Elsevier

Journal of Algebra

Volume 565, 1 January 2021, Pages 160-195
Journal of Algebra

Exceptional cycles for perfect complexes over gentle algebras

https://doi.org/10.1016/j.jalgebra.2020.08.013Get rights and content

Abstract

Exceptional cycles in a triangulated category T with Serre duality, introduced by N. Broomhead, D. Pauksztello, and D. Ploog, have a notable impact on the global structure of T. In this paper we show that if T is homotopy-like, then any exceptional 1-cycle is indecomposable and at the mouth (i.e., the middle term of the Auslander-Reiten triangle ending at it is indecomposable); and any object in an exceptional n-cycle with n3 is at the mouth. Let A be an indecomposable gentle k-algebra with Ak. The Hom spaces between string complexes at the mouth are explicitly determined. The main result classifies “almost all” the exceptional n-cycles in Kb(A-proj), using characteristic components and their AG-invariants, except those exceptional 1-cycles which are band complexes. Namely, the mouth of a characteristic component C of Kb(A-proj) forms a unique exceptional cycle in C, up to an equivalent relation ≈; if the quiver of A is not of type A3, this gives all the exceptional n-cycle in Kb(A-proj) with n2, up to ≈; and a string complex is an exceptional 1-cycle if and only if it is at the mouth of a characteristic component with AG-invariant (1,m). However, a band complex at the mouth is possibly not an exceptional 1-cycle.

Introduction

An exceptional cycle in a triangulated category T with Serre duality has been introduced by N. Broomhead, D. Pauksztello, and D. Ploog [15]. It is a generalization of a spherical object (see e.g. [40], [26]), provides an invariant of triangle-equivalences, and closely relates to the global structure of T. Its importance also lies in the fact that it induces an autoequivalences of T ([15, Theorem 4.5]), which is a generalization of tubular mutation in [30].

Gentle algebras, introduced by I. Assem and A. Skowronski [7], have related to different topics. It is closed under derived equivalence by J. Schröer and A. Zimmermann [39], and exactly the class of finite-dimensional algebras A such that the repetitive algebras Aˆ are special biserial ([7], [34]). It appears in D. Vossieck's classification of algebras with discrete derived categories ([41]), and in cluster tilted algebras (see e.g. [4], [2]). The combinatorial description of Auslander-Reiten triangles of Kb(A-proj) has been given by G. Bobiński [10]. Recently, a geometric derived realization and complete derived invariants of gentle algebras are given in [33], [1] and [32].

The aim of this paper is to study exceptional cycles in an indecomposable homotopy-like triangulated category with Serre duality, and to determine all the exceptional cycles in homotopy category Kb(A-proj), where A is an indecomposable finite-dimensional gentle algebra.

Throughout, k is an algebraically closed field, T is a k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor (if no otherwise stated). Let S:TT be the right Serre functor. So, for objects X and Y in T, there is a k-linear isomorphism HomT(X,Y)HomT(Y,S(X)), which is functorial in X and Y, where is the k-dual Homk(,k). Then S is a triangle-equivalence ([13, Appendix]) and T has Auslander-Reiten triangles with S=τ[1] on objects, where τ is the Auslander-Reiten translate (see [35, Theorem I.2.4]).

An indecomposable object of T is said to be at the mouth, if the middle term of the Auslander-Reiten triangle ending at it is indecomposable, or equivalently, the middle term of the Auslander-Reiten triangle starting from it is indecomposable.

A triangulated category T will be said to be homotopy-like, if MM[i] for any indecomposable object MT and for all i0. It is clear that Kb(B) is homotopy-like, where B is an arbitrary additive subcategory of an abelian category, which is closed under direct summands. So Kb(A-proj) is homotopy-like. On the other hand, if Λ is a finite-dimensional self-injective algebra admitting a non-zero Ω-periodic module, then the stable category Λ-mod_ is not homotopy-like. So, if Λ is a non semi-simple finite-dimensional self-injective algebra of finite representation type, then Λ-mod_ is not homotopy-like.

