Auslander-Reiten translations in monomorphism categories
aa r X i v : . [ m a t h . R T ] J a n AUSLANDER-REITEN TRANSLATIONSIN MONOMORPHISM CATEGORIES
BAO-LIN XIONG PU ZHANG ∗ YUE-HUI ZHANG
Abstract.
We generalize Ringel and Schmidmeier’s theory on the Auslander-Reitentranslation of the submodule category S ( A ) to the monomorphism category S n ( A ). Asin the case of n = 2, S n ( A ) has Auslander-Reiten sequences, and the Auslander-Reitentranslation τ S of S n ( A ) can be explicitly formulated via τ of A -mod. Furthermore, if A is a selfinjective algebra, we study the periodicity of τ S on the objects of S n ( A ), andof the Serre functor F S on the objects of the stable monomorphism category S n ( A ).In particular, τ m ( n +1) S X ∼ = X for X ∈ S n (Λ( m, t )); and F m ( n +1) S X ∼ = X for X ∈S n (Λ( m, t )), where Λ( m, t ) , m ≥ , t ≥ , are the selfinjective Nakayama algebras. Key words and phrases. monomorphism category, Auslander-Reiten translation, tri-angulated category, Serre functor
Introduction
Throughout this paper, n ≥ A an Artin algebra, and A -mod the categoryof finitely generated left A -modules. Let S n ( A ) denote the monomorphism category of A (it is usually called the submodule category if n = 2).The study of such a category goes back to G. Birkhoff [B], in which he initiates toclassify the indecomposable objects of S ( Z / h p t i ) (see also [RW]). In [Ar], S n ( R ) is de-noted by C ( n, R ), where R is a commutative uniserial ring; the complete list of C ( n, R )of finite type, and of the representation types of C ( n, k [ x ] / h x t i ), are given by D. Simson[S] (see also [SW]). Recently, after the deep and systematic work of C. M. Ringel and M.Schmidmeier ([RS1] - [RS3]), the monomorphism category receives more attention. X. W.Chen [C] shows that S ( A ) of a Frobenius abelian category A is a Frobenius exact cat-egory. D. Kussin, H. Lenzing, and H. Meltzer [KLM] establish a surprising link betweenthe stable submodule category with the singularity theory via weighted projective linesof type (2 , , p ). In [Z], S n ( X ) is studied for any full subcategory X of A -mod, and it isproved that for a cotilting A -module T , there is a cotilting T n ( A )-module m ( T ) such that S n ( ⊥ T ) = ⊥ m ( T ), where T n ( A ) = A A ··· A A ··· A ... ... ... ... ··· A ! n × n is the upper triangular matrix algebra ∗ The corresponding author.
Supported by the NSF of China (10725104), and STCSM (09XD1402500). of A , and ⊥ T is the left perpendicular category of T . As a consequence, for a Gorensteinalgebra A , S n ( ⊥ A ) is exactly the category of Gorenstein-projective T n ( A )-modules.Ringel and Schmidmeier construct minimal monomorphisms, and then prove that S ( A )is functorially finite in T ( A )-mod. As a result, S ( A ) has Auslander-Reiten sequences.Surprisingly, the Auslander-Reiten translation τ S of S ( A ) can be explicitly formulated as τ S X ∼ = Mimo τ Cok X for X ∈ S ( A ) ([RS2], Theorem 5.1), where τ is the Auslander-Reiten translation of A -mod. Applying this to selfinjective algebras, among others they get τ S X ∼ = X for indecomposable nonprojective object X ∈ S ( A ), where A is a commutativeuniserial algebra.A beautiful theory should have a general version. The aim of this paper is to generalizeRingel and Schmidmeier’s work on S ( A ) to S n ( A ). As in the case of n = 2, S n ( A ) hasAuslander-Reiten sequences, and τ S of S n ( A ) can be formulated in the same form as above:these can be achieved by using the idea in [RS2]. For selfinjective algebras, Sections 3 and4 of this paper contain new considerations. In order to express the higher power of τ S ,we need the concept of a rotation of an object in Mor n ( A -mod), which is defined in [RS2]for n = 2. In the general case, such a suitable definition needs to be chosen from differentpossibilities, and difficulties need to be overcome to justify that it is well-defined. Also,the Octahedral Axiom is needed in computing the higher power of the rotations, which isthe key step in studying the periodicity of τ S and the Serre functor on the objects.We outline this paper. In Section 1 we set up some basic properties of the categoriesMor n ( A ), S n ( A ) and F n ( A ), and of the functors m i , p i , Ker , Cok , Mono , Epi ; and theconstruction of
Mimo . Section 2 is to transfer the Auslander-Reiten sequences of Mor n ( A )to those of S n ( A ); and to give a formula for τ S of S n ( A ) via τ of A -mod (Theorem 2.4).In Section 3, A is a selfinjective algebra, and hence the stable category A -mod is atrianglated category ([H]), and τ is a triangle functor of A -mod. Using the rotation andthe Octahedral Axiom, we get a formula for τ j ( n +1) S X ∈ Mor n ( A -mod) for X ∈ S n ( A )and j ≥ τ S on objects. In particular, τ m ( n +1) S X ∼ = X for X ∈ S n (Λ( m, t )) (Corollary 3.6), whereΛ( m, t ) , m ≥ , t ≥ , are the selfinjective Nakayama algebras.In Section 4, A is a finite-dimensional self-injective algebra over a field. By [Z], S n ( A )is exactly the category of Gorenstein projective T n ( A )-modules, and hence the stablemonomorphism category S n ( A ) is a Hom-finite Krull-Schmidt triangulated category withAuslander-Reiten triangles. By Theorem I.2.4 of I. Reiten and M. Van den Bergh [RV], S n ( A ) has a Serre functor F S . We study the periodicity of F S on the objects of S n ( A )(Theorem 4.3). In particular, F m ( n +1) S X ∼ = X for X ∈ S n (Λ( m, t )) (Corollary 4.4).In order to make the main clue clearer, we put the proofs of Lemmas 1.5 and 1.6 inAppendix 1. USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 3
Note that S n, , S , , S , , S , , S , and S , are the only representation-finite casesamong all S n,t = S n ( k [ x ] / h x t i ) , n ≥ , t ≥ S ,t with t = 2 , , , S n (Λ(2 , n = 3 and 4.1. Basics of morphism categories
We set up some basic properties of several categories and functors, which will be usedthroughout this paper.1.1. An object of the morphism category
Mor n ( A ) is X ( φ i ) = X ... X n ! ( φ i ) , where φ i : X i +1 → X i are A -maps for 1 ≤ i ≤ n −
1; and a morphism f = ( f i ) : X ( φ i ) → Y ( ψ i ) is f ... f n ! , where f i : X i → Y i are A -maps for 1 ≤ i ≤ n , such that the following diagramcommutes X nf n (cid:15) (cid:15) φ n − / / X n − f n − (cid:15) (cid:15) / / · · · / / X φ / / f (cid:15) (cid:15) X f (cid:15) (cid:15) Y n ψ n − / / Y n − / / · · · / / Y ψ / / Y . (1 . X i the i -th branch of X ( φ i ) , and φ i the i -th morphism of X ( φ i ) . It is well-knownthat Mor n ( A ) is equivalent to T n ( A )-mod (see e.g. [Z], 1.4). Let Z ( θ i ) f → Y ( ψ i ) g → X ( φ i ) bea sequence in Mor n ( A ). Then it is exact at Y ( ψ i ) if and only if each sequence Z i f i → Y i g i → X i in A -mod is exact at Y i for each 1 ≤ i ≤ n .By definition, the monomorphism category S n ( A ) is the full subcategory of Mor n ( A )consisting of the objects X ( φ i ) , where φ i : X i +1 → X i are monomorphisms for 1 ≤ i ≤ n − the epimorphism category F n ( A ) is the full subcategory of Mor n ( A ) consisting ofthe objects X ( φ i ) , where φ i : X i +1 → X i are epimorphisms for 1 ≤ i ≤ n −
1. Since S n ( A ) and F n ( A ) are closed under direct summands and extensions, it follows that theyare exact Krull-Schmidt categories, with the exact structure in Mor n ( A ). The kernel functor
Ker : Mor n ( A ) → S n ( A ) is given by X X ... X n − X n ( φ i ) X n Ker( φ ··· φ n − ) ... Ker( φ n − φ n − )Ker φ n − ( φ ′ i ) , where φ ′ : Ker( φ · · · φ n − ) ֒ → X n , and φ ′ i : Ker( φ i · · · φ n − ) ֒ → Ker( φ i − · · · φ n − ) , ≤ i ≤ n − , are the canonical monomorphisms. For a morphism f : X → Y in Mor n ( A ), Ker f : Ker X → Ker Y is naturally defined via a commutative diagram induced from BAO-LIN XIONG PU ZHANG YUE-HUI ZHANG (1.1). We also need the cokernel functor
Cok : Mor n ( A ) → F n ( A ) given by X X ... X n − X n ( φ i ) Coker φ Coker( φ φ ) ... Coker( φ ··· φ n − ) X ( φ ′′ i ) , where φ ′′ i : Coker( φ · · · φ i +1 ) ։ Coker( φ · · · φ i ) , ≤ i ≤ n − , and φ ′′ n − : X ։ Coker( φ · · · φ n − ) are the canonical epimorphisms. It is clear that the restriction of thekernel functor Ker : F n ( A ) → S n ( A ) is an equivalence with quasi-inverse the restrictionof the cokernel functor Cok : S n ( A ) → F n ( A ).1.2. For each 1 ≤ i ≤ n , the functors m i : A -mod → S n ( A ) and p i : A -mod → F n ( A ) aredefined as follows. For M ∈ A -mod,( m i ( M )) j = M, ≤ j ≤ i ;0 , i + 1 ≤ j ≤ n ; ( p i ( M )) j = , ≤ j ≤ n − i ; M, n − i + 1 ≤ j ≤ n. The j -th morphism of m i ( M ) is id M if 1 ≤ j < i , and 0 if i ≤ j ≤ n −
1; and the j -thmorphism of p i ( M ) is 0 if 1 ≤ j < n − i + 1, and id M if n − i + 1 ≤ j ≤ n −
1. For an A -map f : M → N , define m i ( f ) = f ... f ... : M ... M ... → N ... N ... ; p i ( f ) = ... f ... f : ... M ... M → ... N ... N . We have
Ker p i ( M ) = m n − i +1 ( M ) , Cok m i ( M ) = p n − i +1 ( M ) , ≤ i ≤ n. (1 . Lemma 1.1. ( i ) If P runs over all the indecomposable projective A -modules, then m ( P ) , · · · , m n ( P ) are the all indecomposable projective objects in Mor n ( A ) . ( ii ) If I runs over all the indecomposable injective A -modules, then p ( I ) , · · · , p n ( I ) are the all indecomposable injective objects in Mor n ( A ) . ( iii ) The indecomposable projective objects in S n ( A ) are exactly those in Mor n ( A ) . ( iv ) If I runs over all the indecomposable injective A -modules, then m ( I ) , · · · , m n ( I ) are the all indecomposable injective objects in S n ( A ) . ( v ) If P runs over all the indecomposable projective A -modules, then p ( P ) , · · · , p n ( P ) are the all indecomposable projective objects in F n ( A ) . ( vi ) The indecomposable injective objects in F n ( A ) are exactly those in Mor n ( A ) . ( vii ) Let N M and N be the Nakayama functor of Mor n ( A ) and of A -mod, respectively.Then for a projective A -module P , N M m i ( P ) = p n − i +1 ( N P ) , ≤ i ≤ n . USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 5
Proof.
