( A ) restricts to a triangle-exact functor ν : K b ( P ) → K b ( ν P ) , which is an equivalence if P is Hom-reflexive over R . (2) Given a complex X . over A and a bounded complex P . over P , we obtain a bina-tural R -linear isomorphism ˜ β P . ,X . : Hom D ( A ) ( X . , νP . ) → D Hom D ( A ) ( P . , X . ) . Proof. If P is Hom-reflexive over R , then ν : P → ν P is an equivalence; see(5.6), which clearly induces an equivalence ν : K b ( P ) → K b ( ν P ). It remains toprove Statement (2). By definition, we obtain binatural R -linear isomorphisms β P,X : Hom A ( X, νP ) → D Hom A ( P, X ), for all P ∈ P and X ∈ A .Fix a complex X . over A and a bounded complex P . over P . We may define an R -linear map β P . ,X . : Hom C ( A ) ( X . , νP . ) → D Hom C ( A ) ( P . , X . ) by setting β P . ,X . ( ξ . )( ζ . ) = P i ∈ Z ( − i β Pi,Xi ( ξ i )( ζ i ) , for ξ . : X . → νP . and ζ . : P . → X . in C ( A ) . Using the binaturality of β P,X , wesee that β P . ,X . is binatural and β P . ,X . ( ξ . )( ζ . ) = 0 if ξ . or ζ . is null-homotopic. Thisinduces binatural R -linear maps ¯ β P . ,X . : Hom K ( A ) ( X . , νP . ) → D Hom K ( A ) ( P . , X . )such that ¯ β P . ,X . ( ¯ ξ . )(¯ ζ . ) = β P . ,X . ( ξ . )( ζ . ) , which we claim are isomorphisms. Sublemma. If X . is a bounded complex, then ¯ β P . ,X . is an isomorphism. Indeed, we start with the case where w ( X . ) = 1, say X . concentrates at degree0. Suppose first that w ( P . ) = 1 . If P . concentrates at degree 0, then ¯ β P . ,X . canbe identified with β P ,X , which is an isomorphism. Otherwise, ¯ β P . ,X . is a zeroisomorphism. Suppose now that w ( P . ) = s > . By Lemma 1.8, K ( P ) has an exacttriangle Q . / / P . / / L . / / Q . [1] with w ( Q . ) < s and w ( L . ) = 1 . This yieldsan exact triangle νQ . / / νP . / / νL . / / νQ . [1] in K ( A ). Since I R is injective,we obtain a commutative diagram with exact rows( X . , νL . [ − ¯ β L . [ − ,X . (cid:15) (cid:15) / / ( X . , νQ . ) ¯ β Q . ,X . (cid:15) (cid:15) / / ( X . , νP . ) ¯ β P . ,X . (cid:15) (cid:15) / / ( X . , νL . ) ¯ β L . ,X . (cid:15) (cid:15) / / ( X . , νQ . [1]) ¯ β Q . [1] ,X . (cid:15) (cid:15) D ( L . [ − , X . ) / / D ( Q . , X . ) / / D ( P . , X . ) / / D ( L . , X . ) / / D ( Q . [1] , X . ) , where ( M . , N . ) = Hom K ( A ) ( M . , N . ) . By the induction hypothesis on w ( P . ), we seethat ¯ β P . ,X . is an R -linear isomorphism. Consider next the case where w ( X . ) = t > . By Lemma 1.8, K ( A ) has an exact triangle Z . / / X . / / Y . / / Z . [1] , where w ( Z . ) < t and w ( Y . ) = 1 . This yields a commutative diagram with exact rows( Z . [ − , νP . ) ¯ β P . ,Z . [ − (cid:15) (cid:15) / / ( Y . , νP . ) ¯ β P . ,Y . (cid:15) (cid:15) / / ( X . , νP . ) ¯ β P . ,X . (cid:15) (cid:15) / / ( Z . , νP . ) ¯ β P . ,Z . (cid:15) (cid:15) / / ( Y . [1] , νP . ) ¯ β P . ,Y . [1] (cid:15) (cid:15) D ( P . , Z . [ − / / D ( P . , Y . ) / / D ( P . , X . ) / / D ( P . , Z . ) / / D ( P . , Y . [1]) . LMOST SPLIT SEQUENCES 27
