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Abstract

Let be a Hom-finite triangulated 2-Calabi–Yau category with a cluster tilting object. Under some constructibility assumptions on which are satisfied for instance by cluster categories, by generalized cluster categories and by stable categories of modules over a preprojective algebra of Dynkin type, we prove a multiplication formula for the cluster character associated with any cluster tilting object. This formula generalizes those obtained by Caldero–Keller for representation finite path algebras and by Xiao–Xu for finite-dimensional path algebras. We prove an analogous formula for the cluster character defined by Fu–Keller in the setup of Frobenius categories. It is similar to a formula obtained by Geiss–Leclerc–Schröer in the context of preprojective algebras.

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defiDefinition \newnumberedremarkRemark Cluster characters II]Cluster characters II: A multiplication formula \classno18E30 (primary), 16G20, 13F60, 14A10 (secondary)

Introduction

In recent years, the link between Fomin–Zelevinsky’s cluster algebras [FZ02] and the representation theory of quivers and finite-dimensional algebras has been investigated intensely, cf. for example the surveys [BM06], [GLS08b], [Kel10]. In its most tangible form, this link is given by a map taking objects of cluster categories to elements of cluster algebras. Such a map was first constructed by P. Caldero and F. Chapoton [CC06] for cluster categories and cluster algebras associated with Dynkin quivers. Another approach, leading to proofs of several conjectures on cluster algebras in a more general context, can be found in [DWZ08][DWZ10] (for proofs relying on the use of a Caldero–Chapoton map, see [Pla11], [Pla]).

The results of P. Caldero and B. Keller [CK08] yield two multiplication formulae for the Caldero–Chapoton map of cluster categories associated with Dynkin quivers. The first one categorifies the exchange relations of cluster variables and only applies to objects and such that is of dimension . The second one generalizes it to arbitrary dimensions, and yields some new relations in the associated cluster algebras. These relations very much resemble relations in dual Ringel–Hall algebras [Sch08, section 5.5]. Motivated by these results, C. Geiss, B. Leclerc and J. Schröer [GLS07] proved two analogous formulae for module categories over preprojective algebras. In this latter situation, the number of isomorphism classes of indecomposable objects is usually infinite. Generalizations of the first formula were proved in [CK06] for cluster categories associated with any acyclic quiver, and later in [Pal08] for Hom-finite 2-Calabi–Yau triangulated categories. A generalisation to the Hom-infinite case can be found in [Pla11], and a version for quantum cluster algebras in [Qin]. The first generalization of the second multiplication formula, by A. Hubery (see [Hub]), was based on the existence of Hall polynomials which he proved in the affine case [Hub10], generalizing Ringel’s result [Rin90] for Dynkin quivers. Staying close to this point of view, J. Xiao and F. Xu proved in [XX10] a projective version of Green’s formula [Rin96] and applied it to generalize the multiplication formula for acyclic cluster algebras. Another proof of this formula was found by F. Xu in [Xu10], who used the 2-Calabi–Yau property instead of Green’s formula. Our aim in this paper is to generalize the second multiplication formula to more general 2-Calabi–Yau categories for the cluster character associated with an arbitrary cluster tilting object. This in particular applies to the generalized cluster categories introduced by C. Amiot [Ami09] and to stable categories of modules over a preprojective algebra.

Assume that the triangulated category is the stable category of a Hom-finite Frobenius category . Then C. Fu and B. Keller defined a cluster character on , which ”lifts” the one on . We prove that it satisfies the same multiplication formula as the one proved by Geiss–Leclerc–Schröer in [GLS07].

The paper is organized as follows: In the first section, we fix some notations and state our main result: A multiplication formula for the cluster character associated with any cluster tilting object. In section 2, we recall some definitions and prove the ‘constructibility of kernels and cokernels’ in modules categories. We apply these facts to prove that:

  • If the triangulated category has constructible cones (see section 1.4), the sets under consideration in the multiplication formula, and in its proof, are constructible.

  • Stable categories of Hom-finite Frobenius categories have constructible cones.

  • Generalized cluster categories defined in [Ami09] have constructible cones.