For convenience, we include a justification. ( i ) can be seen from the equivalenceMor n ( A ) ∼ = T n ( A )-mod. For ( ii ), see e.g. Lemma 1.3( ii ) in [Z]. ( iii ) follows from ( i ), and( vi ) follows from ( ii ). Using the equivalence Ker : F n ( A ) → S n ( A ) together with ( vi )and (1.2), we see ( iv ). Using the equivalence Cok : S n ( A ) → F n ( A ) together with ( iii )and (1.2), we see ( v ). To see ( vii ), note that if P is indecomposable, then N M m i ( P ) isan indecomposable injective T n ( A )-module, hence by ( ii ) it is of the form p j ( I ). Thus ... I )0 ... = soc( p j ( I )) = soc( N M m i ( P )) = top( m i ( P )) = ... P )0 ... . Thus n − j + 1 = i, soc( I ) = top( P ), which means N M m i ( P ) = p n − i +1 ( N P ). (cid:4) Mono : Mor n ( A ) → S n ( A ) and Epi : Mor n ( A ) → F n ( A ). Thefirst one is given by X X ... X n − X n ( φ i ) X Im φ ... Im( φ ··· φ n − )Im( φ ··· φ n − ) ( φ ′ i ) , where φ ′ : Im φ ֒ → X , and φ ′ i : Im( φ · · · φ i ) ֒ → Im( φ · · · φ i − ) , ≤ i ≤ n − , are thecanonical monomorphisms. The second one is given by X X ... X n − X n ( φ i ) Im( φ ··· φ n − )Im( φ ··· φ n − ) ... Im φ n − X n ( φ ′′ i ) , where φ ′′ i : Im( φ i +1 · · · φ n − ) ։ Im( φ i · · · φ n − ) , ≤ i ≤ n − , and φ ′′ n − : X n ։ Im φ n − are the canonical epimorphisms. Then Epi ∼ = Cok Ker , Mono ∼ = Ker Cok , (1 . Mono X ∼ = X for X ∈ S n ( A ), and Epi Y ∼ = Y for Y ∈ F n ( A ).For an A -map f : X → Y , denote the canonical A -maps X ։ Im f and Im f ֒ → Y by e f and incl, respectively. The following lemma can be similarly proved as in [RS2] for n = 2. Lemma 1.2.
Let X = X ( φ i ) ∈ Mor n ( A ) . Then ( i ) The morphism X f φ ... ^ φ ··· φ n − : X ։ Mono X is a left minimal approximation of X in S n ( A ) . BAO-LIN XIONG PU ZHANG YUE-HUI ZHANG ( ii ) The morphism incl ... incl1 Xn : Epi
X ֒ → X is a right minimal approximation of X in F n ( A ) . X ( φ i ) ∈ Mor n ( A ), we define Mimo X ( φ i ) ∈ S n ( A ) and Mepi X ( φ i ) ∈ F n ( A )as follows (see [Z]). For each 1 ≤ i ≤ n − , fix an injective envelope e ′ i : Ker φ i ֒ → IKer φ i . Then we have an A -map e i : X i +1 → IKer φ i , which is an extension of e ′ i . Define Mimo X ( φ i ) to be X ⊕ IKer φ ⊕···⊕ IKer φ n − X ⊕ IKer φ ⊕···⊕ IKer φ n − ... X n − ⊕ IKer φ n − X n ( θ i ) , where θ i = φ i ··· e i ···
00 1 0 ···
00 0 1 ··· ... ... ... ... ... ··· ( n − i +1) × ( n − i ) . By construction
Mimo X ( φ i ) ∈ S n ( A ). Since e , · · · , e n − are not unique, we need to verifythat Mimo X ( φ i ) is well-defined. This can be seen from Lemma 1.3( i ) below.The object Mepi X ( φ i ) is dually defined. Namely, for each 1 ≤ i ≤ n −
1, fix a projectivecover π ′ i : PCoker φ i ։ Coker φ i , then we have an A -map π i : PCoker φ i → X i , which is alift of π ′ i , and define Mepi X ( φ i ) ∈ F n ( A ) to be X X ⊕ PCoker φ ... X n − ⊕ PCoker φ n − ⊕···⊕ PCoker φ X n ⊕ PCoker φ n − ⊕···⊕ PCoker φ ( σ i ) , where σ i = φ i π i ···
00 0 1 0 ···
00 0 0 1 ··· ... ... ... ... ... ... ··· i × ( i +1) . Remark. ( i ) Mimo X = X for X ∈ S n ( A ) , and Mepi Y = Y for Y ∈ F n ( A ) . ( ii ) If each X i has no nonzero injective direct summands, then Mimo X ( φ i ) has nononzero injective direct summands in S n ( A ) . If each X i has no nonzero projective directsummands, then Mepi X ( φ i ) has no nonzero projective direct summands in F n ( A ) . Thesecan be seen from Lemma 1.1 ( iv ) and ( v ) , respectively. Lemma 1.3.
Let X ∈ Mor n ( A ) . Then ( i ) The morphism (1 , , ··· , ... (1 , : Mimo X ։ X is a right minimal approximation of X in S n ( A ) . ( ii ) The morphism ( ) ... ... : X ֒ → Mepi X is a left minimal approximation of X in F n ( A ) . USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 7
For a proof of Lemma 1.3 we refer to [RS2] for n = 2, and to [Z] in general case. ByLemmas 1.2 and 1.3, and by Auslander and Smalø [AS], we get the following consequence Corollary 1.4.
The subcategories S n ( A ) and F n ( A ) are functorially finite in Mor n ( A ) and hence have Auslander-Reiten sequences. This corollary is the starting point of this paper. From now on, denote by τ , τ M , τ S and τ F the Auslander-Reiten translations of A -mod, Mor n ( A ), S n ( A ) and F n ( A ), respectively.1.5. Let A -mod (resp. A -mod) denote the stable category of A -mod modulo projec-tive A -modules (resp. injective A -modules). Then τ = DTr induces an equivalence A -mod → A -mod with quasi-inverse τ − = TrD ([ARS], p.106). Let Mor n ( A -mod) de-note the morphism category of A -mod. Namely, an object of Mor n ( A -mod) is X ( φ i ) = X ( φ i ) = X ... X n ! ( φ i ) with φ i : X i +1 → X i in A -mod for 1 ≤ i ≤ n −
1; and a morphism from X ( φ i ) to Y ( θ i ) is f ... f n ! , such that the corresponding version of (1.1) commutes in A -mod.Similarly, one has the morphism category Mor n ( A -mod), in which an object is denoted by X ( φ i ) = X ( φ i ) .The following two lemmas will be heavily used in Sections 2 and 3. In order to makethe main clue clearer, we put their proofs in Appendix 1. Lemma 1.5.
Let X ( φ i ) ∈ Mor n ( A ) . ( i ) Let I , · · · , I n be injective A -modules such that X ′ ( φ ′ i ) = X ⊕ I ⊕···⊕ I n ... X n − ⊕ I n X n ( φ ′ i ) ∈ S n ( A ) ,where each φ ′ i is of the form φ i ∗ ··· ∗∗ ∗ ··· ∗ ... ... ... ... ∗ ∗ ··· ∗ ! ( n − i +1) × ( n − i ) . Then X ′ ( φ ′ i ) ∼ = Mimo X ( φ i ) ⊕ J ,where J is an injective object of S n ( A ) . ( ii ) Let P , · · · , P n − be projective A -modules such that X ′′ ( φ ′′ i ) = X X ⊕ P ... X n ⊕ P n − ⊕···⊕ P ( φ ′′ i ) ∈F n ( A ) , where each φ ′′ i is of the form φ i ∗ ··· ∗∗ ∗ ··· ∗ ... ... ... ... ∗ ∗ ··· ∗ ! i × ( i +1) . Then X ′′ ( φ ′′ i ) ∼ = Mepi X ( φ i ) ⊕ L ,where L is a projective object of F n ( A ) . Lemma 1.6.
Let X ( φ i ) , Y ( ψ i ) ∈ Mor n ( A ) . ( i ) If all branches X i and Y i have no nonzero injective direct summands, then Mimo X ( φ i ) ∼ = Mimo Y ( ψ i ) in S n ( A ) if and only if X ( φ i ) ∼ = Y ( ψ i ) in Mor n ( A - mod) . BAO-LIN XIONG PU ZHANG YUE-HUI ZHANG ( ii ) If all X i and Y i have no nonzero projective direct summands, then Mepi X ( φ i ) ∼ = Mepi Y ( ψ i ) in F n ( A ) if and only if X ( φ i ) ∼ = Y ( ψ i ) in Mor n ( A - mod) . The Auslander-Reiten translation of S n ( A )In this section, we first transfer the Auslander-Reiten sequences of Mor n ( A ) to thoseof S n ( A ) and F n ( A ); and then give a formula of the Auslander-Reiten translation τ S of S n ( A ) via τ of A -mod. Results and methods in this section are generalizations of thecorresponding ones in the case of n = 2, due to Ringel and Schmidmeier [RS2].2.1. The following fact is crucial for later use. Lemma 2.1.
Let → X ( φ i ) f → Y g → Z → be an Auslander-Reiten sequence of Mor n ( A ) . ( i ) If Ker Z is not projective, then → Ker X Ker f → Ker Y Ker g → Ker Z → is eithersplit exact, or an Auslander-Reiten sequence of S n ( A ) . ( ii ) If Cok X is not injective, then → Cok X Cok f → Cok Y Cok g → Cok Z → is eithersplit exact, or an Auslander-Reiten sequence of F n ( A ) . Proof.
We only prove ( i ). Put g ′ = Ker g and f ′ = Ker f . By Snake Lemma, 0 → Ker X f ′ → Ker Y g ′ → Ker Z is exact. Assume that g ′ is not a split epimorphism. We claimthat g ′ is right almost split. Let v : W → Ker Z be a morphism in S n ( A ) which is not asplit epimorphism. Applying Cok , we get t ′ = Cok v : Cok W → Cok Ker Z = Epi Z ,which is not a split epimorphism, and hence the composition t : Cok W t ′ → Epi Z σ ֒ → Z is not a split epimorphism. So, there is a morphism s : Cok W → Y such that t = gs .Applying Ker , we get
Ker t = g ′ Ker s with Ker s : W → Ker Y . Since Ker σ = id Ker Z ,we see v = g ′ Ker s . This proves the claim.Since g ′ is right almost split and Ker Z is not projective, it follows that g ′ is epic, andhence f ′ is not a split monomorphism. We claim that f ′ is left almost split. For this, let p : Ker X → B be a morphism in S n ( A ) which is not a split monomorphism. Take an injectiveenvelope ( e i ) : B /B ... B /B n ! ( π i ) ֒ → I ... I n − ! ( b ′ i ) in Mor n − ( A ). Put B ′ = I ... I n − B ( b ′ i ) ∈ Mor n ( A ), where b ′ n − is the composition B π n − ։ B /B n e n − ֒ → I n − . By construction weget a morphism e ... e n − id B : Cok B → B ′ , and Cok B = Epi B ′ . Hence Ker B ′ = B .Put r ′ = ( r ′ i ) = Cok p : Epi X → Cok B . Then we have a morphism e r ′ ... e n − r ′ n − ! : USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 9 Im( φ ··· φ n − )Im( φ ··· φ n − ) ... Im φ n − → I ... I n − ! ( b ′ i ) in Mor n − ( A ). By the canonical monomorphism ( j i ) : Im( φ ··· φ n − )Im( φ ··· φ n − ) ... Im φ n − ֒ → X X ... X n − in Mor n − ( A ), we get a morphism r ... r n − ! : X ... X n − ! ( φ i ) → I ... I n − ! ( b ′ i ) in Mor n − ( A ), such that e i r ′ i = r i j i for 1 ≤ i ≤ n −
1. Then we get amorphism r = ( r i ) : X → B ′ in Mor n ( A ) by letting r n = r ′ n (we only need to check b ′ n − r n = r n − φ n − : b ′ n − r n = b ′ n − r ′ n = e n − π n − r ′ n = e n − r ′ n − φ ′ n − = r n − j n − φ ′ n − = r n − φ n − ). By construction we get Epi r = r ′ : Epi X → Epi B ′ = Cok B . This processcan be figured as follows. X nφ ′ n − (cid:15) (cid:15) (cid:15) (cid:15) id Xn u u jjjjjjjjjjj r ′ n / / B π n − (cid:15) (cid:15) (cid:15) (cid:15) id B / / B b ′ n − (cid:15) (cid:15) X n r n = r ′ n cccccccccccccccccccccccccccccccccccccccc φ n − (cid:15) (cid:15) Im φ n − r ′ n − / / φ ′ n − (cid:15) (cid:15) (cid:15) (cid:15) H h j n − u u kkkkkkk B /B nπ n − (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) e n − / / I n − b ′ n − (cid:15) (cid:15) X n − r n − (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) (cid:15) (cid:15) ... (cid:15) (cid:15) (cid:15) (cid:15) ... b ′ (cid:15) (cid:15) Im( φ · · · φ n − ) r ′ / / φ ′ (cid:15) (cid:15) (cid:15) (cid:15) H h j u u kkkkkkkk B /B π (cid:15) (cid:15) (cid:15) (cid:15) (cid:31) (cid:127) e / / I b ′ (cid:15) (cid:15) X φ (cid:15) (cid:15) r Im( φ · · · φ n − ) r ′ / / H h j u u kkkkkkkk B /B (cid:31) (cid:127) e / / I X r Clearly, r is not a spit monomorphism (otherwise, Epi r = r ′ : Epi X → Epi B ′ = Cok B is a spit monomorphism, and hence p = Ker r ′ is a spit monomorphism). So there isa morphism h : Y → B ′ such that r = hf : X → B ′ . Applying Ker , we get
Ker r =( Ker h ) f ′ , where Ker h : Ker Y → Ker B ′ = B. Since
Ker r = Ker Epi r = Ker r ′ = Ker Cok p = p, we get p = ( Ker h ) f ′ . This proves the claim, and completes the proof. (cid:4)
Proposition 2.2.