By the induction hypothesis, ¯ β P . ,X . is an isomorphism. This proves the sublemma.In general, assume that P i = 0 for i [ m, n ], where m < n . Considering thebrutal truncations M . = κ ≥ m ( X . ) and N . = κ ≤ n ( M . ) with canonical morphisms µ . : M . → X . and π . : M . → N . , we obtain a commutative diagramHom K ( A ) ( X . , νP . ) ¯ β P . ,X . (cid:15) (cid:15) Hom(¯ µ . , νP . ) / / Hom K ( A ) ( M . , νP . ) ¯ β P . , M . (cid:15) (cid:15) Hom K ( A ) ( N . , νP . ) ¯ β P . ,N . (cid:15) (cid:15) Hom(¯ π . , νP . ) o o D Hom K ( A ) ( P . , X . ) D Hom( P . , ¯ µ . ) / / D Hom K ( A ) ( P . , M . ) D Hom K ( A ) ( P . , N . ) . D Hom( P . , ¯ π . ) o o Since P i = 0 for i [ m, n ], it is not difficulty to see that the horizontal mapsare R -linear isomorphisms. Since N . is bounded, by the sublemma, ¯ β P . ,N . is anisomorphism, and so are ¯ β P . , M . and ¯ β P . ,X . . This establishes our claim. Then, byLemma 5.1, we obtain a binatural R -linear isomorphism˜ β P . ,X . = D ( L − P . ,X . ) ◦ ¯ β P . ,X . ◦ L − X . ,νP . : Hom D ( A ) ( X . , νP . ) → D Hom D ( A ) ( P . , X . ) . The proof of the proposition is completed.
Remark.
The isomorphism stated in Proposition 5.7(2) is known for the boundedderived category of a finite dimensional algebra; see [16, Page 350].We are ready to obtain a sufficient condition for the existence of an almost splittriangle in the derived categories of an abelian category with a Nakayama functor.5.8.
Theorem.
Let A be an abelian R -category with P a strictly additive subcate-gory of projective objects of A and ν : P → A a Nakayama functor. If P . ∈ K b ( P ) and νP . ∈ K b ( ν P ) are strongly indecomposable, then D b ( A ) has an almost splittriangle νP . [ − / / M . / / P . / / νP . , which is also almost split in D ( A ) .Proof. Assume that P . ∈ K b ( P ) and νP . ∈ K b ( ν P ) are strongly indecomposable.By Lemma 5.1, P . and νP . are strongly indecomposable in D b ( A ). In view ofProposition 5.7(2), we obtain an isomorphismΦ : Ext D ( A ) ( − , νP . [ − D ( A ) ( − , νP . ) → D Hom D ( A ) ( P . , − ) , which restricts to an isomorphismΨ : Ext D b ( A ) ( − , νP . [ − D b ( A ) ( − , νP . ) → D Hom D b ( A ) ( P . , − ) . Choose a non-zero R -linear form θ : End D b ( A ) ( P . ) → I R , which vanishes on theradical of End D b ( A ) ( P . ). Then, θ = Ψ P . ( δ . ) for some δ . ∈ Ext D b ( A ) ( P . , νP . [ − θ is in the right End D b ( A ) ( P . )-socle of D End D b ( A ) ( P . ), by Theorem 3.6, δ . is an almost-zero extension in D b ( A ). On the other hand, since D b ( A ) is a fulltriangulated subcategory of D ( A ), we see that δ . ∈ Ext D ( A ) ( P . , νP . [ − P . ( δ . ) = θ , which lies in the right End D ( A ) ( P . )-socle of D End D ( A ) ( P . ). Hence, δ . is an almost-zero extension in D ( A ). Therefore, δ . defines an almost split triangle νP . [ − / / M . / / P . / / νP . in D b ( A ), which is also an almost split trianglein D ( A ). The proof of the theorem is completed. Example.
Let A be an R -algebra. By Lemma 5.5, there exists a Nakayama functor ν A : proj A → Mod A , and hence, Theorem 5.8 applies in D (Mod A ).As an application of Theorem 5.8, we shall describe some almost split trianglesin the derived categories of all modules over a locally finite dimensional algebra.5.9. Corollary.