Throughout, A is an indecomposable finite-dimensional gentle k-algebra. Let A-mod be the category of finitely generated left A-modules, Db(A) the bounded derived category of A-mod, and Kb(A-proj) (Kb(A-inj), respectively) the bounded homotopy category of finitely generated projective (injective, respectively) A-modules. The Nakayama functor HomA(,A) induces componentwisely an equivalence S:Kb(A-proj)Kb(A-inj), so that for PKb(A-proj) and XDb(A) there is a k-linear functorial isomorphism in both arguments (see [21, p. 37]):HomDb(A)(P,X)HomDb(A)(X,S(P)). Since A is a Gorenstein algebra (see [19]), Kb(A-proj)=Kb(A-inj) in Db(A) and S is the Serre functor of Kb(A-proj) (see [22]). So Kb(A-proj) is a k-linear, Hom-finite Krull-Schmidt triangulated category with Serre functor S, and having Auslander-Reiten triangles.

Let d be an integer. An object ET is a d-Calabi-Yau object if S(E)E[d]. In general d-Calabi-Yau objects are not closed under taking direct summands (see [18]). For objects X,YT, let Hom(X,Y) be the complex of k-vector spaces with Homi(X,Y)=HomT(X,Y[i]) and with zero differentials. Thus, forgetting the differential, Hom(X,Y)=iHomT(X,Y[i])[i].

By definition, an exceptional 1-cycle in T is a d-Calabi-Yau object E for some integer d, such that Hom(E,E)kk[d]. An exceptional n-cycle in T with n2 is a sequence (E1,,En) of objects with Hom(Ei,Ei)k for each i and Hom(Ei,Ej)=0 for ji and ji+1, and there are integers mi such that S(Ei)Ei+1[mi] for each i, where En+1:=E1.

An exceptional 1-cycle is also called a spherical object for example in [26] and [27]. Note that “a spherical object” has (slight) different meanings in the literatures, see for example [40], [29], [17]. Here we follow [15], using the unified terminology of an exceptional n-cycle with n1.

Let E be an exceptional 1-cycle which is d-Calabi-Yau. Then d is unique. If d0, or d=0 and EndT(E)k[x]/x2 as algebras, then E is indecomposable; if d=0 and EndT(E)k×k as algebras, then E is decomposable. However we have

Theorem 1.1

Let T be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Assume that T is homotopy-like. Then

  • (1)

    Any exceptional 1-cycle is indecomposable, and at the mouth.

  • (2)

    Any object in an exceptional n-cycle with n3 is at the mouth.

Remark

(1) If T is not indecomposable, then an exceptional 1-cycle may be decomposable and not at the mouth. For example, kk is an exceptional 1-cycle in Db(k)×Db(k).

(2) An object in an exceptional 2-cycle may be not at the mouth. For example, let A be the path algebra of the quiver 123. Then Db(A) has an exceptional 2-cycle (P(2),I(2)), where P(2) (respectively, I(2)) is the indecomposable projective (respectively, injective) A-module at vertex 2. However, P(2) and I(2) are not at the mouth of Db(A).

The proof of Theorem 1.1 will be given in Section 3. The main ideas in the proof are to use a special non-zero non isomorphism from an indecomposable object M to S(M), constructed in [35] (see Lemma 3.1), and the mapping cone of the composition of morphisms in a homotopy cartesian square (see Lemma 3.3, Lemma 3.4).

To determine the exceptional cycles in Kb(A-proj), we need the notion, the shape, and the AG-invariant of a characteristic component of Kb(A-proj).