Let → X f → Y g → Z → be an Auslander-Reiten sequence of Mor n ( A ) . ( i ) If Z ∈ F n ( A ) , and Z is not projective in F n ( A ) , then → Epi X Epi f → Epi Y Epi g → Z → . is an Auslander-Reiten sequence of F n ( A ) ; and → Ker X Ker f → Ker Y Ker g → Ker Z → . is an Auslander-Reiten sequence of S n ( A ) . ( ii ) If X ∈ S n ( A ) , and X is not injective in S n ( A ) , then → X Mono f −→ Mono Y Mono g −→ Mono Z → is an Auslander-Reiten sequence of S n ( A ) ; and → Cok X Cok f → Cok Y Cok g → Cok Z → is an Auslander-Reiten sequence of F n ( A ) . Proof.
We only show ( i ). Since Ker : F n ( A ) → S n ( A ) is an equivalence, and Z isnot projective in F n ( A ), it follows that Ker Z is not projective, and hence by Lemma2.1, (2.2) is either split exact, or an Auslander-Reiten sequence in S n ( A ). Applying theequivalence Cok : S n ( A ) → F n ( A ), and using Epi ∼ = Cok Ker and Z ∈ F n ( A ), wesee that (2.1) is either split exact, or an Auslander-Reiten sequence in F n ( A ). While Epi Y → Z is the composition of the canonical monomorphism Epi
Y ֒ → Y and the rightalmost split morphism Y → Z , so Epi Y → Z is not a split epimorphism, hence (2.1) isan Auslander-Reiten sequence in F n ( A ), so is (2 . (cid:4) τ S and τ M . Corollary 2.3. ( i ) If Z ∈ S n ( A ) , then τ S Z ∼ = Ker τ M Cok Z , and τ −S Z ∼ = Mono τ −M Z . ( ii ) If Z ∈ F n ( A ) , then τ F Z ∼ = Epi τ M Z , and τ −F Z ∼ = Cok τ −M Ker Z . Proof.
We only prove the first formula of ( i ). Assume that Z is indecomposable. If Z isprojective, then Z = m i ( P ) by Lemma 1.1( iii ), where P is an indecomposable projective A -module. By the definition of τ M and a direct computation, we have τ M Cok Z = τ M Cok m i ( P ) = τ M p n − i +1 ( P ) = ∗ ... ∗ ! , it follows that Ker τ M Cok Z = 0 = τ S Z .Assume that Z ∈ S n ( A ) is not projective. Since Cok : S n ( A ) → F n ( A ) is an equivalence, Cok Z ∈ F n ( A ) is not projective. By Lemma 1.1( i ) and ( v ), Cok Z is an indecomposablenonprojective object in Mor n ( A ). Replacing Z by Cok Z in (2.2), we get the assertion by Ker Cok Z ∼ = Z . (cid:4) Example.
Let k be a field, A = k [ x ] / h x i , and S be the simple A -module. Denoteby i : S ֒ → A and π : A ։ S the canonical A -maps. Then we have the Auslander-Reiten USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 11 sequence in Mor ( A )0 / / (cid:16) AS (cid:17) ( i, (cid:18) (cid:19) ( π ) i ! / / (cid:16) S (cid:17) ⊕ (cid:16) SS ⊕ AA (cid:17) (( ) , ( − ,π ) ) (cid:18) − i (cid:19) , − ,π )1 !! / / (cid:16) SSA (cid:17) ( π, / / . By (2 . F ( A )0 / / (cid:16) SS (cid:17) (1 , (cid:18) (cid:19)(cid:18) ii (cid:19) / / (cid:16) S (cid:17) ⊕ (cid:16) SAA (cid:17) (1 ,π ) (cid:18)(cid:18) − i (cid:19) , (cid:18) π (cid:19)(cid:19) / / (cid:16) SSA (cid:17) ( π, / / . By (2 . S ( A )0 / / (cid:16) SS (cid:17) (0 , (cid:18) (cid:19)(cid:18) i (cid:19) / / (cid:16) SSS (cid:17) (1 , ⊕ (cid:16) AS (cid:17) (0 ,i ) (cid:18)(cid:18) i (cid:19) , (cid:18) − − (cid:19)(cid:19) / / (cid:16) ASS (cid:17) (1 ,i ) / / . i ), τ S is formulated via τ M of Mor n ( A ). However, τ M is usuallymore complicated than τ . The rest of this section is to give a formula of τ S via τ .Before stating the main result, we need a notation. For X ( φ i ) ∈ Mor n ( A -mod), define τ X ( φ i ) = τX ... τX n ! ( τφ i ) ∈ Mor n ( A -mod) . Consider the full subcategory given by { Y ( ψ i ) = τX ... τX n ! ( ψ i ) ∈ Mor n ( A ) | Y ( ψ i ) ∼ = τ X ( φ i ) } . Any object in this full subcategory will be denoted by τ X ( φ i ) (we emphasize that thisconvention will cause no confusions). So, for X ( φ i ) ∈ Mor n ( A ) we have τ X ( φ i ) ∼ = τ X ( φ i ) . By Lemma 1.6( i ), Mimo τ X ( φ i ) is a well-defined object in S n ( A ), and there are iso-morphisms Mimo τ X ( φ i ) ∼ = τ X ( φ i ) ∼ = τ X ( φ i ) in Mor n ( A -mod) . If A is selfinjective, thenMor n ( A -mod) = Mor n ( A -mod) , so the isomorphism above is read as follows, which isneeded in the next section Mimo τ X ( φ i ) ∼ = τ X ( φ i ) ∼ = τ X ( φ i ) . (2 . τ − X ( φ i ) , and Mepi τ − X ( φ i ) ∈ F n ( A ) is well-defined.The following result is a generalization of Theorem 5.1 of Ringel and Schmidmeier [RS2]. Theorem 2.4.
Let X ( φ i ) ∈ S n ( A ) . Then ( i ) τ S X ( φ i ) ∼ = Mimo τ Cok X ( φ i ) . ( ii ) τ −S X ( φ i ) ∼ = Ker Mepi τ − X ( φ i ) . Proof.
We only prove ( i ). Recall Cok X ( φ i ) = Coker φ Coker( φ φ ) ... Coker( φ ··· φ n − ) X ( φ ′ i ) . Fix a minimalprojective presentation Q n d n → P e → X →
0. Then we get the following commutativediagram with exact rows Q n d n / / s n − (cid:15) (cid:15) P e / / X / / φ ′ n − (cid:15) (cid:15) (cid:15) (cid:15) Q n − d n − / / (cid:15) (cid:15) P φ ′ n − e / / Coker( φ · · · φ n − ) / / (cid:15) (cid:15) (cid:15) (cid:15) s (cid:15) (cid:15) ... ... φ ′ (cid:15) (cid:15) (cid:15) (cid:15) Q d / / P φ ′ ··· φ ′ n − e / / Coker( φ ) / / . Q i ։ Ker( φ ′ i · · · φ ′ n − e ) is a projective cover, and d i is the composition Q i ։ Ker( φ ′ i · · · φ ′ n − e ) ֒ → P . Applying the Nakayama functor N = D Hom A ( − , A A ), we getthe following commutative diagram0 / / τ X σ n / / α n − (cid:15) (cid:15) N Q n N d n / / N s n − (cid:15) (cid:15) N P / / τ Coker( φ · · · φ n − ) σ n − / / α n − (cid:15) (cid:15) N Q n − N d n − / / N s n − (cid:15) (cid:15) N P ... (cid:15) (cid:15) ... (cid:15) (cid:15) ...0 / / τ Coker( φ φ ) σ / / α (cid:15) (cid:15) N Q N d / / N s (cid:15) (cid:15) N P / / τ Coker φ σ / / N Q N d / / N P. (2 . Step 1.
By (2 . n L i =1 m i ( Q i ) d ... , d d ... , ··· , d n d n d n ... d n / / m n ( P ) φ ′ ··· φ ′ n − e ... φ ′ n − ee / / Cok X ( φ i ) / / . .
4) we have Im d n ⊆ Im d n − ⊆ · · · ⊆ Im d ,and hence Q i ⊕ · · · ⊕ Q n ( d i , ··· ,d n ) −→ P φ ′ i ··· φ ′ n − e −→ Coker( φ i · · · φ n − ) → USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 13 obtain a minimal projective presentation from (2 . n L i =1 m i ( Q i ). By Lemma 1.1( i ), this direct summand is of the form n − L i =1 m i ( Q ′ i ) where Q ′ i is a direct summand of Q i , ≤ i ≤ n −
1, since φ ′ ··· φ ′ n − e ... φ ′ n − ee is already minimal and Q n d n → P e → X → N M , we get the exact sequence0 / / τ M Cok X ( φ i ) ⊕ N M ( n − L i =1 m i ( Q ′ i )) / / N M ( n L i =1 m i ( Q i )) / / N M m n ( P ) . By Lemma 1.1( vii ) this exact sequence can be written as0 / / τ M Cok X ( φ i ) ⊕ n − L i =1 p n − i +1 ( N Q ′ i ) / / n L i =1 p n − i +1 ( N Q i ) d / / p ( N P ) (2 . d = ... N d , · · · , ... N d n . Step 2.