Let Λ = kQ/J be a locally finite dimensional algebra over a field k , where Q is locally finite and J is weakly admissible. (1) If P . ∈ K b (proj Λ ) is indecomposable, then D b (Mod Λ ) has an almost split tri-angle ν Λ P . [ − / / M . / / P . / / ν Λ P . , which is almost split in D (Mod Λ ) . (2) If I . ∈ K b (inj Λ ) is indecomposable, then D b (Mod Λ ) has an almost split triangle I . / / M . / / ν - Λ I . [1] / / I . [1] , which is almost split in D (Mod Λ ) .Proof. Since A is locally finite dimensional, proj Λ is Hom-finite; see [12, (3.2)], andso is K b (proj Λ ). Since the idempotents in D b (Mod Λ ) split; see [23, Corollary A],so do the idempotents in K b (proj Λ ). Thus, K b (proj Λ ) is Krull-Schmidt; see [26,(1.1)]. Moreover, the Nakayama functor ν Λ : proj Λ → Mod Λ induces an equivalence ν Λ : K b (proj Λ ) → K b (inj Λ ) with a quasi-inverse ν - Λ : K b (inj Λ ) → K b (proj Λ ); see(5.7). If P . ∈ K b (proj Λ ) is indecomposable, then so is ν Λ P . ∈ K b (inj Λ ). ByTheorem 5.8, there exists an almost split triangle as stated in Statement (1). Onthe other hand, if I . ∈ K b (inj Λ ) is indecomposable, then so is ν - Λ I . [1] ∈ K b (proj Λ ).By Theorem 5.8, there exists an almost split triangle as stated in Statement (2).The proof of the corollary is completed.Similarly, we can describe some almost split triangles in the derived categoriesof all modules over a reflexive noetherian algebra.5.10. Theorem.
Let A be a reflexive noetherian R -algebra. Consider a stronglyindecomposable complex M . ∈ D b (Mod A ) . (1) If M . is a complex over mod + A , then D b (Mod A ) has an almost split triangle N . / / L . / / M . / / N . [1] if and only if M . has a finite projective resolution P . over proj A ; and in this case, N . ∼ = ν A P . [ − , a complex over mod − A. (2) If M . is a complex over mod − A , then D b (Mod A ) has an almost split trian-gle M . / / L . / / N . / / M . [1] if and only if M . has a finite injective co-resolution I . over inj A ; and in this case, N . ∼ = ν - A I . [1] , a complex over mod + A .Proof. Since A A is R -reflexive, by Lemma 1.3, we see that proj A is Hom-reflexiveover R . Thus, by Proposition 5.7(1), the Nakayama functor ν A : proj A → Mod Λ induces an equivalence ν A : K b (proj A ) → K b (inj A ), which has a quasi-inverse ν - A : K b (inj A ) → K b (proj A ). In particular, a complex P . ∈ K b (proj A ) is stronglyindecomposable if and only if ν A P . ∈ K b (inj A ) is strongly indecomposable. More-over, by Theorem 1.4, mod + A is an abelian category with enough projective modulesin proj A and mod − A is an abelian category with enough injective modules in inj A ,and consequently, every bounded complex over mod + A has a projective resolutionover proj A and every bounded complex over mod − A has an injective co-resolutionover inj A ; see [11, (7.5)]. Now, Statements (1) and (2) follow immediately fromTheorems 5.2 and 5.8. The proof of the theorem is completed. Example. If R is a product of noetherian complete local commutative rings,then every noetherian R -algebra is reflexive; see [4, Section 5].In case R is noetherian complete local, the finiteness of the global dimension ofa noetherian R -algebra is related to the existence of almost split triangles in itsderived category.5.11. Corollary.
Let A be a noetherian R -algebra, where R is a product of com-mutative noetherian complete local rings. The following statements are equivalent. (1) The global dimension of A is finite. (2) Every indecomposable complex in D b (mod + A ) is the ending term of an almostsplit triangle in D b (Mod A ) . LMOST SPLIT SEQUENCES 29 (3)
Every indecomposable object in D b (mod − A ) is the starting term of an almostsplit triangle in D b (Mod A ) . Proof.
First of all, A is a reflexive noetherian R -algebra. Moreover, mod + A is aKrull-Schmidt abelian subcategory of RMod A ; see [4, Section 5], and by Theorem1.4(2), so is mod − A . Thus, D b (mod A ) and D b (mod- A ) are Krull-Schmidt; see [23,Corollary B]. Since mod + A has enough projective modules in proj A and mod − A hasenough injective modules in inj A , we see that D b (mod A ) and D b (mod- A ) are fulltriangulated subcategories of D (Mod A ); see [8, (1.11)].Let A be of finite global dimension. Then, every bounded complex over mod + A has a finite projective resolution over proj A and every bounded complex over mod- A has a finite injective co-resolution over inj A ; see [11, (7.5)]. Thus, Statements (2)and (3) follow from Theorem 5.10. Conversely, assume that Statement (3) holds.Since mod − A is Krull-Schmidt, we deduce from Theorem 5.10(2) that every modulein mod − A is of finite injective dimension. By Proposition 1.4(2), every modulein mod + A op is of finite projective dimension, and hence, A op is of finite globaldimension; see [30, (9.12)]. Being left and right noetherian as a ring, A is of finiteglobal dimension; see [30, (9.23)]. The proof of the corollary is completed.To conclude, we shall describe all possible almost split triangles in the boundedderived category of an abelian category with a Nakayama functor, enough projectiveobjects and enough injective objects.5.12. Theorem.