An indecomposable object of Kb(A-proj) is either a string complex or a band complex (see [9], [10]). The description of Auslander-Reiten triangles of Kb(A-proj) ([10, Main Theorem]) shows that a connected component C of Kb(A-proj) either consists of string complexes, or consists of band complexes. It will be called a characteristic component, if C contains a string complex at the mouth (thus C consists of string complexes). It is known that C is of the formZAn(n2),ZA,ZA/τn(n1). See Proposition 4.3. Since up to shift there are only finitely many string complexes at the mouth, by the shape of C, there exists a unique pair (n,m) of integers, such that τnXX[mn], for any indecomposable object X at the mouth of C. This pair (n,m) will be called the invariant of Avella-Alaminos and Geiss (or the AG-invariant in short) of C. For details see Section 4.

In [8] a characteristic component and its invariant have been defined for Aˆ-mod_, where Aˆ is the repetitive algebra. Since A is Gorenstein, the Happel embedding Kb(A-proj)Aˆ-mod_ preserves the Auslander-Reiten components ([25, Corollary 5.3]). Thus, a characteristic component and its AG invariant here coincide with the ones in [8] (but we include the component of type ZAn).

Since objects in an exceptional n-cycle in Kb(A-proj) are at the mouth (with a unique exception in the case n=2), the dimension of the Hom spaces between string complexes at the mouth will play a central role in determining exceptional cycles in Kb(A-proj). This is given as follows.

Theorem 1.2

Let A be an indecomposable finite-dimensional gentle algebra, M and M string complexes at the mouth. ThendimkHomKb(A-proj)(M,M)={2,ifMS(M)M;1,ifMis isomorphic to eitherS(M)orM,butMis not isomorphic to both of them;0,ifMS(M)andMM. In particular,dimkEnd(M)=dimkHomKb(A-proj)(M,S(M))={2,ifMS(M);1,ifMS(M).

Corollary 1.3

Let A be an indecomposable finite-dimensional gentle algebra, C and C different characteristic components of Kb(A-proj), up to shift. Then Hom(X,Y)=0 for XC and YC.

The proofs will be given in Section 5. The main tools used are Lemma 3.1, the combinatorial description of morphisms between indecomposable objects in Db(A), given by K.K. Arnesen, R. Laking and D. Pauksztello in [3] (see Subsection 5.1), and the bijections between the set of permitted threads and the set of forbidden threads, given by D. Avella-Alaminos and C. Geiss in [8] (see also [11]. See Subsection 2.5).

For two exceptional n-cycles (E1,,En) and (E1,,En) with n1, define (E1,,En)(E1,,En) if and only if there are integers t1,,tn,s with 0sn1, such that(E1,,En)=(Eσs(1)[t1],,Eσs(n)[tn]) where σ is the cyclic permutation (12n). Then ≈ is an equivalent relation on the set En of all the exceptional n-cycles in T with n1.

The following main result classifies all the exceptional n-cycle in Kb(A-proj). It turns out that such an exceptional cycle is exactly a truncation of the τ-orbit of any indecomposable object at the mouth of a characteristic component, with few exceptions.

Theorem 1.4

Let A=kQ/I be a finite-dimensional gentle algebra with Ak, where Q is a finite connected quiver such that the underlying graph of Q is not of type A3.

  • (1)

    Let C be a characteristic component of Kb(A-proj) with AG-invariant (n,m), and X an indecomposable object at the mouth of C. Then(X,τX,,τn1X) is the unique exceptional cycle in C, up to the equivalent relation. (Thus, it does not depend on m.)

  • (2)

    Any exceptional n-cycle in Kb(A-proj) with n2 is given in (1), up to.

  • (3)

    Any object in an exceptional cycle in Kb(A-proj) is indecomposable and at the mouth.

A string complex E is an exceptional 1-cycle if and only if E is at the mouth of a characteristic component of AG-invariant (1,d). In this case E is a d-Calabi-Yau object.

If a band complex E is an exceptional 1-cycle, then E is at the mouth of a homogeneous tube.