Write Y ( θ i ) = τ M Cok X ( φ i ) ⊕ n − L i =1 p n − i +1 ( N Q ′ i ). By taking the i -th branchesand the n -th branches of terms in (2 . / / Y n ( ab ) / / θ i ··· θ n − (cid:15) (cid:15) i L j =1 N Q j ⊕ n L j = i +1 N Q j (1 , (cid:15) (cid:15) (( N d , ··· , N d i ) , ( N d i +1 , ··· , N d n )) / / N P (cid:15) (cid:15) / / Y i ∼ = / / i L j =1 N Q j / / . In particular, Y n = Ker( N d , · · · , N d n ). The upper exact sequence means that Y n − b / / a (cid:15) (cid:15) n L j = i +1 N Q j ( N d i +1 , ··· , N d n ) (cid:15) (cid:15) i L j =1 N Q j ( N d , ··· , N d i ) / / N P is a pull back square, for each 1 ≤ i ≤ n −
1. It follows thatKer( θ i · · · θ n − ) = Ker a = Ker( N d i +1 , · · · , N d n ) , ≤ i ≤ n − , and hence Ker Y ( θ i ) = Ker( N d , ··· , N d n )Ker( N d , ··· , N d n ) ... Ker( N d n − , N d n )Ker N d n ( θ ′ i ) . We explicitly compute
Ker Y ( θ i ) below. Step 3.
By (2.5) we get the following commutative diagram with exact rows:0 / / τ X γ n / / β n − (cid:15) (cid:15) N Q n N d n / / ( ) (cid:15) (cid:15) N P / / τ Coker( φ · · · φ n − ) ⊕ N Q n γ n − / / β n − (cid:15) (cid:15) N Q n − ⊕ N Q n ( N d n − , N d n ) / / (cid:16) E (cid:17) (cid:15) (cid:15) N P ... (cid:15) (cid:15) ... (cid:15) (cid:15) ...0 / / τ Coker( φ φ ) ⊕ n L i =3 N Q i γ / / β (cid:15) (cid:15) N Q ⊕ n L i =3 N Q i ( N d , ··· , N d n ) / / (cid:16) E n − (cid:17) (cid:15) (cid:15) N P / / τ Coker φ ⊕ n L i =2 N Q i γ / / N Q ⊕ n L i =2 N Q i ( N d , ··· , N d n ) / / N P (2 . E i is the identity matrix, γ i = (cid:16) σ i ( −N s i −N ( s i s i +1 ) ··· −N ( s i ··· s n − ))0 E n − i (cid:17) ( n − i +1) × ( n − i +1) for 1 ≤ i ≤ n, β i = (cid:18) α i (0 0 ··· σ i +1 ( −N s i +1 −N ( s i +1 s i +2 ) ··· −N ( s i +1 ··· s n − ))0 E n − i − (cid:19) ( n − i +1) × ( n − i ) for1 ≤ i ≤ n − . From (2.8) we see
Ker Y ( θ i ) ∼ = τ Coker φ ⊕ n L j =2 N Q j τ Coker( φ φ ) ⊕ n L j =3 N Q j ... τ Coker( φ ··· φ n − ) ⊕N Q n τX ( β i ) . ApplyingLemma 1.5( i ), it is isomorphic to Mimo τ Cok X ( φ i ) ⊕ J, where J is an injective objectin S n ( A ). Thus Mimo τ Cok X ( φ i ) ⊕ J ∼ = Ker Y ( θ i ) Def. ∼ = Ker τ M Cok X ( φ i ) ⊕ Ker ( n − L i =1 p n − i +1 ( N Q ′ i )) Cor. . ∼ = τ S X ( φ i ) ⊕ Ker ( n − L i =1 p n − i +1 ( N Q ′ i )) (1 . ∼ = τ S X ( φ i ) ⊕ n − L i =1 m i ( N Q ′ i ) . Since
Mimo τ Cok X ( φ i ) and τ S X ( φ i ) have no nonzero injective direct summands in S n ( A )(cf. Remark ( ii ) in 1.4), and S n ( A ) is Krull-Schmidt, we get τ S X ( φ i ) ∼ = Mimo τ Cok X ( φ i ) . (cid:4) Example.
Let A , S , i , and π be as in 2.3. Then there are 6 indecomposable non-projective objects in S ( A ). By Theorem 2.4 we have τ S (cid:16) AS (cid:17) (0 ,i ) = Mimo τ (cid:16) SAA (cid:17) (1 ,π ) = Mimo (cid:16) S (cid:17) (0 , = (cid:16) S (cid:17) (0 , τ S (cid:16) S (cid:17) (0 , = Mimo τ (cid:16) SSS (cid:17) (1 , = Mimo (cid:16)
SSS (cid:17) (1 , = (cid:16) SSS (cid:17) (1 , τ S (cid:16) SSS (cid:17) (1 , = Mimo τ (cid:16) S (cid:17) (0 , = Mimo (cid:16) S (cid:17) (0 , = (cid:16) AAS (cid:17) ( i, τ S (cid:16) AAS (cid:17) ( i, = Mimo τ (cid:16) SA (cid:17) ( π, = Mimo (cid:16) S (cid:17) (0 , = (cid:16) AS (cid:17) (1 ,i ) τ S (cid:16) SS (cid:17) (0 , = Mimo τ (cid:16) SS (cid:17) (1 , = Mimo (cid:16) SS (cid:17) (1 , = (cid:16) ASS (cid:17) (1 ,i ) τ S (cid:16) ASS (cid:17) (1 ,i ) = Mimo τ (cid:16) SSA (cid:17) ( π, = Mimo (cid:16) SS (cid:17) (0 , = (cid:16) SS (cid:17) (0 , The Auslander-Reiten quiver of S ( A ) looks like A " " EEEEE AA " " EEEEE
AAA " " DDDDD S < < zzzzz " " DDDDD AS < < yyyyy " " EEEEE o o AAS < < yyyyy " " EEEEE o o SSS o o SS < < yyyyy " " EEEEE
ASS < < yyyyy " " EEEEE o o SS < < zzzzz o o SSS < < yyyyy S < < yyyyy o o S n ( A ) I the full subcategory of S n ( A ) consisting of all the objects whichhave no nonzero injective direct summands of S n ( A ). By Theorem 2.4, we have Corollary 2.5.
Every object in S n ( A ) I has the form Mimo X, where each X i has nononzero injective direct summands. The following result will be used in the next section, whose proof is omitted, since it isthe same as the case of n = 2 (see [RS2], Corollary 5.4). Corollary 2.6.
The canonical functor W : S n ( A ) I → Mor n ( A - mod) given by X ( φ i ) X ( φ i ) is dense, preserves indecomposables, and reflects isomorphisms. Theorem 2.7.
Let X ∈ F n ( A ) . Then τ F X ∼ = Cok Mimo τ X, and τ −F X ∼ = Mepi τ − Ker X. Applications to selfinjective algebras
Throughout this section, A is a selfinjective algebra. Then A -mod = A -mod is a triangu-lated category with the suspension functor Ω − ([H], p.16), where Ω − is the cosyzygy of A .The distinguished triangles of A -mod are exactly triangles isomorphic to those given by allthe short exact sequences in A -mod. Note that Ω and N commute and τ ∼ = Ω N ∼ = N Ω is an endo-equivalence of A -mod ([ARS], p.126), and that τ is a triangle functor.A rotation of an object in Mor ( A -mod) is introduced by Ringel and Schmidmeier[RS2]. The definition of a rotation of an object in Mor n ( A -mod) needs new considerations.We have to take up pages to justify that it is well-defined. Then we get a formula for τ j S X ∈ Mor n ( A -mod) for X ∈ S n ( A ) and j ≥
1. This is applied to the study of theperiodicity of τ S on the objects of S n ( A ). In particular, for the selfinjective Nakayamaalgebras Λ( m, t ) we have τ m ( n +1) S X ∼ = X for X ∈ S n (Λ( m, t )).3.1. Let X ( φ i ) ∈ Mor n ( A -mod). Just choose φ i as representatives for the morphisms φ i in A -mod. Let h i +1 : X i +1 → I i +1 be an injective envelope with cokernel Ω − X i +1 ,1 ≤ i ≤ n −
1. Taking pushout we get the following commutative diagram with exact rows0 / / X i +1 h i +1 / / φ ··· φ i (cid:15) (cid:15) I i +1 / / j i +1 (cid:15) (cid:15) Ω − X i +1 / / / / X g i +1 / / Y i +1 / / Ω − X i +1 / / . This gives the exact sequence0 / / X i +1 (cid:18) φ ··· φ i h i +1 (cid:19) / / X ⊕ I i +1( g i +1 , − j i +1 ) / / Y i +1 / / , (3 . ≤ i ≤ n − / / X i +2 (cid:18) φ ··· φ i +1 h i +2 (cid:19) / / φ i +1 (cid:15) (cid:15) X ⊕ I i +2 ( g i +2 , − j i +2 ) / / ( ∗ ) (cid:15) (cid:15) Y i +2 / / ψ i (cid:15) (cid:15) / / X i +1 (cid:18) φ ··· φ i h i +1 (cid:19) / / X ⊕ I i +1 ( g i +1 , − j i +1 ) / / Y i +1 / / . (3 . g i +1 = ψ i g i +2 , ≤ i ≤ n −
2. Put ψ n − = g n . Then g i +1 = ψ i · · · ψ n − , ≤ i ≤ n −
1. By the construction of a distinguished triangle in A -mod,we get distinguished triangles from (3.1) X i +1 φ ··· φ i −→ X → Y i +1 → Ω − X i +1 , ≤ i ≤ n − , (3 . X n / / φ n − (cid:15) (cid:15) X ψ n − / / Y n / / ψ n − (cid:15) (cid:15) Ω − X n (cid:15) (cid:15) X n − / / φ n − (cid:15) (cid:15) X / / Y n − / / ψ n − (cid:15) (cid:15) Ω − X n − (cid:15) (cid:15) ... φ (cid:15) (cid:15) ... ... ψ (cid:15) (cid:15) ... (cid:15) (cid:15) X φ / / X / / Y / / Ω − X . (3 . rotation Rot X ( φ i ) of X ( φ i ) is defined to be( X ψ n − / / Y n / / · · · ψ / / Y ) ∈ Mor n ( A -mod) USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 17 (here and in the following, for convenience we write the rotation in a row). We remarkthat
Rot X ( φ i ) is well-defined: if X ( φ i ) ∼ = Y ( θ i ) in Mor n ( A -mod) with all X i and Y i havingno nonzero injective direct summands, then Mimo X ( φ i ) ∼ = Mimo Y ( θ i ) by Lemma 1.6( i ),and hence Rot X ( φ i ) ∼ = Rot Y ( θ i ) , by Lemma 3.1 below. Lemma 3.1.
Let X ( φ i ) ∈ Mor n ( A ) . Then Rot X ( φ i ) ∼ = Cok Mimo X ( φ i ) in Mor n ( A - mod) . Before proving Lemma 3.1, for later convenience, we restate Claim 2 in § Lemma 3.2.
Let / / A (cid:16) fh (cid:17) / / B ⊕ I / / C / / be an exact sequence with I an injective A -module. Then there is an injective A -module J such that I = IKer f ⊕ J , and that thefollowing diagram with exact rows commutes / / A (cid:18) fe (cid:19) / / B ⊕ IKer f ⊕ J (cid:18) w (cid:19) (cid:15) (cid:15) / / C ′ ⊕ J ≀ (cid:15) (cid:15) / / / / A (cid:18) feh ′ (cid:19) / / B ⊕ IKer f ⊕ J / / C / / , where h = ( eh ′ ) , e : A → IKer f is an extension of the injective envelope Ker f ֒ → IKer f , h ′ : A → J satisfies h ′ Ker f = 0 , and C ′ = Coker ( fe ) . Proof of Lemma 3.1.
We divide the proof into three steps.
Step 1.