Let A be an abelian R -category with a Nakayama functor ν : P → A ,where P is a Hom-reflexive strictly additive subcategory of projective objects of A such that A has enough projective objects in P and enough injective objects in ν P . (1) If Z . ∈ D b ( A ) is strongly indecomposable, then there exists an almost splittriangle X . / / Y . / / Z . / / X . [1] in D b ( A ) if and only if Z . has a finiteprojective resolution P . over P ; and in this case, X . ∼ = νP . . (2) If X . ∈ D b ( A ) is strongly indecomposable, then there exists an almost splittriangle X . / / Y . / / Z . / / X . [1] in D b ( A ) if and only if X . has a finiteinjective co-resolution I . over ν P ; and in this case, Z . ∼ = ν - I . . (3) If every object in A has a finite projective resolution over P ( respectively, injec-tive co-resolution over ν P ) , then D b ( A ) has almost split triangles on the right ( respectively, left ); and the converse holds in case A is Krull-Schmidt.Proof. By Lemma 5.7(1), the Nakayama functor ν : P → A induces an equivalence ν : K b ( P ) → K b ( ν P ) with a quasi-inverse ν - : K b ( ν P ) → K b ( P ) . Since A hasenough projective objects in P and enough injective objects in ν P , every boundedcomplex over A has a projective resolution over P and an injective co-resolutionover ν P ; see [11, (7.5)]. In view of Theorems 5.2 and 5.8, we see easily that thefirst two statements hold true.Next, assume that every object in A has a finite projective resolution over P .Then, every bounded complex over A has a finite resolution over P ; see [11, (7.5)].By Statement (1), D b ( A ) has almost split triangles on the right. Conversely, sup-pose that A has almost split triangles on the right. In particular, every stronglyindecomposable object in A is the ending term of an almost split triangle in D b ( A ),and by Statement (1), it has a finite projective resolution over P . If A in additionis Krull-Schmidt, then every object in A has a finite projective resolution over P .This proves the first part of Statement (3), and the second part follows dually. Theproof of the theorem is completed. Example. (1) Let A be an artin algebra over a commutative artinian ring R .Then mod A is a Hom-finite abelian R -category with enough projective modules andenough injective modules. Considering the Nakayama functor ν A : proj A → mod A ,we see that Theorem 5.12 applies in D b (mod A ), and in particular, it includesHappel’s results stated in [17].(2) Let Λ = kQ/J be a strongly locally finite dimensional algebra over a field k ,where Q is locally finite and J is locally admissible. Then mod b Λ is a Hom-finiteabelian k -category with enough projective modules in proj Λ and enough injectivemodules in inj Λ . Considering the Nakayama functor ν Λ : proj Λ → mod b Λ , we seethat Theorem 5.12 applies in D b (mod b Λ ). In case Q has no infinite path, everymodule in mod b Λ has a finite projective dimension and a finite injective dimension,and by Theorem 5.12(3), D b (mod b Λ ) has almost split triangles.(3) Let Λ = kQ/J , where Q is a locally finite quiver and J is the ideal in kQ generated by the paths of length two. Then, every module in mod b Λ is of finiteprojective dimension over proj Λ if and only if Q has no right infinite path, andevery module in mod b Λ is of finite injective dimension over inj Λ if and only if Q has no left infinite path. By Theorem 5.12, D b (mod b Λ ) has almost split triangles(on the left, on the right) if and only if Q has no (left, right) infinite path. References [1]
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Shiping Liu, D´epartement de math´ematiques, Universit´e de Sherbrooke, Sherbrooke,Qu´ebec, Canada
E-mail address : [email protected] Hongwei Niu, D´epartement de math´ematiques, Universit´e de Sherbrooke, Sherbrooke,Qu´ebec, Canada
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