Thus, all of the Hom-finite 2-Calabi–Yau triangulated categories related to cluster algebras which have been introduced so far have constructible cones. Notably this holds for cluster categories associated with acyclic quivers, and for the stable categories associated with the exact subcategories of module categories over preprojective algebras constructed in [GLS08a] and  [BIRS09]. In section 3, we prove the main theorem. In the last section, we consider the setup of Hom-finite Frobenius categories. We prove a multiplication formula for the cluster character defined by Fu–Keller in [FK10].

1 Notations and main result

Let be the field of complex numbers. The only place where we will need more than the fact that is an algebraically closed field is proposition 2.1 in section 2.1. See [Joy06, section 3.3] for an explanation, illustrated with an example, of the fact that the theory of constructible functions does not extend to fields of positive characteristic. Let be a Hom-finite, 2-Calabi–Yau, Krull–Schmidt -category which admits a basic cluster tilting object . In order to prove the main theorem, a constructibility hypothesis will be needed. This hypothesis is precisely stated in section 1.3 and it will always be explicitly stated when it is assumed. Stable categories of Hom-finite Frobenius categories satisfy this constructibility hypothesis, cf. section 2.4, so that the main theorem applies to cluster categories (thanks to the construction in [GLSa, Theorem 2.1]), to stable module categories over preprojective algebras… Moreover, the main theorem applies to the generalized cluster categories of [Ami09], cf. section 2.5.

We let denote the endomorphism algebra of in , and we let denote the covariant functor from to co-represented by . We denote the image in of an object in under the cluster character associated with (see [Pal08]) by . Before recalling the formula for , we need to introduce some notation. Let be the Gabriel quiver of , and denote by its vertices. For each vertex , denote by (resp. ) the corresponding simple (resp. projective) module. For any two finite-dimensional -modules and , define

As shown in [Pal08, Section 3], the form descends to the Grothendieck group (that is, it only depends on the dimension vectors of and ). Note that this would not be true for the form in general, since is quite often of infinite global dimension (see [KR07]). For a -module , the projective variety is the Grassmannian of submodules of with dimension vector . For any object , there are triangles

with in add (see [KR07, Proposition 2.1]), which are triangulated analogues of projective presentations and injective copresentations respectively. The index and coindex of (with respect to ) are the following classes in :

For some properties of the index, see [Pal08], and for a more thorough study, see [DK08], [FK10] and [Pla, Section 4.2]. Then we have:

where the sum runs over all classes . For any two objects and in , and any morphism in , we denote the isomorphism class of objects appearing in a triangle of the form

by (the middle term of ). Note that any two such objects are isomorphic.

1.1 -stratification

Let and be objects in . If an object of belongs to the class for some morphism in , we let denote the set of all isomorphism classes of objects such that:

  • is the middle term of some morphism in ,

  • and

  • for all in , we have .

The equality of classes yields an equivalence relation on the ‘set’ of middle terms of morphisms in . Fix a set of representatives for this relation. Further, we denote the set of all with by , and the set of such that by . It will be proven in section 2.3 that if the cylinders of the morphisms are constructible with respect to in the sense of section 1.3 below, then the sets are constructible, and the set is finite.

Remark that if belongs to , then . Hence the fibers of the map sending to are finite unions of sets . Therefore, the sets are constructible; we have

for some finite set , and

is a refinement of the previous decomposition.

1.2 The variety

Let be a finite dimensional -vector space. We denote by the set of morphisms of -algebras from to . Since is finitely generated, the set is a closed subvariety of some finite product of copies of .

Let be a finite quiver, and let be a tuple of non-negative integers. A -dimensional matrix representation of in is given by

  • a right -module structure on for each vertex of and

  • a -linear map for each arrow of .

Clearly, for fixed , the -dimensional matrix representations of in form an affine variety on which the group acts by changing the bases in the spaces . We write for the set of orbits.

1.3 Constructible cones

Let be the quiver: . Let , and be objects of . Let be the -tuple of integers

Let be the map from to

sending a morphism to the orbit of the exact sequence of -modules

where is a triangle in . The cylinders over the morphisms are constructible with respect to if the map lifts to a constructible map

(see section 2.1). The category is said to have constructible cones if this holds for arbitrary objects , and .