Recall
Mimo X ( φ i ) = X ⊕ n − L l =1 IKer φ l ... X n − ⊕ IKer φ n − X n ( θ i ) with θ i = φ i ··· e i ···
00 1 ··· ... ... ··· ... ··· ( n − i +1) × ( n − i ) ,e i : X i +1 → IKer φ i is an extension of the injective envelope Ker φ i ֒ → IKer φ i , and Cok Mimo X ( φ i ) = Coker( θ )Coker( θ θ ) ... Coker( θ ··· θ n − ) X ⊕ n − L l =1 IKer φ l ( θ ′ i ) . Since θ · · · θ i = diag ( α i , E n − i − ) : X i +1 ⊕ n − L l = i +1 IKer φ l → X ⊕ n − L l =1 IKer φ l , where α i = φ ··· φ i e φ ··· φ i ... e i − φ i e i : X i +1 → X ⊕ i L l =1 IKer φ l , and E n − i − is the identity matrix, we get the following commutative diagram with exact rows, 2 ≤ i ≤ n − / / X i +1 α i / / φ i (cid:15) (cid:15) X ⊕ i L l =1 IKer φ l π i / / ( E i , (cid:15) (cid:15) Coker( θ · · · θ i ) / / θ ′ i − (cid:15) (cid:15) / / X i α i − / / X ⊕ i − L l =1 IKer φ l π i − / / Coker( θ · · · θ i − ) / / . π n − = θ ′ n − .Applying Lemma 3.2 to the upper exact sequence of (3.5) for 1 ≤ i ≤ n −
1, weget injective A -modules J i +1 such that i L l =1 IKer φ l = IKer( φ · · · φ i ) ⊕ J i +1 and that thefollowing diagram with exact rows commutes0 / / X i +1 (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J i +1 (cid:18) w i (cid:19) (cid:15) (cid:15) (cid:18) b i b i
00 0 1 (cid:19) / / Z i +1 ⊕ J i +1 / / ≀ β i (cid:15) (cid:15) / / X i +1 φ ··· φ i a i d i ! / / X ⊕ IKer( φ · · · φ i ) ⊕ J i +1 π i / / Coker( θ · · · θ i ) / / . α i = (cid:16) φ ··· φ i a i d i (cid:17) , a i : X i +1 → IKer( φ · · · φ i ) is an extension of the injective envelopeKer( φ · · · φ i ) ֒ → IKer( φ · · · φ i ), d i : X i +1 → J i +1 satisfies d i Ker( φ · · · φ i )=0, and Z i +1 =Coker (cid:0) φ ··· φ i a i (cid:1) . Thus by (3.6) and (3.5) we get the following commutative diagram withexact rows for 2 ≤ i ≤ n − / / X i +1 (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J i +1 (cid:18) w i (cid:19) (cid:15) (cid:15) (cid:18) b i b i
00 0 1 (cid:19) / / Z i +1 ⊕ J i +1 / / β i ≀ (cid:15) (cid:15) / / X i +1 φ ··· φ i a i d i ! / / φ i (cid:15) (cid:15) X ⊕ IKer( φ · · · φ i ) ⊕ J i +1 π i / / (cid:18) ∗ ∗ ∗ ∗ (cid:19) (cid:15) (cid:15) Coker( θ · · · θ i ) / / θ ′ i − (cid:15) (cid:15) / / X i φ ··· φ i − a i − d i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J i π i − / / (cid:18) w i − (cid:19) − (cid:15) (cid:15) Coker( θ · · · θ i − ) / / β − i − ≀ (cid:15) (cid:15) / / X i φ ··· φ i − a i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J i (cid:18) b i − b i −
00 0 1 (cid:19) / / Z i ⊕ J i / / . Taking the first and the last rows, we get the following commutative diagram with exactrows:0 / / X i +1 φ i (cid:15) (cid:15) (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J i +1 (cid:18) c i c i ∗∗ ∗ ∗ (cid:19) (cid:15) (cid:15) (cid:18) b i b i
00 0 1 (cid:19) / / Z i +1 ⊕ J i +1 / / β − i − θ ′ i − β i (cid:15) (cid:15) / / X i φ ··· φ i − a i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J i (cid:18) b i − b i −
00 0 1 (cid:19) / / Z i ⊕ J i / / , USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 19 where for later convenience we write β − i − θ ′ i − β i = (cid:0) f i − ∗∗ ∗ (cid:1) , ≤ i ≤ n − . (3 . J i and J i +1 , we get the following commutative diagram with exact rows, 2 ≤ i ≤ n −
1, 0 / / X i +1 φ i (cid:15) (cid:15) (cid:16) φ ··· φ i a i (cid:17) / / X ⊕ IKer( φ · · · φ i ) (cid:16) c i c i (cid:17) (cid:15) (cid:15) ( b i , b i ) / / Z i +1 / / f i − (cid:15) (cid:15) / / X i (cid:16) φ ··· φ i − a i − (cid:17) / / X ⊕ IKer( φ · · · φ i − ) ( b i − , b i − ) / / Z i / / . (3 . Step 2.
Now we consider the rotation
Rot X ( φ i ) . Recall from the beginning of thissubsection that h i +1 : X i +1 → I i +1 is an injective envelope. For 1 ≤ i ≤ n −
1, applyingLemma 3.2 to (3.1), we get injective A -modules J ′ i +1 such that I i +1 = IKer( φ · · · φ i ) ⊕ J ′ i +1 and that the following diagram with exact rows commutes (cf. (3.6))0 / / X i +1 (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J ′ i +1 (cid:18) b i b i
00 0 1 (cid:19) / / w ′ i ! (cid:15) (cid:15) Z i +1 ⊕ J ′ i +1 / / ≀ β ′ i (cid:15) (cid:15) / / X i +1 φ ··· φ i a i d ′ i ! / / X ⊕ IKer( φ · · · φ i ) ⊕ J ′ i +1 ( g i +1 , − j i +1 ) / / Y i +1 / / , (3 . h i +1 = (cid:16) a i d ′ i (cid:17) , and d ′ i : X i +1 → J ′ i +1 , satisfying d ′ i Ker( φ · · · φ i ) = 0. Thus by (3.9)and (3.2) we get the following commutative diagram with exact rows for 2 ≤ i ≤ n − / / X i +1 (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J ′ i +1 w ′ i ! (cid:15) (cid:15) (cid:18) b i b i
00 0 1 (cid:19) / / Z i +1 ⊕ J ′ i +1 / / β ′ i ≀ (cid:15) (cid:15) / / X i +1 φ ··· φ i a i d ′ i ! / / φ i (cid:15) (cid:15) X ⊕ IKer( φ · · · φ i ) ⊕ J ′ i +1 ( g i +1 , − j i +1 ) / / (cid:18) ∗ ∗ ∗ ∗ (cid:19) (cid:15) (cid:15) Y i +1 / / ψ i − (cid:15) (cid:15) / / X i φ ··· φ i − a i − d ′ i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J ′ i ( g i , − j i ) / / w ′ i − ! − (cid:15) (cid:15) Y i / / β ′− i − ≀ (cid:15) (cid:15) / / X i φ ··· φ i − a i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J ′ i (cid:18) b i − b i −
00 0 1 (cid:19) / / Z i ⊕ J ′ i / / . Taking the first and the last rows, we get the following commutative diagram with exactrows: / / X i +1 φ i (cid:15) (cid:15) (cid:18) φ ··· φ i a i (cid:19) / / X ⊕ IKer( φ · · · φ i ) ⊕ J ′ i +1 (cid:18) c ′ i c ′ i ∗∗ ∗ ∗ (cid:19) (cid:15) (cid:15) (cid:18) b i b i
00 0 1 (cid:19) / / Z i +1 ⊕ J ′ i +1 / / β ′− i − ψ i − β ′ i (cid:15) (cid:15) / / X i φ ··· φ i − a i − ! / / X ⊕ IKer( φ · · · φ i − ) ⊕ J ′ i (cid:18) b i − b i −
00 0 1 (cid:19) / / Z i ⊕ J ′ i / / , where for later convenience we write β ′− i − ψ ′ i − β ′ i = (cid:16) f ′ i − ∗∗ ∗ (cid:17) , ≤ i ≤ n − . (3 . J ′ i and J ′ i +1 , we get the following commutative diagram with exact rows for2 ≤ i ≤ n − / / X i +1 φ i (cid:15) (cid:15) (cid:16) φ ··· φ i a i (cid:17) / / X ⊕ IKer( φ · · · φ i ) (cid:16) c ′ i c ′ i (cid:17) (cid:15) (cid:15) ( b i , b i ) / / Z i +1 / / f ′ i − (cid:15) (cid:15) / / X i (cid:16) φ ··· φ i − a i − (cid:17) / / X ⊕ IKer( φ · · · φ i − ) ( b i − , b i − ) / / Z i / / . Comparing the above diagram with (3 . f i − f ′ i factors through an injective A -module for each 1 ≤ i ≤ n − Step 3.
Now we get the following diagram, where the first row can be consideredas
Cok Mimo X ( φ i ) (we identify X ⊕ n − L l =1 IKer φ l with X ⊕ IKer( φ · · · φ n − ) ⊕ J n ; andidentify X ⊕ I n with X ⊕ IKer( φ · · · φ n − ) ⊕ J ′ n ): X ⊕ n − L l =1 IKer φ l θ ′ n − / / ≀ (cid:18) w n − (cid:19) − (cid:15) (cid:15) Coker( θ · · · θ n − ) θ ′ n − / / β − n − ≀ (cid:15) (cid:15) · · · / / Coker( θ θ ) θ ′ / / β − ≀ (cid:15) (cid:15) Coker θ β − (cid:15) (cid:15) X ⊕ IKer( φ · · · φ n − ) ⊕ J n (cid:18) b n − b n −
00 0 1 (cid:19) / / (cid:18) (cid:19) (cid:15) (cid:15) Z n ⊕ J n (cid:16) f n − ∗∗ ∗ (cid:17) / / ( ) (cid:15) (cid:15) · · · / / Z ⊕ J (cid:16) f ∗∗ ∗ (cid:17) / / ( ) (cid:15) (cid:15) Z ⊕ J ( ) (cid:15) (cid:15) X ⊕ IKer( φ · · · φ n − ) ⊕ J ′ n (cid:18) b n − b n −
00 0 1 (cid:19) / / ≀ w ′ n − ! (cid:15) (cid:15) Z n ⊕ J ′ n (cid:16) f ′ n − ∗∗ ∗ (cid:17) / / β ′ n − ≀ (cid:15) (cid:15) · · · / / Z ⊕ J ′ (cid:16) f ′ ∗∗ ∗ (cid:17) / / β ′ ≀ (cid:15) (cid:15) Z ⊕ J ′ β ′ ≀ (cid:15) (cid:15) X ⊕ I n ( ψ n − , − j n ) / / Y n ψ n − / / · · · / / Y ψ / / Y (note that the squares in the first two rows commute in A -mod: the left square comesfrom (3.6); and the remaining commutative squares come from (3.7). Also, note that thesquares in the last two rows commute in A -mod: the left square comes from (3.9); and theremaining commutative squares come from (3.10)). However, the squares in the middlemay not commute in A -mod; and the point is that they commute in A -mod, as we explainbelow. USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 21
Note that the left square in the middle commutes by a direct computation. Since f i − f ′ i factors through an injective A -module, 1 ≤ i ≤ n −
2, we realize that the remaining n − A -mod. It follows that the above diagram commutesin A -mod. It is clear that the vertical morphisms are isomorphisms in A -mod. Regardingthe above diagram in A -mod, the first row is exactly Cok Mimo X ( φ i ) , and the last rowis exactly Rot X ( φ i ) . Thus, Rot X ( φ i ) ∼ = Cok Mimo X ( φ i ) in Mor n ( A -mod). This com-pletes the proof. (cid:4) X ( φ i ) ∈ Mor n ( A ). For 1 ≤ k < i < j ≤ n , by (3.3) and the Octahedral Axiomwe get the following commutative diagram with first two rows and the last two columnsbeing distinguished triangles in A -mod:Ω Y ij / / (cid:15) (cid:15) X j φ i ··· φ j − / / X i / / φ k ··· φ i − (cid:15) (cid:15) Y ij (cid:15) (cid:15) Ω Y kj / / X j φ k ··· φ j − / / X k / / (cid:15) (cid:15) Y kj (cid:15) (cid:15) Y ki (cid:15) (cid:15) Y ki (cid:15) (cid:15) Ω − X i / / Ω − Y ij . (3 . ≤ m ≤ n we prove the following formula by induction Rot m X ( φ i ) = (Ω − ( m − Y m − m → Ω − ( m − Y m − m → · · · → Ω − ( m − Y m → Ω − ( m − X m → Ω − ( m − Y mn → Ω − ( m − Y mn − → · · · → Ω − ( m − Y mm +1 ) (3 . . − ( m − X m is the ( n − m + 1)-st branch of Rot m X ( φ i ) .By definition (3.12) holds for m = 1. Assume that it holds for 1 ≤ m ≤ n − . Considerthe following commutative diagram with rows of distinguished trianglesΩ − ( m − Y m − m / / (cid:15) (cid:15) Ω − ( m − Y mm +1 / / Ω − ( m − Y m − m +1 / / (cid:15) (cid:15) Ω − ( m − Y m − m (cid:15) (cid:15) ... (cid:15) (cid:15) ... ... (cid:15) (cid:15) ... (cid:15) (cid:15) Ω − ( m − Y m / / (cid:15) (cid:15) Ω − ( m − Y mm +1 / / Ω − ( m − Y m +1 / / (cid:15) (cid:15) Ω − ( m − Y m (cid:15) (cid:15) Ω − ( m − X m / / (cid:15) (cid:15) Ω − ( m − Y mm +1 / / Ω − m X m +1 / / (cid:15) (cid:15) Ω − m X m (cid:15) (cid:15) Ω − ( m − Y mn / / (cid:15) (cid:15) Ω − ( m − Y mm +1 / / Ω − m Y m +1 n / / (cid:15) (cid:15) Ω − m Y mn (cid:15) (cid:15) ... (cid:15) (cid:15) ... ... (cid:15) (cid:15) ... (cid:15) (cid:15) Ω − ( m − Y mm +2 / / Ω − ( m − Y mm +1 / / Ω − m Y m +1 m +2 / / Ω − m Y mm +22 BAO-LIN XIONG PU ZHANG YUE-HUI ZHANG where the l -th row (1 ≤ l ≤ m −
1) is from the fourth column of (3 .