N. Broomhead, D. Pauksztello, and D. Ploog [15, 5.1] found earlier the assertion (1) for the gentle algebras Λ(r,n,m) with n>r: these are exactly derived-discrete algebras of finite global dimension which is not of Dynkin type, up to derived equivalence.

Remark

(1) If A=k, then Kb(A-proj)=Db(k) has no mouths, and (k,k) is the unique exceptional cycle in Kb(A-proj), up to ≈. So, Theorem 1.4(2) does not hold for A=k.

(2) If A=kQ/I is a gentle algebra such that the underlying graph of Q is of type A3, then Kb(A-proj)Db(k(123)), and (P(3),I(3)=P(1),I(1),S(2)) and (P(2),I(2)) are all the exceptional cycles in Db(k(123)), up to ≈. So, Theorem 1.4(1) and (2) do not hold in this case.

(3) A band complex at the mouth of a homogeneous tube is not necessarily an exceptional 1-cycle. See Example 7.2.

The proof of Theorem 1.4 will be given in Section 6. The main tools used in the proof are Theorem 1.2, Lemma 6.2, and Theorem 1.1.

Section snippets

Exceptional cycles

Definition 2.1

([15]) An exceptional n-cycle in T with n2 is a sequence (E1,,En) of objects satisfying the conditions:

  • (E1)

    Hom(Ei,Ei)k for each i;

  • (E2)

    there are integers mi such that S(Ei)Ei+1[mi] for each i, where En+1:=E1;

  • (E3)

    Hom(Ei,Ej)=0, unless j=i or j=i+1. (This condition vanishes if n=2.)

The sequence (m1,,mn) of integers in the definition is unique, and we will call (E1,,En) an exceptional cycle with respect to (m1,,mn). It is clear that (see e.g. [20, Lemma 2.2]) a sequence (E1,,En) of objects in T with n

The following observation is essentially due to I. Reiten and M. Van den Bergh [35].

Lemma 3.1

Let T be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Suppose that T is homotopy-like and TDb(k). Then for any indecomposable object M of T, there are no Auslander-Reiten triangles of the form τM0MS(M); and there is a non-zero morphism f:MS(M) which is not an isomorphism.

Remark

In Db(k) any non-zero morphism kS(k)=k is an isomorphism, and k[1]0kIdkk is an

Characteristic components of Kb(A-proj)

Let A be an indecomposable finite-dimensional gentle k-algebra.

Proof of Theorem 1.2 and Corollary 1.3

A key observation for proving Theorem 1.2 is Lemma 5.2 below. One of the tools in the proof of Lemma 5.2 is a description of morphisms between string complexes in the bounded complex category Cb(A-proj) of A-proj, via single maps, double maps, and graph maps, introduced in [3].

Exceptional 2-cycles of the form (E,E)

We first point out the following fact.

Proposition 6.1

Let T be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Assume that T is homotopy-like. Then there exists an exceptional 2-cycle of the form (E,E) in T if and only if TDb(k).

Proof

It is clear that (k,k) is an exceptional cycle in Db(k). Assume that (E,E) is an exceptional cycle in T. Then S(E)E[t] for some integer t, by (E2). Since HomT(E,E[t])=HomT(E,S(E))End(E)k, by (E1) one has t=0. Thus there is an

Examples

Example 7.1

Let A=k[x]/x2. Since A is symmetric, Kb(A-proj) is a 0-Calabi-Yau category (in the sense of [28]), and hence each object is 0-Calabi-Yau. The indecomposable objects in Kb(A-proj) are of form Xl[t] with tZ and l0, where Xl is the following complex with each differential given by multiplication by x: The Auslander-Reiten quiver of Kb(A-proj) has a unique component of the type ZA, which is a characteristic component with AG invariant (1,0), as given below. All the exceptional 1-cycles are A[t]

Acknowledgement

We sincerely thank the referee for carefully reading the manuscript and helpful suggestions.

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    Supported by the NNSFC (National Natural Science Foundation of China) Grant No. 11971304.

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