1.4 Main result

Let be a constructible function from an algebraic variety over to any abelian group, and let be a constructible subset of this variety. Then one defines “the integral of on with respect to the Euler characteristic” to be

cf. for example the introduction of [Lus97]. Our aim in this paper is to prove the following:

Theorem 1.1

Let be any cluster tilting object in . Let and be two objects such that the cylinders over the morphisms and are constructible with respect to . Then we have:

where denotes the class in of a non zero morphism in .

The statement of the theorem is inspired from [GLS07], cf. also [XX10]. We will prove it in section 3. Our proof is inspired from that of P. Caldero and B. Keller in [CK08]. Note that in contrast with the situation considered there, in the above formula, an infinite number of isomorphism classes may appear.

2 Constructibility

2.1 Definitions

Let be a topological space. A locally closed subset of is the intersection of a closed subset with an open one. A constructible subset is a finite (disjoint) union of locally closed subsets. The family of constructible subsets is the smallest one containing all open (equivalently: closed) subsets of and stable under taking finite intersections and complements. A function from to an abelian group is constructible if it is a finite -linear combination of characteristic functions of constructible subsets of . Equivalently, is constructible if it takes a finite number of values and if its fibers are constructible subsets of .

For an algebraic variety , the ring of constructible functions from to is denoted by . The following proposition will be used, as in [XX10], in order to prove lemma 2.4 of section 2.3.

Proposition 2.1

[Dim04, Proposition 4.1.31] Associated with any morphism of complex algebraic varieties , there is a well-defined push-forward homomorphism . It is determined by the property

for any closed subvariety in and any point .

Let and be algebraic varieties. A map is said to be constructible if there exists a decomposition of into a finite disjoint union of locally closed subsets , such that the restriction of to each is a morphism of algebraic varieties. Note that the composition of two constructible maps is constructible, and that the composition of a constructible function with a constructible map is again a constructible function.

2.2 Kernels and cokernels are constructible

In section 2.1 of [Xu10], it is shown that the kernel and cokernel of a morphism of modules over a path algebra are constructible. In this section, we give direct proofs in the more general case where is replaced by a finite dimensional algebra .

Let and be two finite dimensional vector spaces over the field , of respective dimensions and . Let be a linear subspace of . Define to be the set of all morphisms such that .

Lemma 2.2

The set is a locally closed subset of .

{proof}

Let be a basis of , and let be a basis of whose first vectors form a basis of . Let be such that . Let be a -linear map, and denote by its matrix in the bases and . Denote by the submatrix of formed by its first rows and by the one formed by its last rows. For , let be the set of all subsets of of cardinality .

The map belongs to if and only if:

  • There exists in such that the submatrix has a non-zero determinant and

  • if the last entries of a linear combination of columns of vanish, then the combination itself vanishes.

Condition b) is equivalent to the inclusion and so to the inclusion . Therefore, condition b) can be restated as condition b’):

  • For all , and all , the determinant of the submatrix of obtained by taking lines in and columns in vanishes.

Let be the set of all maps that satisfy condition a) with respect to the index set , and let be the set of all maps that satisfy condition b’). For all , the set is an open subset of and the set is a closed subset of . Since we have the equality:

the set is locally closed in .

Let be the quiver: .

Lemma 2.3

Let be a finite dimensional algebra, and let and be finitely generated -modules of dimensions and respectively. The map from to which sends a morphism to the orbit of the representation lifts to a constructible map from to .

Dually, the map from to which sends a morphism to the orbit of the representation lifts to a constructible map from to .

{proof}

Let us prove the first assertion. We keep the notations of the proof of lemma 2.2. For a subset of , let be the linear subspace of generated by . Then is the union of its intersections with each , for . It is thus enough to consider the restriction of the map to , where is a given linear subspace of . Since the set is the union of the locally closed subsets , for , we can fix such a and only consider the restriction of to . Let be a morphism in and assume that is in . Then the cokernel of the -linear map is and the projection of onto along is given by the matrix , where is the submatrix and is the submatrix . Moreover, if we denote by the structure of -module of , then the structure of -module of induced by is given by , for all .

2.3 Constructibility of

Let , and be as in section 1. Recall that denotes the endomorphism algebra . This algebra is the path algebra of a quiver with ideal of relations . Recall that we denote by the vertices of .

The following lemma is a particular case of [Dim04, Proposition 4.1.31], and was already stated in [XX10] for hereditary algebras.

