Exceptional cycles for perfect complexes over gentle algebras☆
Introduction
An exceptional cycle in a triangulated category 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 . Its importance also lies in the fact that it induces an autoequivalences of ([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 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 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 , where A is an indecomposable finite-dimensional gentle algebra.
Throughout, k is an algebraically closed field, is a k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor (if no otherwise stated). Let be the right Serre functor. So, for objects X and Y in , there is a k-linear isomorphism , which is functorial in X and Y, where is the k-dual . Then S is a triangle-equivalence ([13, Appendix]) and has Auslander-Reiten triangles with on objects, where τ is the Auslander-Reiten translate (see [35, Theorem I.2.4]).
An indecomposable object of 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 will be said to be homotopy-like, if for any indecomposable object and for all . It is clear that is homotopy-like, where is an arbitrary additive subcategory of an abelian category, which is closed under direct summands. So 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 is not homotopy-like. So, if Λ is a non semi-simple finite-dimensional self-injective algebra of finite representation type, then 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, the bounded derived category of A-mod, and (, respectively) the bounded homotopy category of finitely generated projective (injective, respectively) A-modules. The Nakayama functor induces componentwisely an equivalence , so that for and there is a k-linear functorial isomorphism in both arguments (see [21, p. 37]): Since A is a Gorenstein algebra (see [19]), in and S is the Serre functor of (see [22]). So 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 is a d-Calabi-Yau object if . In general d-Calabi-Yau objects are not closed under taking direct summands (see [18]). For objects , let be the complex of k-vector spaces with and with zero differentials. Thus, forgetting the differential, .
By definition, an exceptional 1-cycle in is a d-Calabi-Yau object E for some integer d, such that . An exceptional n-cycle in with is a sequence of objects with for each i and for and , and there are integers such that for each i, where .
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 .
Let E be an exceptional 1-cycle which is d-Calabi-Yau. Then d is unique. If , or and as algebras, then E is indecomposable; if and as algebras, then E is decomposable. However we have
Theorem 1.1 Let be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Assume that is homotopy-like. Then Any exceptional 1-cycle is indecomposable, and at the mouth. Any object in an exceptional n-cycle with is at the mouth.
Remark (1) If is not indecomposable, then an exceptional 1-cycle may be decomposable and not at the mouth. For example, is an exceptional 1-cycle in . (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 . Then has an exceptional 2-cycle , where (respectively, ) is the indecomposable projective (respectively, injective) A-module at vertex 2. However, and are not at the mouth of .
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 , 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 , we need the notion, the shape, and the AG-invariant of a characteristic component of .
An indecomposable object of is either a string complex or a band complex (see [9], [10]). The description of Auslander-Reiten triangles of ([10, Main Theorem]) shows that a connected component C of 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 form 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 of integers, such that , for any indecomposable object X at the mouth of C. This pair 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 -, where is the repetitive algebra. Since A is Gorenstein, the Happel embedding 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 ).
Since objects in an exceptional n-cycle in are at the mouth (with a unique exception in the case ), the dimension of the Hom spaces between string complexes at the mouth will play a central role in determining exceptional cycles in . This is given as follows.
Theorem 1.2 Let A be an indecomposable finite-dimensional gentle algebra, M and string complexes at the mouth. Then In particular,
Corollary 1.3 Let A be an indecomposable finite-dimensional gentle algebra, C and different characteristic components of , up to shift. Then for and .
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 , 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 and with , define if and only if there are integers with , such that where σ is the cyclic permutation . Then ≈ is an equivalent relation on the set of all the exceptional n-cycles in with .
The following main result classifies all the exceptional n-cycle in . 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 be a finite-dimensional gentle algebra with , where Q is a finite connected quiver such that the underlying graph of Q is not of type . Let C be a characteristic component of with AG-invariant , and X an indecomposable object at the mouth of C. Then is the unique exceptional cycle in C, up to the equivalent relation ≈. (Thus, it does not depend on m.) Any exceptional n-cycle in with is given in (1), up to ≈. Any object in an exceptional cycle in 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 . 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 with : these are exactly derived-discrete algebras of finite global dimension which is not of Dynkin type, up to derived equivalence.
Remark (1) If , then has no mouths, and is the unique exceptional cycle in , up to ≈. So, Theorem 1.4(2) does not hold for . (2) If is a gentle algebra such that the underlying graph of Q is of type , then , and and are all the exceptional cycles in , 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 with is a sequence of objects satisfying the conditions: for each i; there are integers such that for each i, where ; , unless or . (This condition vanishes if .)
The sequence of integers in the definition is unique, and we will call an exceptional cycle with respect to . It is clear that (see e.g. [20, Lemma 2.2]) a sequence of objects in with
The following observation is essentially due to I. Reiten and M. Van den Bergh [35].
Lemma 3.1 Let be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Suppose that is homotopy-like and . Then for any indecomposable object M of , there are no Auslander-Reiten triangles of the form ; and there is a non-zero morphism which is not an isomorphism.
Remark In any non-zero morphism is an isomorphism, and is an
Characteristic components of
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 of , via single maps, double maps, and graph maps, introduced in [3].
Exceptional 2-cycles of the form
We first point out the following fact.
Proposition 6.1 Let be an indecomposable k-linear Hom-finite Krull-Schmidt triangulated category with Serre functor. Assume that is homotopy-like. Then there exists an exceptional 2-cycle of the form in if and only if . Proof It is clear that is an exceptional cycle in . Assume that is an exceptional cycle in . Then for some integer t, by (E2). Since , by (E1) one has . Thus there is an
Examples
Example 7.1 Let . Since A is symmetric, is a 0-Calabi-Yau category (in the sense of [28]), and hence each object is 0-Calabi-Yau. The indecomposable objects in are of form with and , where is the following complex with each differential given by multiplication by x: The Auslander-Reiten quiver of has a unique component of the type , which is a characteristic component with AG invariant , as given below. All the exceptional 1-cycles are
Acknowledgement
We sincerely thank the referee for carefully reading the manuscript and helpful suggestions.
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