11) by taking j = m +1 , i = m, k = m − l , and then applying Ω − ( m − ; the m -th row is from the first row of (3 . j = m + 1 , i = m , and then applying Ω − ( m − ; the l -th row ( m + 1 ≤ l ≤ n − .
11) by taking j = n + m + 1 − l, i = m + 1 , k = m , and thenapplying Ω − m . By the definition of the rotation (cf. (3.4)), this proves (3.12) for m + 1.3.3. For X ( φ i ) ∈ Mor n ( A -mod), define Ω − X ( φ i ) to be Ω − X ... Ω − X n ! (Ω − φ i ) ∈ Mor n ( A -mod) . Lemma 3.3.
Let X ( φ i ) ∈ Mor n ( A ) . Then Rot j ( n +1) X ( φ i ) = Ω − j ( n − X ( φ i ) , ∀ j ≥ . Proof.
By taking m = n in (3 . Rot n X ( φ i ) = (Ω − ( n − Y n − n → Ω − ( n − Y n − n → · · · → Ω − ( n − Y n → Ω − ( n − X n ) . We have the following commutative diagram with rows being distinguished trianglesΩ − ( n − Y n − n / / (cid:15) (cid:15) Ω − ( n − X n ( − n − Ω − ( n − φ n − / / Ω − ( n − X n − / / Ω − ( n − φ n − (cid:15) (cid:15) Ω − ( n − Y n − n (cid:15) (cid:15) Ω − ( n − Y n − n / / (cid:15) (cid:15) Ω − ( n − X n ( − n − Ω − ( n − φ n − φ n − / / Ω − ( n − X n − / / (cid:15) (cid:15) Ω − ( n − Y n − n (cid:15) (cid:15) ... (cid:15) (cid:15) ... ... Ω − ( n − φ (cid:15) (cid:15) ... (cid:15) (cid:15) Ω − ( n − Y n / / (cid:15) (cid:15) Ω − ( n − X n ( − n − Ω − ( n − φ ··· φ n − / / Ω − ( n − X / / Ω − ( n − φ (cid:15) (cid:15) Ω − ( n − Y n (cid:15) (cid:15) Ω − ( n − Y n / / Ω − ( n − X n ( − n − Ω − ( n − φ ··· φ n − / / Ω − ( n − X / / Ω − ( n − Y n where the l -th row (1 ≤ l ≤ n −
1) is from the first row of (3 .
11) by taking j = n, i = n − l ,and then applying Ω − ( n − (note that ( − ( n − arises from applying Ω − ( n − ). Using(3.4) we get that Rot n +1 X ( φ i ) is( Ω − ( n − X n ( − n − Ω − ( n − φ n − / / Ω − ( n − X n − − ( n − φ n − / / · · · Ω − ( n − φ / / Ω − ( n − X ) ∼ = ( Ω − ( n − X n Ω − ( n − φ n − / / Ω − ( n − X n − − ( n − φ n − / / · · · Ω − ( n − φ / / Ω − ( n − X )= Ω − ( n − X ( φ i ) . From this and induction the assertion follows. (cid:4) τ is a triangle functor, by construction we see Rot τ X ( φ i ) ∼ = τ Rot X ( φ i ) . (3 . USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 23
Theorem 3.4.
Let A be a selfinjective algebra, X ( φ i ) ∈ S n ( A ) . Then there are thefollowing isomorphisms in Mor n ( A - mod)( i ) τ j S X ( φ i ) ∼ = τ j Rot j X ( φ i ) for j ≥ . In particular, τ S X ( φ i ) ∼ = τ Cok X ( φ i ) . ( ii ) τ s ( n +1) S X ( φ i ) ∼ = τ s ( n +1) Ω − s ( n − X ( φ i ) , ∀ s ≥ . Proof. ( i ) First, we claim that there are the following isomorphisms in Mor n ( A -mod):( Cok Mimo τ ) j Y ( ψ i ) ∼ = τ j Rot j Y ( ψ i ) , ∀ Y ( ψ i ) ∈ Mor n ( A -mod) , ∀ j ≥ . (3 . Cok Mimo τ ) j Y ( ψ i ) ∼ = Cok Mimo τ ( Cok Mimo τ ) j − Y ( ψ i ) Lemma . ∼ = Rot τ ( Cok Mimo τ ) j − Y ( ψ i )(2 . ∼ = Rot τ ( Cok Mimo τ ) j − Y ( ψ i ) Induction ∼ = Rot τ j Rot j − Y ( ψ i )(3 . ∼ = τ j Rot j Y ( ψ i ) . Now, we have the following isomorphisms in Mor n ( A -mod): τ j S X ( φ i ) Theorem . ∼ = ( Mimo τ Cok ) j X ( φ i ) ∼ = Mimo τ ( Cok Mimo τ ) j − Cok X ( φ i )(2 . ∼ = τ ( Cok Mimo τ ) j − Cok X ( φ i )(3 . ∼ = τ j Rot j − Cok X ( φ i ) ∼ = τ j Rot j − Cok Mimo X ( φ i ) Lemma . ∼ = τ j Rot j X ( φ i ) , where we have used Mimo X ( φ i ) = X ( φ i ) since X ( φ i ) ∈ S n ( A ).( ii ) This follows from Lemma 3.3 and ( i ) by taking j = s ( n + 1). (cid:4) n ( A -mod) to Mor n ( A -mod) . Before stating the main result,we need a notation. Let X ( φ i ) ∈ Mor n ( A ). For positive integers r and t , the object τ r Ω − t X ( φ i ) ∈ Mor n ( A -mod) is already defined (cf. 2.4 and 3.3). As in 2.4, we consider the full subcategory of Mor n ( A ) given by { Y ( ψ i ) = τ r Ω − t X ... τ r Ω − t X n ! ( ψ i ) ∈ Mor n ( A ) | Y ( ψ i ) ∼ = τ r Ω − t X ( φ i ) } . Any object in this subcategory will be denoted by τ r Ω − t X ( φ i ) (we emphasize that thisconvention will cause no confusions). So, we have τ r Ω − t X ( φ i ) ∼ = τ r Ω − t X ( φ i ) . By Lemma1.6( i ), Mimo τ r Ω − t X ( φ i ) ∈ S n ( A ) is a well-defined object, and there are the followingisomorphisms in Mor n ( A -mod) Mimo τ r Ω − t X ( φ i ) ∼ = τ r Ω − t X ( φ i ) ∼ = τ r Ω − t X ( φ i ) . (3 . Theorem 3.5.
Let A be a selfinjective algebra, and X ( φ i ) ∈ S n ( A ) . Then we have τ s ( n +1) S X ( φ i ) ∼ = Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) , s ≥ . (3 . Proof.
By Theorem 3.4( ii ) we have τ s ( n +1) S X ( φ i ) ∼ = τ s ( n +1) Ω − s ( n − X ( φ i ) ∼ = Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) . Since
Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) ∈ S n ( A ) I (cf. Remark ( ii ) in 1.4), the assertion followsfrom Corollary 2.6. (cid:4) m, t ), which is defined below. Let Z m bethe cyclic quiver with vertices indexed by the cyclic group Z /m Z of order m , and witharrows a i : i −→ i + 1 , ∀ i ∈ Z /m Z . Let k Z m be the path algebra of the quiver Z m , J theideal generated by all arrows, and Λ( m, t ) := k Z m /J t with m ≥ , t ≥
2. Any connectedselfinjective Nakayama algebra over an algebraically closed field is Morita equivalent toΛ( m, t ) , m ≥ , t ≥
2. Note that Λ( m, t ) is a Frobenius algebra of finite representationtype, and that Λ( m, t ) is symmetric if and only if m | ( t − m, t ) see [ARS], p.197. In the stable category Λ( m, t )-mod, we have thefollowing information on the orders of τ and Ω (see 5.1 in [CZ]) o ( τ ) = m ; o (Ω) = m, t = 2; m ( m,t ) , t ≥ , (3 . m, t ) is the g.c.d of m and t . By (3.16) and (3.17) we get the following Corollary 3.6.
For an indecomposable nonprojective object X ( φ i ) ∈ S n (Λ( m, t )) , m ≥ , t ≥ , there are the following isomorphisms: ( i ) If n is odd, then τ m ( n +1) S X ( φ i ) ∼ = X ( φ i ) ; ( ii ) If n is even, then τ m ( n +1) S X ( φ i ) ∼ = X ( φ i ) . USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 25
Example.
Let A = kQ/ h δα, βγ, αδ − γβ i , where Q is the quiver 2 • α / / • β / / δ o o • γ o o Then A is selfinjective with τ ∼ = Ω − and Ω ∼ = id on the object of A -mod. The Auslander-Reiten quiver of A is (cid:30) (cid:30) ====
12 31 (cid:27) (cid:27) GGGG AAAA @ @ (cid:1)(cid:1)(cid:1)(cid:1) o o GGGG o o % % LLLLL o o
12 3
GGGG ; ; AAAA > > }}}} o o % % LLLLL sssss C C (cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6) o o
12 3 o o ; ; xxxx (cid:30) (cid:30) ==== > > }}}} o o ; ; o o sssss o o @ @ (cid:1)(cid:1)(cid:1)(cid:1) Let X ( φ i ) be an indecomposable nonprojective object in S n ( A ). By (3.16), for s ≥ τ s ( n +1) S X ( φ i ) ∼ = Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) ∼ = Mimo Ω − sn X ( φ i ) in S n ( A ). Thenby Remark ( i ) in 1.4 we get( i ) if n ≡ , or 3 (mod6), then τ n +1 S X ( φ i ) ∼ = X ( φ i ) ; and( ii ) if n ≡ ± , or ± τ n +1) S X ( φ i ) ∼ = X ( φ i ) .4. Serre functors of stable monomorphism categories
Throughout this section, A is a finite-dimensional selfinjective algebra over a field. Westudy the periodicity of the Serre functor F S on the objects of the stable monomorphismcategory S n ( A ). In particular, F m ( n +1) S X ∼ = X for X ∈ S n (Λ( m, t )).4.1. Let A be a Hom-finite Krull-Schmidt triangulated k -category with suspension func-tor [1]. For the Auslander-Reiten triangles we refer to [H]. In an Auslander-Reiten trian-gle X → Y → Z → X [1], the indecomposable object X is uniquely determined by Z . Write X = f τ A Z , and extend the action of f τ A to arbitrary objects, and put f τ A f τ A is not a functor. By Theorem I.2.4 of [RV], A has a Serre functor F ifand if A has Auslander-Reiten triangles; if this is the case, F and [1] f τ A coincide onthe objects of A . If X f → Y g → Z h → X [1] is an Auslander-Reiten triangle, then so is X [1] − f [1] −→ Y [1] − g [1] −→ Z [1] − h [1] −→ X [2]. It follows that F Z ∼ = ([1] f τ A ) Z ∼ = ( f τ A [1]) Z, ∀ Z ∈ A . (4 . A is self-injective, by Corollary 4.1( ii ) of [Z], S n ( A ) is exactly the category ofGorenstein projective T n ( A )-modules, hence it is a Frobenius category whose projective-injective objects are exactly all the projective T n ( A )-modules. Thus, the stable category S n ( A ) of S n ( A ) modulo projective objects is a Hom-finite Krull-Schmidt triangulated category with suspension functor Ω − S = Ω − S n ( A ) . Since S n ( A ) has Auslander-Reiten se-quences, it follows that S n ( A ) has Auslander-Reiten triangles, and hence it has a Serrefunctor F S = F S n ( A ) , which coincides with Ω − S f τ S on the objects of S n ( A ).In order to make the following computation more clear, we denote by Q : S n ( A ) →S n ( A ) the natural functor. Then f τ S QZ = Q τ S Z = τ S Z, ∀ Z ∈ S n ( A ) . (4 . W : S n ( A ) I → Mor n ( A -mod) givenby X ( φ i ) X ( φ i ) . Lemma 4.1.
The functor W induces a functor f W : S n ( A ) → Mor n ( A - mod) satisfying f W Q | S n ( A ) I = W ; and f W reflects isomorphisms. Proof.
The definition of f W is clear by the requirement f W Q | S n ( A ) I = W . We need tocheck that it is well-defined. If X ( φ i ) and Y ( ψ i ) are indecomposable and nonprojective in S n ( A ), and X ( φ i ) ∼ = Y ( ψ i ) in S n ( A ), then X ( φ i ) ∼ = Y ( ψ i ) in S n ( A ), and hence X ( φ i ) ∼ = Y ( ψ i ) in Mor n ( A -mod), i.e., f W is well-defined on objects. For any morphism f = ( f i ) : X ( φ i ) → Y ( ψ i ) in S n ( A ) which factors through a projective object of S n ( A ), by Lemma 1.1( iii ), themorphism ( f i ) = 0 in Mor n ( A -mod). Thus f W is well-defined.Assume that f W X ∼ = f W Y in Mor n ( A -mod) for X, Y ∈ S n ( A ) . We may write X = QX ′ , Y = QY ′ with X ′ , Y ′ ∈ S n ( A ) I . Then
W X ′ ∼ = W Y ′ in Mor n ( A -mod). By Corollary2.6, W reflects isomorphisms, thus X ′ ∼ = Y ′ in S n ( A ) I , and hence X ∼ = Y in S n ( A ). (cid:4) Lemma 4.2.
For X = X ( φ i ) ∈ S n ( A ) , we have the following isomorphism in Mor n ( A - mod) f W Ω S X ∼ = Ω f W X. (4 . Proof.
Let 0 → Ω S X → P → X → S n ( A ) with P projective.Taking the i -th branches we see that (Ω S X ) i = Ω X i ⊕ P ′ i for some projective A -module P ′ i . It follows that (Ω S X ) i = Ω X i in A -mod. Write Ω S X as (Ω S X ) ( ψ i ) . By the followingcommutative diagram with exact columns0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (Ω S X ) n ψ n − / / (cid:15) (cid:15) (Ω S X ) n − / / (cid:15) (cid:15) · · · / / (Ω S X ) ψ / / (cid:15) (cid:15) (Ω S X ) (cid:15) (cid:15) P n / / (cid:15) (cid:15) P n − / / (cid:15) (cid:15) · · · / / P / / (cid:15) (cid:15) P (cid:15) (cid:15) X n φ n − / / (cid:15) (cid:15) X n − / / (cid:15) (cid:15) · · · / / X φ / / (cid:15) (cid:15) X (cid:15) (cid:15) ψ i = Ω φ i , and hence the assertion follows. (cid:4) USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 27 a and b , let [ a, b ] denote the l.c.m of a and b . The main resultof this section is Theorem 4.3.
Let A be a selfinjective algebra, and F S be the Serre functor of S n ( A ) .Then we have an isomorphism in S n ( A ) for X ( φ i ) ∈ S n ( A ) and for s ≥ F s ( n +1) S X ( φ i ) ∼ = Mimo τ s ( n +1) Ω − sn X ( φ i ) . (4 . Moreover, if d and d are positive integers such that τ d M ∼ = M and Ω d M ∼ = M for each indecomposable nonprojective A -module M, then F N ( n +1) S X ( φ i ) ∼ = X ( φ i ) , where N = [ d ( n +1 ,d ) , d (2 n,d ) ] . Proof.
We have isomorphisms in Mor n ( A -mod) for s ≥ f W F s ( n +1) S X ( φ i ) (4 . ∼ = f W Ω − s ( n +1) S f τ S s ( n +1) X ( φ i )(4 . ∼ = Ω − s ( n +1) f W f τ S s ( n +1) X ( φ i )(4 . ∼ = Ω − s ( n +1) f W Q τ s ( n +1) S X ( φ i )(3 . ∼ = Ω − s ( n +1) f W Q
Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) Lemma . ∼ = Ω − s ( n +1) W Mimo τ s ( n +1) Ω − s ( n − X ( φ i ) ∼ = Ω − s ( n +1) τ s ( n +1) Ω − s ( n − X ( φ i )(3 . ∼ = Ω − s ( n +1) τ s ( n +1) Ω − s ( n − X ( φ i ) ∼ = τ s ( n +1) Ω − sn X ( φ i )(3 . ∼ = f W Mimo τ s ( n +1) Ω − sn X ( φ i ) . Now (4.4) follows from Lemma 4.1. Since d | N ( n + 1) , d | (2 N n ), taking s = N in (4.4)we get F N ( n +1) S X ( φ i ) ∼ = Mimo τ N ( n +1) Ω − Nn X ( φ i ) ∼ = Mimo X ( φ i ) = X ( φ i ) . (cid:4) Note that the conditions on τ and Ω in Theorem 4.3 hold in particular for representation-finite selfinjective algebras.4.5. Applying Theorem 4.3 to the selfinjective Nakayama algebras Λ( m, t ), we get Corollary 4.4.
Let F S be the Serre functor of S n (Λ( m, t )) with m ≥ , t ≥ , and X bean arbitrary object in S n (Λ( m, t )) . Then ( i ) If t = 2 , then F N ( n +1) S X ∼ = X , where N = m ( m,n − . ( ii ) If t ≥ , then F N ( n +1) S X ∼ = X , where N = m ( m,t,n +1) . Proof. ( i ) In this case τ = Ω, and o ( τ ) = m = o (Ω), by (3.17). Put N = m ( m,n − . Itfollows from (4.4) that F N ( n +1) S X ∼ = Mimo τ N ( n +1) Ω − Nn X ∼ = Mimo Ω − N ( n − X ∼ = Mimo Ω − m n − m,n − X ∼ = Mimo X = X. ( ii ) This follows from Theorem 4.3 by taking d = m and d = m ( m,t ) . By a computationin elementary number theory, we get[ d ( n + 1 , d ) , d (2 n, d ) ] = [ m ( n + 1 , m ) , m ( m,t ) (2 n, m ( m,t ) ) ] = m ( m, t, n + 1) . (cid:4) Appendix 1: Proofs of Lemmas 1.5 and 1.6
We give proofs of Lemmas 1.5 and 1.6.
Lemma 5.1.
Let X ( φ i ) ∈ Mor n ( A ) , I , · · · , I n be injective A -modules such that X ′ ( φ ′ i ) = X ⊕ I ⊕···⊕ I n ... X n − ⊕ I n X n ( φ ′ i ) ∈ S n ( A ) , where φ ′ i = φ i ··· ∗ ··· ∗ ··· ... ... ... ... ... ∗ ··· ( n − i +1) × ( n − i ) . Then X ′ ( φ ′ i ) ∼ = Mimo X ( φ i ) ⊕ J , where J is an injective object of S n ( A ) . Moreover, J = n − L i =1 m i ( Q i ) ,where Q i is an injective A -module such that Q i ⊕ IKer φ i ∼ = I i +1 , ≤ i ≤ n − . Proof.
It is clear that the morphism (1 , , ··· , ... (1 , : X ′ ( φ ′ i ) ։ X ( φ i ) is a right approximationof X ( φ i ) in S n ( A ) (this can be proved as Lemma 1.3( i ), see [Z], Lemma 2.3). By Lemma1.3( i ), there is an object J ∈ S n ( A ) such that X ′ ( φ ′ i ) ∼ = Mimo X ( φ i ) ⊕ J . Comparing thebranches we get J n = 0 and I i +1 ⊕ · · · ⊕ I n ∼ = IKer φ i ⊕ · · · ⊕ IKer φ n − ⊕ J i , ∀ ≤ i ≤ n − . (5 . Q n − = J n − . From I n ∼ = IKer φ n − ⊕ J n − we see that Q n − is an injective A -module. Since J ∈ S n ( A ), Q n − is a submodule of J n − , thus J n − = Q n − ⊕ Q n − .By I n − ⊕ I n ∼ = IKer φ n − ⊕ IKer φ n − ⊕ J n − in (5.1), we see I n − ∼ = IKer φ n − ⊕ Q n − .Repeating this process we see that J i is of the form J i = Q i ⊕ · · · ⊕ Q n − with Q i beinginjective A -modules, and Q i ⊕ IKer φ i ∼ = I i +1 , 1 ≤ i ≤ n −
1. Thus J = n − L i =1 m i ( Q i ) is aninjective object of S n ( A ). (cid:4) Now we can prove Lemma 1.5 (cf. Claim 2 in § n = 2). Proof of Lemma 1.5.
We just prove ( i ). Since the A -map φ ′ i : X i +1 ⊕ I i +2 ⊕· · ·⊕ I n → X i ⊕ I i +1 ⊕ I i +2 ⊕ · · · ⊕ I n is monic, the restriction of φ ′ i on I i +2 ⊕ · · · ⊕ I n is also monic, USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 29 and hence it is a split monomorphism. Hence X ′ ( φ ′ i ) is isomorphic to X ′′ ( φ ′′ i ) = X ⊕ I ⊕···⊕ I n ... X n − ⊕ I n X n ( φ ′′ i ) ∈ S n ( A ) , where φ ′′ i = φ i ··· ∗ ··· ∗ ··· ... ... ... ... ... ∗ ··· ( n − i +1) × ( n − i ) . Then the assertion follows from Lemma 5.1. (cid:4)
Lemma 5.2.
Let X ( f i ) , X ( g i ) ∈ Mor n ( A ) such that f i − g i factors through an injective A -module, ≤ i ≤ n − , and h i : X i → I i be an injective envelope, ≤ i ≤ n . Set X ′ ( f ′ i ) = X ⊕ I ⊕···⊕ I n ... X n − ⊕ I n X n ( f ′ i ) where f ′ i = f i ··· h i +1 ···
00 1 0 ··· ... ... ... ... ... ··· ( n − i +1) × ( n − i ) , and X ′ ( g ′ i ) = X ⊕ I ⊕···⊕ I n ... X n − ⊕ I n X n ( g ′ i ) where g ′ i = g i ··· h i +1 ···
00 1 0 ··· ... ... ... ... ... ··· ( n − i +1) × ( n − i ) . Then X ′ ( f ′ i ) ∼ = X ′ ( g ′ i ) in S n ( A ) . Proof.
For 1 ≤ i ≤ n −
1, it is clear that f i − g i : X i +1 → X i factors through the injectiveenvelope h i +1 : X i +1 → I i +1 , and hence there is an A -map u i : I i +1 → X i such that g i − f i = u i h i +1 . The following commutative diagram shows X ′ ( f ′ i ) ∼ = X ′ ( g ′ i ) . X n f ′ n − / / α n =1 Xn X n − ⊕ I n / / α n − (cid:15) (cid:15) · · · / / X ⊕ I ⊕ · · · ⊕ I nα (cid:15) (cid:15) f ′ / / X ⊕ I ⊕ · · · ⊕ I nα (cid:15) (cid:15) X n g ′ n − / / X n − ⊕ I n / / · · · / / X ⊕ I ⊕ · · · ⊕ I n g ′ / / X ⊕ I ⊕ · · · ⊕ I n where α i = u i g i u i +1 ··· ( g i g i +1 ··· g n − u n − )0 1 h i +1 u i +1 ··· ( h i +1 g i +1 ··· g n − u n − )0 0 1 ··· ( h i +2 g i +2 ··· g n − u n − ) ... ... ... ... ... ··· h n − u n − ··· ( n − i +1) × ( n − i +1) , ≤ i ≤ n . (cid:4) Lemma 5.3.
Let X ( f i ) , X ( g i ) ∈ Mor n ( A ) such that f i − g i factors through an injective A -module, ≤ i ≤ n − . If each X i has no nonzero injective direct summands, then Mimo X ( f i ) ∼ = Mimo X ( g i ) in S n ( A ) . Proof.
Consider X ′ ( f ′ i ) and X ′ ( g ′ i ) defined in Lemma 5.2, which are isomorphic in S n ( A ).By Lemma 5.1, there exist injective A -modules Q f,i and Q g,i such that IKer f i ⊕ Q f,i ∼ = I i +1 ∼ = IKer g i ⊕ Q g,i , ≤ i ≤ n − , and Mimo X ( f i ) ⊕ n − M i =1 m i ( Q f,i ) ∼ = X ′ ( f ′ i ) ∼ = X ′ ( g ′ i ) ∼ = Mimo X ( g i ) ⊕ n − M i =1 m i ( Q g,i ) . By Claim 3 in § f i ∼ = IKer g i , ≤ i ≤ n −
1. Thus Q f,i ∼ = Q g,i , and n − L i =1 m i ( Q f,i ) ∼ = n − L i =1 m i ( Q g,i ), from which the assertion follows. (cid:4) Now we can prove Lemma 1.6 (cf. Theorem 4.2 of [RS2] for the case of n = 2). Proof of Lemma 1.6.
We only prove ( i ). If Mimo X ( φ i ) ∼ = Mimo Y ( ψ i ) in S n ( A ),then by the construction of Mimo , X ( φ i ) ∼ = Y ( ψ i ) in Mor n ( A -mod). Conversely, assumethat X ( φ i ) ∼ = Y ( ψ i ) in Mor n ( A -mod). Since all X i and Y i have no nonzero injective directsummands, there are A -isomorphisms x i : X i → Y i , such that each x i φ i − ψ i x i +1 factorsthrough an injective A -module, 1 ≤ i ≤ n −
1. Then each φ i − x − i ψ i x i +1 factors throughan injective A -module, 1 ≤ i ≤ n −
1. By Lemma 5.3,
Mimo X ( x − i ψ i x i +1 ) ∼ = Mimo X ( φ i ) .Since Y ( ψ i ) ∼ = X ( x − i ψ i x i +1 ) , we get Mimo Y ( ψ i ) ∼ = Mimo X ( φ i ) . (cid:4) Appendix 2: Auslander-Reiten quivers of some monomorphism categories
We include the Auslander-Reiten quivers of some representation-finite monomorphismcategories.6.1. By Simson [S], S n, , S , , S , , S , , S , and S , are the only representation-finitecases among all S n,t = S n ( k [ x ] / h x t i ) , n ≥ , t ≥
2. In [RS3], Ringel and Schmidmeier givethe Auslander-Reiten quivers of S ,t , t = 2 , , ,
5. In the following we give the remainingcases, namely, S n, ( n ≥ S , , and S , .( i ) There are n indecomposable projective objects and n ( n +1)2 indecomposable nonpro-jective objects in S n, . For the Auslander-Reiten quivers of S , see 2.5. The Auslander-Reiten quiver of S , is as follows, where A = k [ x ] / h x i and S is the simple A -module. A (cid:27) (cid:27) AA (cid:27) (cid:27) AAA (cid:27) (cid:27) AAAA (cid:27) (cid:27) S C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) AS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o AAS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o AAAS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o SSSS o o SS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) ASS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o AASS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o SSS C C (cid:7)(cid:7)(cid:7) o o SSS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) ASSS C C (cid:7)(cid:7)(cid:7) (cid:27) (cid:27) o o SS C C (cid:7)(cid:7)(cid:7)(cid:7) o o SSSS C C (cid:7)(cid:7)(cid:7) S C C (cid:7)(cid:7)(cid:7)(cid:7) o o ( ii ) Let A = k [ x ] / h x i . Denote by M and S the two indecomposable nonprojective A -modules, where S is simple. There are 3 indecomposable projective objects and 24 USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 31 indecomposable nonprojective objects in S , , whose Auslander-Reiten quiver is as follows. A " " EEEEE
AAA " " DDDD M " " DDDD < < yyyyy AS " " EEEE o o AAM " " EEEEE < < yyyyy o o MMM ! ! CCC o o M " " DDDD o o MS ! ! DDD < < yyyyy A ⊕ AS ⊕ AM " " DD < < yyy o o MMS ! ! CCC = = {{{{ o o S ⊕ AMM ! ! DD < < zzz o o MS o o S ⊕ AS ⊕ MM " " DDD = = zzz (cid:25) (cid:25) M ⊕ AS ⊕ AM " " EEE < < zz (cid:25) (cid:25) o o M ⊕ AS ⊕ MS " " EEE = = zzz (cid:25) (cid:25) o o S ⊕ AMS ! ! CCC = = {{ (cid:24) (cid:24) o o S ⊕ AS ⊕ MM " " DDD = = zzz (cid:25) (cid:25) o o AMM < < yyy MM < < yyy o o ASS = = {{{ o o SS < < zzz o o AMM o o S ⊕ AS ⊕ AM E E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) " " EEEE
MSS E E (cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11)(cid:11) " " EEEEE o o S ⊕ AM F F (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ! ! CCC o o AMS E E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) " " DDDDD o o S ⊕ AS ⊕ AM o o AAS < < zzzz SSS < < yyyyy o o S < < yyyy o o AM = = {{{{ " " DDDD o o AAS < < zzzz o o AA < < yyyyy ( iii ) Let A, M, S be as in ( ii ). There are 4 indecomposable projective objects and 80indecomposable nonprojective objects in S , , whose Auslander-Reiten quiver looks like.6.2. Consider the selfinjective Nakayama algebra Λ = Λ(2 , S = k / / o o , S = 0 / / k o o , P = k / / k o o , P = k / / k. o o There are 4 indecomposable projective objects and 6 indecomposable nonprojective objectsin S (Λ), with the Auslander-Reiten quiver as follows. P ! ! DDD P P ! ! DDD S = = {{{ ! ! CCC P S = = {{{ ! ! DDD o o S S ! ! DDD o o S ! ! CCC o o S S = = {{{ S = = {{{ ! ! DDD o o P S = = {{{ ! ! DDD o o S S o o P = = {{{ P P = = {{{ There are 6 indecomposable projective objects and 12 indecomposable nonprojectiveobjects in S (Λ), with the Auslander-Reiten quiver as follows. P (cid:31) (cid:31) >>>> P P (cid:31) (cid:31) >>>> P P P (cid:31) (cid:31) >>>> S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) P S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o P P S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o S S S (cid:31) (cid:31) >>>> o o S (cid:31) (cid:31) >>>> o o S S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) P S S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o S S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o P S S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o S S (cid:31) (cid:31) >>>> o o S S S ? ? (cid:0)(cid:0)(cid:0)(cid:0) S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o P S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o P P S (cid:31) (cid:31) >>>> ? ? (cid:0)(cid:0)(cid:0)(cid:0) o o S S S o o P ? ? (cid:0)(cid:0)(cid:0)(cid:0) P P ? ? (cid:0)(cid:0)(cid:0)(cid:0) P P P ? ? (cid:0)(cid:0)(cid:0)(cid:0) References [Ar] D. M. Arnold, Abelian groups and representations of finite partially ordered sets, Canad. Math. Soc.Books in Math., Springer-Verlag, New York, 2000.[ARS] M. Auslander, I. Reiten, S. O. Smalø, Representation theory of Artin algebras, Cambridge Studiesin Adv. Math. 36., Cambridge Univ. Press, 1995.[AS] M. Auslander, S. O. Smalø, Almost split sequences in subcategories, J. Algebra 69(1981), 426-454.[B] G. Birkhoff, Subgroups of abelian groups, Proc. Lond. Math. Soc. II, Ser. 38(1934), 385-401.[C] X. W. Chen, Stable monomorphism category of Frobenius category, avaible in arXiv: math. RT0911.1987, 2009.[CZ] C. Cibils, P.Zhang, Calabi-Yau objects in triangulated categories, Trans. Amer. Math. Soc. 361(2009),6501-6519.[H] D. Happel, Triangulated categories in representation theory of finite dimensional algebras, Lond.Math. Soc. Lecture Notes Ser. 119, Cambridge Univ. Press, 1988.[KLM] D. Kussin, H. Lenzing, H. Meltzer, Nilpotent operators and weighted projective lines, avaible inarXiv: math. RT 1002.3797.[RV] I. Reiten, M. Van den Bergh, Noetherian hereditary abelian categories satisfying Serre duality, J.Amer. Math. Soc. 15(2)(2002), 295-366.[RW] F. Richman, E. A. Walker, Subgroups of p -bounded groups, in: Abelian groups and modules, TrendsMath., Birkh¨auser, Basel, 1999, 55-73. USLANDER-REITEN TRANSLATIONS IN MONOMORPHISM CATEGORIES 33 [RS1] C. M. Ringel, M. Schmidmeier, Submodules categories of wild representation type, J. Pure Appl.Algebra 205(2)(2006), 412-422.[RS2] C. M. Ringel, M. Schmidmeier, The Auslander-Reiten translation in submodule categories, Trans.Amer. Math. Soc. 360(2)(2008), 691-716.[RS3] C. M. Ringel, M. Schmidmeier, Invariant subspaces of nilpotent operators I, J. rein angew. Math.614(2008), 1-52.[S] D. Simson, Representation types of the category of subprojective representations of a finite poset over K [ t ] / ( t m ) and a solution of a Birkhoff type problem, J. Algebra 311(2007), 1-30.[SW] D. Simson, M. Wojewodzki, An algorithmic solution of a Birkhoff type problem, Fundamenta Infor-maticae 83(2008), 389-410.[Z] P. Zhang, Monomorphism categories, cotilting theory, and Gorenstein-projective modules, preprint(2009). Avaible in arXiv: math. RT 1101.3872, 2011. Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R.Chinaxiongbaolin @ gmail.com, pzhang @ sjtu.edu.cn, zyh @@