Lemma 2.4

For any two dimension vectors and with , the function

is constructible.

{proof}

Let be the closed subset of

formed by those pairs for which the subspaces , , form a subrepresentation. Apply proposition 2.1 to the first projection and remark that .

Corollary 2.5

Let and be objects in , and let be in . Assume that the cylinders over the morphisms are constructible. Then the function

is constructible.

{proof}

By our hypothesis, the map sending to the image of its middle term in , where the union is over the dimension vectors not greater than , lifts to a constructible map from to . The claim therefore follows from lemma 2.4.

Let be a triangle in , and denote by the class of in the Grothendieck group .

Lemma 2.6

We have:

{proof}

Let lift . Using respectively proposition 2.2, lemma 2.1.(2), lemma 7 and section 3 of [Pal08], we have the following equalities:

Corollary 2.7

Let and be two objects such that the cylinders over the morphisms are constructible. The map which sends to the coindex (or to the index) of its middle term is constructible.

{proof}

Note that is at most the sum of the dimension vectors of and , so that by lemma 2.6 the map takes a finite number of values. By our hypothesis and lemma 2.3, there exists a constructible map:

which lifts the map sending to the isomorphism class of the structure of -module on . Moreover, the map sending a module in to in only depends on the dimension vector of and thus is constructible. Therefore, the map is constructible.

Proposition 2.8

Let be such that the cylinders over the morphisms are constructible. Then the sets are constructible subsets of . Moreover, the set is a finite disjoint union of such constructible subsets.

{proof}

Fix a triangle in . Then is in if and only if

  • and

  • For all , .

Therefore, the claim follows from corollary 2.5 and corollary 2.7.

2.4 Stable categories have constructible cones

In this section, we assume moreover that is the stable category of a Hom-finite, Frobenius, Krull–Schmidt category , which is linear over the algebraically closed field . Our aim is to prove that such a category has constructible cones.

Let denote the ideal in of morphisms factoring through a projective-injective object. Let , and be objects of the category . Fix a -linear section of the projection induced by the canonical functor . Fix a conflation in , with being projective-injective in , and, for any in , consider its pull-back via :

Via , this diagram induces a triangle in .

For any , we have a commutative diagram with exact rows:

Fix and a morphism . Denote by the endomorphism algebra of in the category of morphisms of , and by the set of dimension vectors such that , , and .

Lemma 2.9

There exists a constructible map

which lifts the map sending to the orbit of the matrix representation of in given by .

{proof}

By definition of a pull-back, the map is a kernel for the map , with appropriate signs. Moreover, the morphism is a kernel for . Therefore, lemma 2.3 in section 2.2 applies and such a constructible map exists.

Denote by the set of dimension vectors such that:

, , and .

Proposition 2.10

There exists a constructible map

which lifts the map sending to the orbit of the representation

{proof}

Let be an inflation from to a projective-injective object in . This inflation induces a commutative diagram of modules over the endomorphism algebra of in the Frobenius category of inflations of :

The map which sends to the orbit of the diagram lifts to a constructible one. This is proved by repeating the proof of lemma 2.9 for the functor

instead of and using lemma 2.3 for .

By applying lemma 2.3 to , we see that the vertical cokernel of diagram is constructible as a -module. Now the claim follows because the terms of the cokernel are -modules and is also the stable endomorphism algebra of in the Frobenius category of inflations of .

2.5 Generalized cluster categories have constructible cones

Let be a Jacobi-finite quiver with potential in (cf. section 3.3 of [Ami09]), and let be the Ginzburg dg algebra associated with (cf. section 4.2 of [Gin]). The perfect derived category is the thick subcategory of the derived category generated by . The finite dimensional derived category is the full subcategory of whose objects are the dg modules whose homology is of finite total dimension. An object belongs to if and only if is finite dimensional for each object of .

Lemma [Keller–Yang]

  • The category is contained in .

  • An object of belongs to if and only if it is quasi-isomorphic to a dg -module of finite total dimension.

  • The category is equivalent to the localization of the homotopy category of right dg -modules of finite total dimension with respect to its subcategory of acyclic dg modules.

Note that the previous lemma, taken from the appendix of [KY11], is stated above under some restrictions which do not appear there.

Recall that the generalized cluster category associated with , defined in [Ami09], is the localization of the category by the full subcategory . It is proved in [Ami09] that the canonical t-structure on restricts to a t-structure on . We will denote this t-structure by . Denote by the full subcategory of defined by:

Recall from [Ami09] that the canonical functor from to induces a -linear equivalence from to and that the functor induces an equivalence from to .

Fix an object in . Without loss of generality, assume that belongs to . Note that the canonical cluster tilting object does belong to .

Lemma 2.11

Let be an object of . If is left orthogonal to , which happens for instance when is in or in , then there is a functorial isomorphism

{proof}

Let be left orthogonal to . By [Ami09, Proposition 2.8], we have Moreover, for any , we have

Let . The object belongs to and belongs to , so that the -Calabi–Yau property (see [Kel08]) implies that the morphism space is isomorphic to the dual of . This latter vanishes since belongs to . The same argument shows that the space also vanishes. Therefore applying the functor to the triangle

yields an isomorphism .

Lemma 2.12

Let and assume that belongs to . Then the functor induces a bijection

{proof}

By assumption, is left orthogonal to the subcategory . Therefore, the space is isomorphic to , and thus to . Since and are perfect over , their images under are quasi-isomorphic to dg modules of finite total dimension.

Proposition 2.13

Let be the Ginzburg dg algebra associated with a Jacobi-finite quiver. Then the category has constructible cones.

{proof}

We write for the ideal of generated by the arrows of the Ginzburg quiver, and p for the left adjoint to the canonical functor . Let , and be dg modules of finite total dimension. Since is finite dimensional, there exists a quasi-isomorphism , where is of finite total dimension and such that any morphism may be represented by a fraction:

We thus obtain a surjection Fix a -linear section of this surjection. Choose such that and vanish. Then for the cone of any morphism from to , we have . For being any one of , , we thus have isomorphisms

where denotes the finite dimensional quotient of p by . The category is the stable category of a Hom-finite Frobenius category. By section 2.4, the category has constructible cones: There exists a constructible map (associated with ) as in section 1.3. By composing this map with the section , we obtain a map as required.

Proposition 2.14

Let be the Ginzburg dg algebra associated with a Jacobi-finite quiver. Then the generalized cluster category has constructible cones.

{proof}

Let and be in . Up to replacing them by isomorphic objects in , we may assume that belongs to and to . The projection then induces an isomorphism . Let be in , and let be a triangle in . Let us denote the sets of morphisms by . There is a commutative diagram

where the morphisms in the first two and in the last two columns are isomorphisms by lemma 2.11, and so is the middle one by the five lemma. Note that belongs to , so that, by lemma 2.12, we have isomorphisms:

and

for and thus also for being the middle term of any triangle in . Let and let be a triangle in . Let be the morphism in corresponding to and let be a triangle in . Then the sequence obtained from by applying the functor is isomorphic to the one obtained from by applying the functor . By proposition 2.13, the cylinders of the morphisms are constructible with respect to .

3 Proof of theorem 1.1

Let be a cluster tilting object of . Let and be two objects in , such that the cylinders of the morphisms and are constructible with respect to . Let us begin the proof with some notations and some considerations on constructibility. Let be a morphism in for some , and let be a triangle in . The image of under lifts the orbit of the matrix representation of in given by (the definition of the constructible map is given in section 1.3). In all of this section, we will take the liberty of denoting by , and the image . Denote by the dimension vector . For any object in and any non-negative , and in , let be the subset of

formed by the pairs such that is a submodule of of dimension vector , and , where , and are given by . We let

  • denote the union of all with and and

  • denote the union of all with .

Lemma 3.1

The sets are constructible.

{proof}

Denote by the dimension vector , and fix a dimension vector . Consider the map induced by which sends a pair in to . By our assumption, this map (exists and) is constructible. Therefore, the subset of

formed by the pairs such that is a submodule of is a constructible subset. We denote by this constructible subset. We thus have a constructible function sending the pair to . This function induces a constructible function , and the set is the fiber of above .

The fiber above the class of the first projection is and thus all fibers have Euler characteristics equal to that of . Therefore we have:

Define to be the variety . Consider the following map:

By our assumption, the map is constructible.

Let be the subvariety of formed by the points in the image of , and let be the complement of in .

We can now start to compute: