On cluster algebras with coefficients and 2-Calabi-Yau categories

Building on work by Geiss-Leclerc-Schroer and by Buan-Iyama-Reiten-Scott we investigate the link between certain cluster algebras with coefficients and suitable 2-Calabi-Yau categories. These include the cluster-categories associated with acyclic quivers and certain Frobenius subcategories of module categories over preprojective algebras. Our motivation comes from the conjectures formulated by Fomin and Zelevinsky in `Cluster algebras IV: Coefficients'. We provide new evidence for Conjectures 5.4, 6.10, 7.2, 7.10 and 7.12 and show by an example that the statement of Conjecture 7.17 does not always hold.

Our emphasis here is on cluster algebras with coefficients. More precisely, we investigate certain symmetric cluster algebras of geometric type with coefficients. Coefficients are of great importance both in geometric examples of cluster algebras [27] [28] [8] [41] [23] and in the study of duality phenomena [18] as shown in [21]. Following [2], we consider two types of categories which allow us to incorporate coefficients into the representation-theoretic model: 1) 2-Calabi-Yau Frobenius categories; 2) 2-Calabi-Yau 'subtriangulated' categories, i.e. full subcategories of the form ⊥ (ΣD) of a 2-Calabi-Yau triangulated category C, where D is a rigid functorially finite subcategory of C and Σ the suspension functor of C. In both cases, we establish the link between the category and its associated cluster algebra using (variants of) cluster characters in the sense of Palu [36]. For subtriangulated categories, we use the restriction of the cluster characters constructed in [36]. For Frobenius categories, we construct a suitable variant in section 3 (Theorem 3.3).
The work of Geiss-Leclerc-Schröer [26] [23] and Buan-Iyama-Reiten-Scott [2] provides us with large classes of 2-Calabi-Yau Frobenius categories and of 2-Calabi-Yau subtriangulated categories which admit cluster structures in the sense of [2]. Our general results imply that for these classes, the 2-Calabi-Yau categories do yield 'categorifications' of the corresponding cluster algebras with coefficients (Theorems 5.4 and 6.3). As an application, we show that Conjectures 7.2, 7.10 and 7.12 of [21] hold for these cluster algebras (Proposition 5.5 and Theorem 6.3). Let us recall the statements of these conjectures:

cluster monomials are linearly independent;
Date: October 16, 2007. Last modified on December 28, 2008. 7.10 different cluster monomials have different g-vectors and the g-vectors of the cluster variables in any given cluster form a basis of the ambient lattice; 7.12 the g-vectors of a cluster variable with respect to two neighbouring clusters are related by a certain piecewise linear transformation (so that the g-vectors equal the g † -vectors of [13]). In the case of cluster algebras with principal coefficients admitting a categorification by a 2-Calabi-Yau subtriangulated category, we obtain a representation-theoretic interpretation of the F -polynomial defined in section 3 of [21], cf. Theorem 6.5. This interpretation implies in particular that Conjecture 5.4 of [21] holds in this case: The constant coefficient of the F -polynomial equals 1. We also deduce that the multidegree of the F -polynomial associated with a rigid indecomposable equals the dimension vector of the corresponding module (Proposition 6.6). By combining this with recent work by Buan-Marsh-Reiten [5], cf. also [17], we obtain a counterexample to Conjecture 7.17 (and 6.11) of [21]. We point out that the corresponding computations were already present in G. Cerulli's work [12]. Following a suggestion by A. Zelevinsky, we show that, by assuming the existence of suitable categorifications, instead of the equality claimed in Conjecture 7.17, one does have an inequality: The multidegree of the F -polynomial is greater or equal to the denominator vector (Proposition 6.8). We also show in certain cases that the transformation rule for g-vectors predicted by Conjecture 6.10 of [21] does hold (Proposition 6.9).
Let us emphasize that our proofs for certain cluster algebras of some of the conjectures of [21] depend on the existence of suitable Hom-finite 2-Calabi-Yau categories with a clustertilting object. This hypothesis imposes a finiteness condition on the corresponding cluster algebra (to the best of our knowledge, it is not known how to express this condition in combinatorial terms). The construction of such 2-Calabi-Yau categories is a non trivial problem for which we rely on [6] in the acyclic case and on [26] [23] [2] and [1] in the non acyclic case. As A. Zelevinsky has kindly informed us, many of the conjectures of [21] will be proved in [16] in full generality building on [35] and [15].
Acknowledgments. We thank A. Zelevinsky for stimulating discussions and for informing us about the ongoing work in [16] and [12]. We are grateful to A. Buan and I. Reiten for sharing the results on dimension vectors obtained in [5]. We thank J. Schröer and O. Iyama for their interest and for motivating discussions. The first-named author gratefully acknowledges a 5-month fellowship of the Liegrits network (MRTN-CT 2003-505078) during which this research was carried out. Both authors thank the referee for his great help in making this article more readable.

Recollections
2.1. Cluster algebras. In this section, we recall the construction of cluster algebras of geometric type with coefficients from [21]. For an integer x, we use the notations The tropical semifield on a finite family of variables u j , j ∈ J, is the abelian group (written multiplicatively) freely generated by the u j , j ∈ J, endowed with the addition ⊕ defined by Let 1 ≤ r ≤ n be integers. Let P be the tropical semifield on the indeterminates x r+1 , . . . , x n . Let QP be the group algebra on the abelian group P. It identifies with the algebra of Laurent polynomials with rational coefficients in the variables x r+1 , . . . , x n . Let F be the field of fractions of the ring of polynomials with coefficients in QP in r indeterminates. A seed in F is a pair (B, x) formed by • an n×r-matrixB with integer entries whose principal part B (the submatrix formed by the first r rows) is antisymmetric; • a free generating set x = {x 1 , . . . , x r } of the field F.
The matrixB is called the exchange matrix and the set x the cluster of the seed (B, x). Let 1 ≤ s ≤ r be an integer. The seed mutation in the direction s transforms the seed (B, x) into the seed µ s (B, x) = (B ′ , x ′ ), where • the entries b ′ ij ofB ′ are given by • The cluster x ′ = {x ′ 1 , . . . x ′ r } is given by x ′ j = x j for j = s whereas x ′ s ∈ F is determined by the exchange relation

Mutation in a fixed direction is an involution.
Let T r be the r-regular tree, whose edges are labeled by the numbers 1, . . . , r so that the r edges emanating from each vertex carry different labels. A cluster pattern is the assignment of a seed (B t , x t ) to each vertex t of T r such that the seeds assigned to vertices t and t ′ linked by an edge labeled s are obtained from each other by the seed mutation µ s . Fix a vertex t 0 of the r-regular tree T r . Clearly, a cluster pattern is uniquely determined by the initial seed (B t 0 , x t 0 ), which can be chosen arbitrarily.
Fix a seed (B, x) and let (B t , x t ), t ∈ T r be the unique cluster pattern with initial seed (B, x). The clusters associated with (B, x) are the sets x t , t ∈ T r . The cluster variables are the elements of the clusters. The cluster algebra A(B) = A(B, x) is the ZP-subalgebra of F generated by the cluster variables. Its ring of coefficients is ZP. It is a 'cluster algebra without coefficients' if r = n and thus ZP = Z.

2.2.
Cluster algebras from ice quivers. As we have seen in the previous subsection, our cluster algebras are given by certain integer matricesB. Such matrices can also be encoded by 'ice quivers': A quiver is a quadruple Q = (Q 0 , Q 1 , s, t), where Q 0 is a set (the set of vertices), Q 1 is a set (the set of arrows) and s and t are two maps Q 1 → Q 0 (taking an arrow to its source respectively to its target). An ice quiver is a pair (Q, F ) consisting of a quiver Q and a subset F of its set of vertices (F is the set of frozen vertices).
Let (Q, F ) be an ice quiver such that the set Q 0 is the set of natural numbers from 1 to n, the set Q 1 is finite and the set F is the set of natural numbers from r + 1 to n for some 1 ≤ r ≤ n. The associated integer matrixB(Q, F ) is the n × r matrix whose entry b ij equals the number of arrows from i to j minus the number of arrows from j to i. The cluster algebra with coefficients A(Q, F ) is defined as the cluster algebra A(B(Q, F )). The matrixB(Q, F ) determines the ice quiver (Q, F ) if 1) Q does not have loops (arrows from a vertex to itself) and 2) Q does not have 2-cycles (pairs of distinct arrows α, β such that s(α) = t(β) and t(α) = s(β)) and 3) there are no arrows between any vertices of F . Given integers 1 ≤ r ≤ n each integer matrixB with antisymmetric principal part B (formed by the first r rows ofB), is obtained as the matrix associated with a unique ice quiver satisfying these properties. The mutation of ice quivers satisfying conditions 1)-3) is defined via the mutation of the corresponding integer matrices recalled in section 2.1.

2.3.
Krull-Schmidt categories. An additive category has split idempotents if each idempotent endomorphism e of an object X gives rise to a direct sum decomposition Y ⊕Z ∼ → X such that Y is a kernel for e. A Krull-Schmidt category is an additive category where the endomorphism rings of indecomposable objects are local and each object decomposes into a finite direct sum of indecomposable objects (which are then unique up to isomorphism and permutation). Each Krull-Schmidt category has split idempotents. We write indec(C) for the set of isomorphism classes of indecomposable objects of a Krull-Schmidt category C.
Let C be a Krull-Schmidt category. An object X of C is basic if every indecomposable of C occurs with multiplicity ≤ 1 in X. In this case, X is fully determined by the full additive subcategory add(X) whose objects are the direct factors of finite direct sums of copies of X. The map X → add(X) yields a bijection between the isomorphism classes of basic objects and the full additive subcategories of C which are stable under taking direct factors and only contain finitely many indecomposables up to ismorphism.
Let k be an algebraically closed field. A k-category is a category whose morphism sets are endowed with structures of k-vector spaces such that the composition maps are bilinear. A k-category is Hom-finite if all of its morphism spaces are finite-dimensional. A k-linear category is a k-category which is additive. Let C be a k-linear Hom-finite category with split idempotents. Then C is a Krull-Schmidt category. Let T be an additive subcategory of C stable under taking direct factors. The quiver Q = Q(T ) of T is defined as follows: The vertices of Q are the isomorphism classes of indecomposable objects of T and the number of arrows from the isoclass of T 1 to that of T 2 equals the dimension of the space of irreducible morphisms where rad denotes the radical of T , i.e. the ideal such that rad(T 1 , T 2 ) is formed by all non isomorphisms from T 1 to T 2 .
2.4. 2-Calabi-Yau triangulated categories. Let k be an algebraically closed field. Let C be a k-linear triangulated category with suspension functor Σ. We assume that C is Hom-finite and has split idempotents. Thus, it is a Krull-Schmidt category. For objects X, Y of C and an integer i, we define Let d be an integer. Following [42], cf. also [32], we define the category C to be d-Calabi-Yau if there exists a family of linear forms such that the bilinear forms are non degenerate and satisfy Σ p f, g = (−1) pq Σ q g, f for all f in C(Y, Σ q X) and all g ∈ C(X, Σ p Y ), where p + q = d.
Let us assume that C is 2-Calabi-Yau. A cluster-tilting subcategory of C is a full additive subcategory T ⊂ C which is stable under taking direct factors and such that • for each object X of C, the functors C(X, ?) : T → mod k and C(?, X) : T op → mod k are finitely generated; • an object X of C belongs to T iff we have Ext 1 (T, X) = 0 for all objects T of T . A cluster-tilting object is a basic object T of C such that add(T ) is a cluster-tilting subcategory. Equivalently, an object T is cluster-tilting if it is rigid and if each object X satisfying Ext 1 (T, X) = 0 belongs to add(T ). The following definition is taken from section 1 of [2]. Recall that C is a Hom-finite k-linear triangulated category with split idempotents which is 2-Calabi-Yau.
). The cluster-tilting subcategories of C determine a cluster structure on C if the following hold: 0) There is at least one cluster-tilting subcategory in C. 1) For each cluster-tilting subcategory T ′ of C and each indecomposable M of T ′ , there is a unique (up to isomorphism) indecomposable M * not isomorphic to M and such that the additive subcategory is a cluster-tilting subcategory; 2) In the situation of 1), there are triangles  We refer to section 1, page 11 of [2] for a list of classes of examples where this assumption holds. In particular, this list contains the cluster categories associated with finite quivers without oriented cycles and the stable categories of preprojective algebras of Dynkin quivers. We refer the reader to the surveys [4] [39] [30] [31] for more information on cluster categories and to the survey [24] for more information on stable categories of Dynkin quivers.
2.5. Cluster characters. The notion of cluster character is due to Palu [37]. Under suitable assumptions, cluster characters allow one to pass from 2-Calabi-Yau categories to cluster algebras. We recall the definition and the main construction from [37]. Let k be an algebraically closed field and C a k-linear Hom-finite triangulated category with split idempotents which is 2-Calabi-Yau. Let R be a commutative ring. A cluster character on C with values in R is a map ζ : obj(C) → R such that is one-dimensional) and are non-split triangles, then we have Assume that C has a cluster-tilting object T which is the direct sum of r pairwise non isomorphic indecomposable summands T 1 , . . . T r . In a vast generalization of Caldero-Chapoton's work [9], Palu has shown in [37] that there is a canonical cluster-character Let us recall Palu's construction. First we need to introduce some more notation. Let C be the endomorphism algebra of T . Let mod C denote the category of k-finite-dimensional right C-modules. For each 1 ≤ i ≤ r, the morphism space C(T, T i ) becomes an indecomposable projective right C-module denoted by P i . Its By Theorem 11 of [37], the map (L, M ) → L, M a induces a well-defined bilinear form on the Grothendieck group K 0 (mod C). By [34], for any X ∈ C, we have triangles The index and coindex of X with respect to T are defined to be the classes in K 0 (add T ) where e runs through the positive elements of the Grothendieck group K 0 (mod C) and Gr e (C(T, M )) denotes the variety of submodules U of the right C-module C(T, M ) such that the class of U is e and χ is the Euler characteristic (of the underlying topological space if k = C or of l-adic cohomology if k is arbitrary).
2.6. From 2-CY categories to cluster algebras without coefficients. In this section, we show how certain 2-Calabi-Yau triangulated categories can be linked to cluster algebras without coefficients via cluster characters. All we need to do is to combine the results recalled in sections 2.4 and 2.5. In the main part of the paper, we will concentrate on the case where our cluster algebras do have coefficients.
Let k be an algebraically closed field and C a Hom-finite k-linear 2-Calabi-Yau triangulated category with split idempotents as defined in section 2.4. Let T be a cluster-tilting object in C. Assume that T is the direct sum of r pairwise non isomorphic indecomposable objects T 1 , . . . , T r . Let ζ T : obj(C) → Q(x 1 , . . . , x r ) be Palu's cluster character associated with T as recalled in section 2.5. In particular, we have Now assume that the cluster-tilting subcategories define a cluster structure on C (cf. section 2.4). A cluster-tilting object T ′ is reachable from T if add(T ′ ) is obtained from add(T ) be a finite sequence of mutations as defined in 2.4. A rigid object M is reachable from T if it lies in add(T ′ ) for a cluster-tilting object T ′ reachable from T . Let Q be the quiver of the endomorphism algebra C of T , or equivalently, the quiver of the category add(T ). We consider Q as an ice quiver with empty set of frozen vertices and denote by A(Q) the associated cluster algebra without coefficients (defined by specializing the construction of 2.2 to the case where the set of frozen vertices is empty). It is the subalgebra of Q(x 1 , . . . , x r ) generated by the cluster variables. Proof. Clearly, part a) follows from part b). Let us prove part b). Let T r be the r-regular tree and let t 0 be a fixed vertex of T r . Let B be the antisymmetric matrix associated with the quiver Q and let x be the initial cluster x 1 , . . . , x r . Let (B t , x t ), t ∈ T r , be the unique cluster pattern with initial seed (B t 0 , x t 0 ) = (B, x) (cf. section 2.1). To each vertex t of T, we assign a cluster-tilting object T t with indecomposable direct summands T t,1 , . . . , T t,r such that 1) we have T t 0 = T and 2) if t is linked to t ′ by an edge labeled s, then T t ′ is obtained from T t by mutating at the summand T t,s . It follows from point 1) of the definition of a cluster structure that T t is well-defined for each vertex t of T. Moreover, it follows from point 4) of the same definition that for each vertex t of T, the antisymmetric matrix B t corresponds to the quiver of the category add(T t ) under the bijection of section 2.2. We claim that for each vertex t of T, the cluster character takes the shift ΣT t,i of the indecomposable direct summand T t,i of T t to the cluster variable x t,i , 1 ≤ i ≤ r. Indeed, this holds for t = t 0 by equation (2.2). Now assume that it holds for some vertex t and that t is linked to a vertex t ′ by an edge labeled s. We know that the indecomposable summands of T t ′ are the T t ′ ,i = T t,i for i = s and a new summand T ′ t,s which is not isomorphic to T t,s . By part a) of lemma 2.2, the extension space between T t,s and T t ′ ,s is one-dimensional and we have non split triangles Here, the middle terms are sums of copies of the T t,i , i = s, and the multiplicities are determined by the quivers of the endomorphism algebras of T and T ′ , as indicated in part b) of lemma 2.2. More precisely, if b t ij denotes the (i, j)-entry of the exchange matrix, is ] + . Now if we use points 2) and 3) of the definition of a cluster character, we see that the induction hypothesis and equation (2.1) yield the exchange relation 2.7. Frobenius categories. A Frobenius category is an exact category in the sense of Quillen [38] which has enough projectives and enough injectives and where an object is projective iff it is injective. By definition, such a category is endowed with a class of admissible exact sequences 0 → L → M → N → 0.
Following [22] we will call the left morphism L → M of such a sequence an inflation, the right morphism a deflation and, sometimes, the whole sequence a conflation. Let E be a Frobenius category. Its associated stable category E is the quotient of E by the ideal of morphisms factoring through a projective-injective object. It was shown by Happel [29] that E has a canonical structure of triangulated category. We have Let k be an algebraically closed field and E a Hom-finite Frobenius category with split idempotents. Suppose that E is a 2-Calabi-Yau Frobenius category, i.e. its associated stable category C = E is 2-Calabi-Yau in the sense of section 2.4. A cluster-tilting subcategory of E is a full additive subcategory T ⊂ E which is stable under taking direct factors and such that • for each object X of E, the functors E(X, ?) : T → mod k and E(?, X) : T op → mod k are finitely generated; • an object X of E belongs to T iff we have Ext 1 E (T, X) = 0 for all objects T of T . Clearly if these conditions hold, each projective-injective object of E belongs to T . A cluster-tilting object is a basic object T of E such that add(T ) is a cluster-tilting subcategory. Equivalently, an object T is cluster-tilting if it is rigid and if each object X satisfying Ext 1 E (T, X) = 0 belongs to add(T ). The following definition is taken from section 1 of [2]. Recall that E is a k-linear Hom-finite Frobenius category with split idempotents such that the associated stable category C = E is 2-Calabi-Yau.
is a cluster-tilting subcategory; 2) In the situation of 1), there are conflations where g and t are minimal right T ′ ∩ T ′′ -approximations and f and s are minimal left T ′ ∩ T ′′ -approximations. 3) For any cluster-tilting subcategory T ′ , the quiver Q(T ′ ) does not have loops nor 2-cycles.
be removing all arrows between projective vertices. The cluster tilting subcategory T ′′ = µ M (T ′ ) of 1) is the mutation of T ′ at the non projective indecomposable object M . The mutation of a cluster-tilting object T is defined via the mutation of the cluster-tilting subcategory add(T ).
Lemma 2.5. Suppose that the cluster-tilting subcategories determine a cluster structure on E. Then, in the situation of condition 2) of the above definition 2.4, the following hold: a) The space We omit the proof of the lemma since it is entirely parallel to that of lemma 2.2. Large classes of examples of Frobenius categories where the cluster-tilting objects define a clusterstructure are obtained in [23] and [3], cf. the survey [24] and example 5.3 below. For an extension of the theory from the antisymmetric to the antisymmetrizable case, we refer to [14].

Cluster characters for 2-Calabi-Yau Frobenius categories
Let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = E is 2-Calabi-Yau (cf. section 2.4).
are non-split triangles, then we have From now on, we assume in addition that E contains a cluster-tilting object T . Using T we will construct a cluster character on E and link it to Palu's cluster character associated with the image of T in the triangulated category C = E (cf. section 2.5).
Let C be the endomorphism algebra of T (in E) and C = End C (T ). Let Let T i , 1 ≤ i ≤ n, be the pairwise non isomorphic indecomposable direct summands of T . We choose the numbering of the T i so that T i is projective exactly for r < i ≤ n for some integer 1 ≤ r ≤ n. For 1 ≤ i ≤ n, let S i be the top of the indecomposable projective P i = F T i . Note that C and C are finite dimensional k-algebras, so finitely presented modules are the same as finitely generated modules. As in section 4 of [34], we identify Mod C with the full subcategory of Mod C formed by the modules without composition factors isomorphic to one of the S i , r < i ≤ n. Let DC be the unbounded derived category of the abelian category Mod C of all right C-modules. Let per C be the perfect derived category of C, i.e. the full subcategory of DC whose objects are all the complexes quasi-isomorphic to bounded complexes of finitely generated projective C-modules. Let D b (mod C) the bounded derived category of mod C identified with the full subcategory of DC whose objects are all complexes whose total homology is finite-dimensional over k. As shown in section 4 of [34], we have the following embeddings We have a bilinear form where K 0 (per C) (resp. K 0 (D b (mod C))) is the Grothendieck group of per C (resp. D b (mod C)) and Σ is the shift functor of D b (mod C).
For arbitrary finitely generated C-modules L and N , put be the truncated Euler forms on the split Grothendieck group K sp 0 (mod C). By the proposition below, if L is a C-module, then L, N 3 only depends on the dimension vector dim L in K 0 (mod C). We put dim L, N 3 = L, N 3 .
a) The restriction of the map Proof. a) We need to show that for arbitrary finitely generated C-modules L, N with dim L = dim N , we have [L] = [N ] in K 0 (per C). Let 0 = L s ⊂ L s−1 ⊂ · · · ⊂ L 0 = L and 0 = N s ⊂ N s−1 ⊂ · · · ⊂ N 0 = N be composition series of L and N respectively. By [34], we know that every C-module has projective dimension at most 3 in mod C. Assume for simplicity that L s−1 = S 1 , L s−2 /L s−1 = S 2 . Denote by P * i a minimal projective resolution of S i . Then we have the following commutative diagram where the middle term is a projective resolution of L s−2 . In this way, we inductively construct projective resolutions for L and N . If m i is the multiplicity of S i in the composition factors of L and N , then we obtain projective resolutions of L and N of the form Let P L (resp. P N ) be the projective resolution complex of L (resp. N ). We have L ∼ = P L and N ∼ = P N in per C, which implies By a), we have [P L ] = [P N ] in K 0 (per C), which implies the equality.
One should note that the truncated Euler form , 3 does not descend to the Grothendieck group K 0 (mod C) in general (except if the global dimension of C is not greater than 3), cf. remark 3.5.
Using the bilinear forms introduced so far, for M ∈ E, we define the Laurent polynomial Here we consider Ext 1 E (T, M ) as a right C-module via the natural action of C = End E (T ) on the first argument; the sum ranges over all the elements of the Grothendieck group; for a C-module L, the notation Gr e (L) denotes the projective variety of submodules of L whose class in the Grothendieck group is e; for an algebraic variety V , the notation χ(V ) denotes the Euler characteristic (of the underlying topological space of V if k = C and of l-adic cohomology if k is arbitrary).
M is Palu's cluster character associated with the cluster-tilting object T , cf. section 2.5.
The following theorem shows that M → X ′ M is a cluster character on E and that, if we specialize the 'coefficients' x r+1 , . . . , x n to 1, it specializes to the composition of Palu's cluster character M → X M with the suspension functor M → ΣM . Notice that this theorem does not involve cluster algebras (but paves the way for establishing a link with cluster algebras when E admits a cluster structure, cf. Theorem 5.4 below). Theorem 3.3. As above, let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents such that E is Hom-finite, the stable category C = E is 2-Calabi-Yau and E contains a cluster-tilting object T . For an object M of E, let X ′ M and X M be the Laurent polynomials defined above.

Now by the definition, we have
Therefore, we only need to show that the exponents of x i , 1 ≤ i ≤ r, in the corresponding terms of X ΣM and X ′ M are equal. There exists a triangle in C We may and will assume that this triangle is minimal, i.e. does not admit a non zero direct factor of the form Since E is Frobenius, we can lift this triangle to a short exact sequence in E Applying the functor F to this short exact sequence, we get a projective resolution of F M as a C-module, On the other side, we have the following minimal triangle Next we will show that S i , e a = − e, S i 3 . Let N be a C-module such that dim N = e. Note that N and the S i , 1 ≤ i ≤ r, are C-modules and that all of them are finitely presented C-modules. Therefore, they lie in the perfect derived category per(C). Thus, we can use the relative 3-Calabi-Yau property of per(C) (cf. [34]) to deduce that S i , e a = − e, S i 3 . We have N ), c) is proved in exactly the same way as Corollary 3.7 in [9]. d) Let be the non-split conflations in E, and the associated triangles in C. For any classes e, f , g in the Grothendieck group K 0 (mod C), let X e,f be the variety whose points are the C-submodules E ⊂ GΣE such that the dimension vector of (GΣi) −1 E equals e and the dimension vector of (GΣp)E equals f . Similarly, let Y f,e be the variety whose points are the C-submodules E ⊂ GΣE ′ such that the dimension vector of (GΣi ′ ) −1 E equals f and the dimension vector of (GΣp ′ )E equals e. Put X g e,f = X e,f ∩ Gr g (GΣE) , . Since C is a 2-CY triangulated category, by section 5.1 of [37] we also have Therefore, part d) is a consequence of the following lemma. Proof. We have the following commutative diagram as in section 4 of [37] ( where i, j, k are monomorphisms, β is an epimorphism and One can easily show that ker GΣi = ker α. We have an exact sequence If we apply F = Hom E (T, ?) to the short exact sequence we get the long exact sequences of C-modules Remark 3.5. If C has finite global dimension, the Grothendieck group K 0 (mod C) has the Euler form , . We can then define a Laurent polynomial X f M as follows

One can show that in this case
In fact, if gldimC < ∞, then the perfect derived category per(C) equals D b (mod C), and S i belongs to per(C) for all i. Thus, we have 4. Index and g-vector 4.1. Index. As in section 3, we let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = E is 2-Calabi-Yau (cf. section 2.4). Moreover, we assume that E admits a cluster-tilting object T and we write C = End E (T ) and C = End C (T ). Let D(Mod C) be the derived category of C-modules, D − (mod C) the right bounded derived category of mod C, H − (P) the right bounded homotopy category of finitely generated projective C-modules. It is well known that there is an equivalence Proposition 4.1. For an arbitrary C-module Z which is also a finitely presented C-module we have a canonical isomorphism Proof. For arbitrary X ∈ D − (mod C), by the equivalence, we have a P X ∈ H − (P) such that X ∼ = P X in D − (mod C). Assume that P X has the following form Clearly, the complex P X is the direct limit of the complexes X i . We write hocolim for the total left derived functor of the functor of taking the direct limit. Since taking direct limits over filtered systems is an exact functor, the functor hocolim is simply induced by the direct limit functor. Thus, we have P X ∼ = hocolim X i in D(Mod C). Note that by Proposition 4 of [34], Z belongs to per C, i.e. Z is compact in D(Mod C). So we have By the definition of X i , we know that X i ∈ per C. Since per C is a full subcategory of D(Mod C), by the relative 3-Calabi-Yau property of per C, we have the following It is easy to see that this colimit is a stationary system, i.e. ∃ N such that for i > N , we Thus, we have Note that since D − (mod C) is a full subcategory of D(Mod C), we get the isomorphism For each X ∈ E, there is a unique minimal conflation (up to isomorphism) The following result is easily deduced from Theorem 2.3 of [13].

g-vector.
Let us recall the definition of g-vectors from section 7 of [21]. Let 1 < r ≤ n be integers. LetB = (b ij ) be an n × r matrix with integer entries, whose principal part B (i.e. the submatrix formed by the first r rows) is antisymmetric. Let A(B) be the cluster algebra with coefficients associated withB, cf. the end of section 2.1. Let z be an element of A(B). Suppose that we can write z as z = R( y 1 , . . . , y r ) where R( y 1 , . . . , y r ) is a primitive rational polynomial. If rankB = r, then the g-vector of z is defined by g(z) = (g 1 , . . . , g r ).
Note that rankB = r implies that the g-vector is well-defined. As in the previous section, we let k be an algebraically closed field and E a k-linear Frobenius category with split idempotents. We assume that E is Hom-finite and that the stable category C = E is 2-Calabi-Yau (cf. section 2.4). Moreover, we assume that E admits a cluster-tilting object T and we write C = End E (T ) and C = End C (T ). Let T 1 , T 2 , . . ., T n be the pairwise non isomorphic indecomposable direct summands of T numbered in such a way that T i is projective iff r < i ≤ n. We define B(T ) = (b ij ) n×n to be the antisymmetric matrix associated with the quiver of the endomorphism algebra of T . Let B(T ) 0 be the submatrix formed by the first r columns of B(T ). We suppose that we have rank B(T ) 0 = r. In analogy with the definition of g-vectors in a cluster algebra, for M ∈ E, if we can write X ′ M as where R( y 1 , . . . , y r ) is a primitive rational polynomial, then we define the g-vector g T (X ′ M ) of M with respect to T to be g T (X ′ M ) = (g 1 , . . . , g r ). As in the cluster algebra case, this is well-defined since rank B(T ) 0 = r. Proposition 4.3. Assume that rank B(T ) 0 = r. For arbitrary M ∈ E, the g-vector g T (X ′ M ) is well-defined and its i-th coordinate is given by

Corollary 4.4. As above, let E be a Hom-finite k-linear Frobenius category such that its stable category C = E is 2-Calabi-Yau and assume that
• E admits a cluster-tilting object T with indecomposable direct summands T 1 , . . . , T n numbered in such a way that T i is projective iff r < i ≤ n, where 1 < r ≤ n is an integer; • the first r columns of the antisymmetric matrix B(T ) associated with the quiver of the algebra C = End E (T ) are linearly independent. Then the following hold.
a) The map M → X ′ M induces an injection from the set of isomorphism classes of non projective rigid indecomposables of E into the set Q(x 1 , . . . , x n ). b) Let I be a finite set and T i , i ∈ I, cluster tilting objects of E. Suppose that for each i ∈ I, we are given an object M i which belongs to add T i and does not have non zero projective direct factors. If the M i are pairwise non isomorphic, then the X ′ M i are linearly independent.
Proof. a) clearly follows from b). Let us prove b). First, we will show that we can assign a degree to each x i such that for every 1 ≤ i ≤ r the degree of y i is 1. Indeed, it suffices to put deg(x i ) = k i , where the k i are rationals such that we have (k 1 , k 2 , . . . , k n )B(T ) 0 = (1, 1, . . . , 1).
Since rank B(T ) 0 = r, this equation does admit a solution. Thus, the term of strictly minimal total degree in Suppose that the X ′ M i are linearly dependent, i.e. there is a non-empty subset I ′ of I and rationals c i , i ∈ I ′ , which are all non zero such that If we consider the terms of minimal total degree of the polynomial above, we find  Next we will investigate the relation between the indices of an exchange pair. Recall that F is the functor Hom E (T, ?) : The F -exact sequences define a new exact structure on the additive category E. For each X, we have an F -exact conflation This shows that E endowed with the F -exact sequences has enough projectives and that its subcategory of projectives is add T . Moreover, if we denote by Ext i F (X, Z) the i-th extension groups of the category E endowed with the F -exact sequences, then Ext 1 F (X, Z) is the cohomology at Hom E (T 1 , Z) of the complex Proof. Using Lemma 4.6, the proof is the same as that of proposition 15.4 in [23].
Then we have and exactly one of these cases occurs. Let Then h is a linear combination of the columns of B(T ) 0 .
Proof. The first part follows from Proposition 4.7 directly, because the index is additive on F -exact sequences. Since (R k , R * k ) is an exchange pair, we have

For simplicity, we write
By comparing the minimal total degree we get that n i=1 x is a monomial in y i , 1 ≤ i ≤ r, which implies the result.

Frobenius 2-Calabi-Yau realizations
Recall the bijection defined in section 2.2 between antisymmetric integer n × n-matrices and finite quivers without loops nor 2-cycles with vertex set {1, 2, . . . , n}: The quiver Q corresponds to the matrix B iff b ij > 0 exactly when there are arrows from i to j in Q and in this case their number is b ij .
We call an n × n antisymmetric integer matrix B acyclic if the corresponding quiver Q does not have oriented cycles. Two matrices B and B ′ are called mutation equivalent if we can obtain B ′ from B by a series of matrix mutations followed by conjugation with a permutation matrix.
Let 0 ≤ r < n be positive integers and let (Q, F ) be an ice quiver (cf. section 2.2) with vertex set Q 0 = {1, . . . , n} and set of frozen vertices F = {r + 1, . . . , n}. We defineB to be the n × r matrix formed by the first r columns of the skew-symmetric matrix associated with Q and we let A(Q, F ) = A(B) be the cluster algebra with coefficients associated with B, cf. sections 2.1 and 2.2.

Definition 5.1. A Frobenius 2-Calabi-Yau realization of the cluster algebra A(B) is a Frobenius category E with a cluster tilting object T as in section 3 such that
1) E has a cluster structure in the sense of [2], cf. section 2.7; 2) T has exactly n indecomposable pairwise non isomorphic summands T 1 , T 2 , . . ., T n and among these, precisely T r+1 , . . ., T n are projectives; 3) The matrixB equals the matrix formed by the first r columns of the antisymmetric matrix associated with the quiver of the endomorphism algebra of T in E.

Remark 5.2. Suppose we have a Frobenius 2-CY realization of a cluster algebra A(Q, F ) as above. Let 1 ≤ s ≤ r. Then by Lemma 2.5 b), we have conflations
Here the middle terms are the sums Therefore, none of the first r vertices of Q can be a source or a sink. 3 . Let Λ be the completion of the preprojective algebra of ∆ and W the Weyl group associated with ∆. Let w be the element of W given by the reduced word s 2 s 1 s 2 s 3 s 2 . Let e i , i = 1, 2, 3, be the primitive idempotents corresponding to the vertices of ∆. Let I i = Λ(1 − e i )Λ. By Theorem II.2.8 of [2], the category Sub Λ/I w formed by all Λ-submodules of finite direct sums of copies of Λ/I w is a Frobenius category whose associated stable category is 2-Calabi-Yau; moreover, it contains the cluster-tilting object T = Λ/I 2 ⊕ Λ/I 2 I 1 ⊕ Λ/I 2 I 1 I 2 ⊕ Λ/I 2 I 1 I 2 I 3 ⊕ Λ/I w .
According to Proposition II. 1.11 of [loc. cit.], in this decomposition, each direct factor differs from the preceding one by one indecomposable direct summand T i , 1 ≤ i ≤ 5, and among these, exactly T 3 , T 4 and T 5 are projective-injective. Moreover, by Theorem II. 4.1 of [loc. cit.], the quiver of the cluster-tilting object is We return to the general setup. Following [13] we define a cluster-tilting object T ′ of E to be reachable from T if it is obtained from T by a finite sequence of mutations. We define an indecomposable rigid objects M to be reachable from T if it occurs as a direct factor of a cluster-tilting object reachable from T . Proof. a) It follows from Theorem 3.3 c) that X ′ M is a cluster variable for each indecomposable rigid M reachable from T and from the existence of a cluster structure on E that the map M → X ′ M is a surjection onto the set of cluster variables. The injectivity of the map M → X ′ M follows from Lemma 4.2 and Proposition 4.3. The second statement follows from the first one and the fact that E has a cluster structure. b) The map is injective by Lemma 4.2. It is surjective thanks to part a) and Proposition 4.3.
Theorem 5.5. Let 1 < r ≤ n be integers and A(B) the cluster algebras with coefficients associated with an initial n × r-matrixB of maximal rank. Suppose that A(B) admits a Frobenius 2-CY realization E with cluster tilting object T . a) Conjecture 7.2 of [21] holds for A, i.e. cluster monomials are linearly independent. b) Conjecture 7.10 of [21] holds for A, i.e. 1) Different cluster monomials have different g-vectors with respect to a given initial seed.
2) The g-vectors of the cluster variables in any given cluster form a Z-basis of the lattice Z r . c) Conjecture 7.12 of [21] holds for A, i.e. if (g 1 , . . . , g r ) and (g ′ 1 , . . . , g ′ r ) are the gvectors of one and the same cluster variable with respect to two clusters t and t ′ related by the mutation at l, then we have LetB be a 2r × r matrix whose principal (i.e. top r × r) part B 0 is mutation equivalent to an acyclic matrix, and whose complementary (i.e. bottom) part is the r × r identity matrix. Let A(B) be the cluster algebra with the initial seed (x,B).
Theorem 5.6. With the above notation, the cluster algebra A(B) does not admit a Frobenius 2-CY realization.
Proof. Suppose that A(B) has a Frobenius 2-CY realization E. Then there is a cluster tilting object T of E with 2r indecomposable direct summands. Then we have B(T ) 0 =B. Since B 0 is mutation equivalent to an acyclic matrix B c by a series of mutations, we have a cluster tilting object T ′ such that the quiver of the stable endomorphism algebra of T ′ corresponds to B c . Let A be the stable endomorphism algebra of T ′ . By the main theorem of [33], we have a triangle equivalence E ≃ C A , where C A is the cluster category of A. Thus the cluster tilting graph of E is connected and every rigid object of E can be extended to a cluster tilting object of E.
Let F = Hom E (T, ?). Let S i , 1 ≤ i ≤ 2r, be the simple modules of End E (T ). For each object M of E, we have the Laurent polynomial As in Proposition 4.3, we can rewrite X ′ M as where e j is the j-th coordinate of e in the basis of the S i , 1 ≤ i ≤ 2r. If the indecomposable object M is rigid and not isomorphic to T i for r < i ≤ 2r, then X ′ M is a cluster variable of A(B). By the definition of the rational function F l,t associated with the cluster variable x l,t in [21], we have 2r j=r+1 x e j−r j ). Put Note that G M is always a polynomial of x i , r + 1 ≤ i ≤ 2r, with constant term 1. By Proposition 5.2 in [21], we know that the polynomial F M is not divisible by x i , r + 1 ≤ i ≤ 2r. Now for i > r, we have F M, S i τ ≥ 0 in general, which implies that F M, S i τ = 0. In particular, F M, S i τ = [ind T (M ) : T i ] = 0, for r + 1 ≤ i ≤ 2r. Consider M = ΣT 1 , which is rigid and indecomposable, so X ′ M is a cluster variable of the cluster algebra A(B). But in the Frobenius category E we have the conflation where P is an injective hull of T 1 , which implies Thus there is always some r + 1 ≤ i ≤ 2r such that [ind T (M ) : T i ] = 0. Contradiction.
Remark 5.7. In the above notation, if B 0 is acyclic, then it is easy to deduce that the cluster algebra A(B) does not have a Frobenius 2-CY realization. Indeed in this case, one of the first r vertices of Q which corresponds toB is always a sink. This is incompatible with the existence of a Frobenius 2-CY realization by remark 5.2.
6. Triangulated 2-Calabi-Yau realizations 6.1. Definitions. Let B = (b ij ) n×n be an antisymmetric integer matrix and A(B) the associated cluster algebra. A 2-Calabi-Yau triangulated category C is called a triangulated 2-Calabi-Yau realization of the matrix B if C admits a cluster tilting object T such that • C has a cluster structure in the sense [2], cf. section 2.4; • T has exactly n non isomorphic indecomposable direct summands T 1 , . . ., T n ; • The antisymmetric matrix B(T ) associated with the quiver of the endomorphism algebra of T equals B. We denote a triangulated 2-CY realization of B by C ⊃ add T .
Let n 1 and n 2 be positive integers. Let B 1 and B 2 be antisymmetric integer n 1 × n 1 resp. n 2 × n 2 -matrices. Let B 21 be an integer n 2 × n 1 -matrix with non negative entries. Let C i ⊃ T i be a triangulated 2-CY realization of B i , i = 1, 2. Let B be the matrix A gluing of C 1 ⊃ T 1 with C 2 ⊃ T 2 with respect to B is a triangulated 2-CY realization C ⊃ T of B endowed with full additive subcategories T ′ 1 and T ′ 2 such that where C 2 is the cluster category of (A 1 ) n 1 and T 2 the image of the subcategory of finitely generated projective modules. It is well known that each acyclic matrix B admits a triangulated 2-CY realization C Q B , where C Q B is the cluster category of the quiver Q B corresponding to B. In the last subsection, we will see that C Q B does admit a principal gluing. Amiot's work [1] provides some evidence for the conjecture: Indeed, if C 1 and C 2 are generalized cluster categories [1] associated with Jacobi-finite quivers with potential [15], it is easy to construct a quiver with potential which provides a gluing as required by the conjecture.
6.2. Cluster algebras with coefficients. Let B be an antisymmetric integer n × nmatrix. Suppose that the matrix B admits a triangulated 2-CY realization C with the cluster tilting subcategory T = add T . Let T i , 1 ≤ i ≤ n, be the non isomorphic indecomposable direct summands of T . By the definition, we have B(T ) = B. The mutations of the matrix B correspond to the mutations of the cluster tilting object T . Fix an integer 0 < r ≤ n and consider the submatrix B 0 of B formed by the first r columns of B. If l ≤ r, then we have where µ l is the mutation in the direction l. Thus we can view the cluster algebra A(B 0 ) with coefficients as a sub-cluster algebra of A(B), cf. Ch. III of [2].
Denote by P the full subcategory of C whose objects are the finite direct sums of copies of T r+1 , . . . , T n . We define a subcategory of C By Theorem I.2.1 of [2], the quotient category U/P is a 2-Calabi-Yau triangulated category and the projection U → U/P induces a bijection between the cluster tilting subcategories of C containing P and the cluster tilting subcategories of U/P. Thus, a mutation of a cluster tilting object in U/P can be viewed as a mutation of a cluster tilting object in U ⊂ C which does not affect the direct summands T i , r < i ≤ n. This exactly corresponds to a mutation of the matrix B in one of the first r directions. In particular, a mutation of the cluster algebra A(B 0 ) corresponds to a mutation of a cluster tilting object in U.
Recall from section 2.5 that on C, we have Palu's cluster character associated with T , which is given by the formula We consider the composition of this map with the shift: We consider the restriction of the map M → X ′ M to the subcategory U. It follows from Proposition 2.3 that if M is an indecomposable rigid object reachable from T in U, then X ′ M is a cluster variable of A(B 0 ). We will rewrite this variable so as to express its g-vector (if it is defined) in terms of the index of M : Let M be an object of U. Then Hom C (T, ΣM ) is an End C (T )-module which vanishes at each vertex r < i ≤ n. Let e be the image of Hom C (T, ΣM ) in the Grothendieck group of mod End C (T ). Let e j be the j-th coordinate of e with respect to the basis S i , 1 ≤ i ≤ n. We have As in section 4, put Then X ′ M can be rewritten as χ(Gr e (Hom C (T, ΣM ))) r j=1 y e j j ).
As in section 4, when rank B 0 = r, we can define the g-vector of M ∈ U with respect to a cluster tilting object T . Thus we have proved part a) of the following proposition. We leave the easy proof of part b) to the reader.
Proposition 6.2. Suppose that rank B 0 = r. Let M be an object of U.
a) The g-vector of X ′ M with respect to the initial cluster is given by In analogy with the definition in section 5, we define a cluster tilting object T ′ of U to be reachable from T if it is obtained from T by a sequence of mutations at indecomposable rigid objects of U not in P. We define an indecomposable rigid object of U to be reachable from T if it is a direct factor of a cluster tilting object reachable from T . Theorem 6.3. Let B be an antisymmetric integer n × n-matrix and 1 ≤ r ≤ n an integer such that the submatrix B 0 of B formed by the first r columns has rank r. Let A = A(B 0 ) be the associated cluster algebra with coefficients. Assume that the matrix B admits a triangulated 2-CY realization given by a triangulated category C with a cluster tilting object T which is the sum of n indecomposable direct factors T 1 , . . . , T n . Denote by P the full subcategory of C whose objects are the finite direct sums of copies of T r+1 , . . . , T n and define the subcategory U of C by d) Conjecture 7.10 of [21] holds for A, i.e. 1) Different cluster monomials have different g-vectors with respect to a given initial seed.
2) The g-vectors of the cluster variables in any given cluster form a Z-basis of the lattice Z r . e) Conjecture 7.12 of [21] holds for A, i.e. if (g 1 , . . . , g r ) and (g ′ 1 , . . . , g ′ r ) are the gvectors of one and the same cluster variable with respect to two clusters t and t ′ related by the mutation at l, then we have where the b ij are the entries of the r × r-matrix B associated with t and we write [x] + for max(x, 0) for any integer x.
Proof. It follows from Proposition 2.3 that the map M → X ′ M is well-defined and surjective onto the set of cluster variables of A(B 0 ). It is injective by Proposition 6.2 b) because rigid objects of U/P are determined by their indices and the map taking a rigid object M of U without non zero direct factors in P to its image in U/P is injective (up to isomorphism). This also implies part b). The same proof as for Corollary 4.4 b) yields the linear independence of the cluster monomials in c). Let us prove that the cluster monomials form a basis of the Z[x r+1 , . . . , x n ]-submodule of A(B 0 ) which they generate. Indeed, over Z, this submodule is spanned by the images X ′ M of all rigid objects of U obtained as direct sums of objects of P and indecomposable rigid objects reachable from T not belonging to P. Such objects M are in particular rigid in T and they can be distinguished (up to isomorphism) by their indices. Now again, the same proof as for Corollary 4.4 b) shows that these X ′ M are linearly independent over Z. Clearly this implies that the cluster monomials form a basis of the Z[x r+1 , . . . , x n ]-submodule of A(B 0 ) which they generate. As in the proof of Theorem 5.5 b), the assertions in part d) follow from the interpretation of the g-vector given in 6.2 b) and the facts that 1) rigid objects of U/P are determined by their indices (Theorem 2.3 of [13]) and 2) the indices of the indecomposable direct factors of a cluster-tilting subcategory T of U/P form a basis of K 0 (T ) (Theorem 2.6 of [13]).
Part e) is proved in exactly the same way as the corresponding statement for cluster algebras with a 2-CY Frobenius realization in Theorem 5.4 c).
The following is the AR quiver of C Q , where P i , 1 ≤ i ≤ 4, are the indecomposable projective kQ-modules.
In this case, the matrix B(T ) 0 is  We have rank B(T ) 0 = 2. Moreover, the cluster algebra A(B(T ) 0 ) has principal coefficients.
6.3. Cluster algebras with principal coefficients. In this subsection, we suppose that 2r = n and that the complementary part of B 0 is the r × r identity matrix. Thus the cluster algebra A(B 0 ) has principal coefficients. Recall that for the matrix B, we have a triangulated 2-CY realization C ⊃ add T and we have fixed P = add(T r+1 ⊕ . . . ⊕ T 2r ). Let Q = add(T 1 ⊕ . . . ⊕ T r ). Let C 1 = U/P and C 2 = ⊥ (ΣQ)/Q be the quotient categories, T 1 = add(π 1 (T 1 ⊕ . . . ⊕ T r )) and T 2 = add(π 2 (T r+1 ⊕ . . . ⊕ T 2r )) the corresponding cluster tilting subcategories, where π 1 and π 2 are the respective projection functors. Then C is a gluing of C 1 ⊃ T 1 with C 2 ⊃ T 2 with respect to the matrix B.
As in section 5, for a cluster variable x l,t of the cluster algebra A(B 0 ) which corresponds to an indecomposable rigid object M ∈ U and not in P, we denote the rational function F l,t defined in section 3 of [21] χ(Gr e (Hom C (T, ΣM ))) 2r j=r+1 x e j−r j ).
The following result is now a consequence of Proposition 3.6 and 5.2 in [21]. We give a proof based on representation theory. Note that conjecture 5.4 of [21] will be proved in full generality in [16]. Proof. We need to show that for each i > r, [ind T (M ) : T i ] is zero. Since X ′ M is a cluster variable and M is indecomposable, we have the following two cases: Case 1: M ∼ = ΣT j for some j ≤ r. We have ind T (M ) = −[T j ], which implies that [ind T (M ) : T i ] = 0. Case 2: M is not isomorphic to ΣT j for any j ≤ r. Recall that by assumption, M is not isomorphic to T j for any j > r. We have the following minimal triangle Since M belongs to U, for each i > r we have Hom C (M, ΣT i ) = 0. If we had [T 1 M : T i ] = 0 for some i > r, then the above minimal triangle would have a non zero direct factor Suppose that we have [T 0 M : T i ] = 0 for some i > r. Applying the functor F = Hom C (T, ?) to the triangle, we get a minimal projective resolution of F M as an End C (T )-module. Note that for i > r, the projective module F T i is also a simple module, which implies that F M is decomposable. Contradiction.
Suppose that the indecomposable rigid object M of C is reachable from T and consider the polynomial F M of Theorem 6.5. We define the f -vector f T (M ) = (f 1 , . . . , f r ) of M with respect to T by where Trop(u 1 , . . . , u r ) is the tropical semifield defined in section 2.1.
Proposition 6.6. Suppose that M is not isomorphic to T i for 1 ≤ i ≤ 2r, and let dim Hom C (T, ΣM ) = (d 1 , . . . , d r ). Then we have Proof. By Theorem 6.5, we have χ(Gr e (Hom C (T, ΣM ))) 2r j=r+1 x e j−r j . Therefore, we obtain Under the assumptions above, we have proved that the dimension vector of Hom C (T, ΣM ) equals the f -vector f T (M ). Conjecture 7.17 of [21] states that the f -vectors coincide with the denominator vectors in general. But by recent work of A. Buan, R. Marsh and I. Reiten [5], the dimension vectors do not always coincide with the denominator vectors. In fact, as shown in [5], for a quiver Q whose underlying graph is an affine Dynkin diagram, the vector dim Hom C Q (T, M ) is different from the denominator vector of X T M if M = R and R is a direct factor of T , where R is a rigid regular indecomposable of maximal quasi-length. This leads to the following minimal counterexample to Conjecture 7.17 in [21]. Let us point out that the corresponding computations already appear in [12]. In subsection 5.5 below, we will show that in many cases, the f -vector is greater or equal to the denominator vector.
6.4. A counterexample. Example 6.7. Let Q be the following quiver e e e e Let A(Q) be the cluster algebra associated with the initial seed given by Q and x = (x 1 , x 2 , x 3 ). Consider the mutations at 3, 2, 1. Let x t 3 be the corresponding cluster. We have and the corresponding F -polynomial is F x t 3 1 = 1 + (1 + y 1 + y 1 y 2 )y 3 + y 1 y 2 y 2 3 .
Then the f -vector of x t 3 1 does not coincide with the denominator vector.
Let us interpret this counterexample in terms of representation theory. Let A 2,1 be the quiver 3 Consider the cluster category C A 2,1 of kA 2,1 . Let P i , 1 ≤ i ≤ 3, be the indecomposable projective modules and S i the corresponding simple modules. Then is a cluster tilting object of C A 2,1 , where τ is the Auslander-Reiten translation functor. The quiver Q T of T looks like τ S 3 | | z z z z q q q q We will show that the cluster category C A 2,1 ⊃ add T admits a principal gluing. For this, consider the following quiver Q 1 : It admits a cluster category C Q 1 . Let T Q 1 = kQ 1 be the canonical cluster tilting object in C Q 1 . Let T ′ = µ 3 (µ 6 (T Q 1 )) be the cluster tilting object obtained by mutations from T Q 1 . Denote the non isomorphic indecomposable direct summands of T ′ by T ′ i , 1 ≤ i ≤ 6. Then the quiver of Q T ′ is T ′ Let P = add(T ′ 4 ⊕ T ′ 5 ⊕ T ′ 6 ). Then U/P is a 2-Calabi-Yau triangulated category and admits a cluster tilting object with the quiver Q T . By the main theorem of [33], we know that there is a triangle equivalence U/P ≃ C A 2,1 . Thus, we see that the matrix B(T ′ ) admits a triangulated 2-CY realization C Q 1 which is the required principal gluing of C A 2,1 ⊃ add T . We may assume that the images of T ′ 1 , T ′ 2 , T ′ 3 coincide with P 1 , P 2 , τ S 3 in C A 2,1 respectively. Denote the shift functor in C Q 1 (resp. C A 2,1 ) by Σ (resp. [1]).
Note that the denominator vector of X ′ N equals the denominator vector of X T τ S 3 . Now the result follows from the Proposition above.
where G is the functor Hom C (T, ?) : C → mod End C (T ). The following proposition is proved in greater generality in [16]. with T 1 M , T 0 ′ M in add T and [T 0 ′ M : T l ] = 0, where (T l ) g l is the sum of g l copies of T l . Applying the functor Hom C (T ′ , ?) to the shift of the above triangle, we get the exact sequence 0 → Hom C (T ′ , Σ(T l ) g l ) → Hom C (T ′ , ΣM ) → Hom C (T ′ , Σ 2 T 1 M ) → . . . Note that Hom C (T ′ , Σ(T l ) g ) ∼ = (S l ) g l , i.e. S l occurs with multiplicity ≥ g l in the socle of Hom C (T ′ , ΣM ). If the multiplicity of S l in the socle of Hom C (T ′ , ΣM ) was > g l , then S l would occur in the socle of Hom C (T ′ , Σ 2 T 1 M ). This is not the case since Hom C (T ′ , Σ 2 T 1 M ) is the sum of injective indecomposables not isomorphic to the injective hull Hom C (T ′ , Σ 2 T l ) of S l . Conversely, if S l occurs in the socle of Hom C (T ′ , ΣM ), thanks to the split idempotents property of C, we have an irreducible morphism α : ΣT l → ΣM in C. Thus, by the definition of the index, we get g l > 0. Moreover, the multiplicity of S l equals g l by the same argument as before.
Assume that g l > 0. For an arbitrary submodule U of Hom C (T ′ , ΣM ), let dim U = (e 1 , . . . , e n ). We will show that Indeed, consider the projective resolution of the simple module S l . . . → ⊕P b il i → P l → S l → 0. Applying the functor Hom End C (T ′ ) (?, U ), we get the exact sequence 0 → Hom(S l , U ) → Hom(P l , U ) → Hom(⊕P b il i , U ) → . . . , which implies the inequality because the dimension of Hom(S l , U ) is less or equal to the multiplicity of S l in the socle of Hom C (T ′ , ΣM ), which equals g l . By Theorem 6.5, we have 6.7. Acyclic cluster algebras with principal coefficients. Let B be an antisymmetric integer r × r-matrix. Assume that B is acyclic. Let Q be the corresponding quiver of B with set of vertices Q 0 = {1, . . . , r} and with set of arrows Q 1 . Let C Q be the cluster category of Q, T = kQ the canonical cluster tilting object of C Q . We claim that the cluster category C Q ⊃ add T admits a principal gluing.
Indeed, we define a new quiverQ = Q ← − Q 0 associated with Q: Its set of vertices is {1, . . . , 2r}, and its arrows are those of Q and new arrows from r + i to i for each vertex i of Q. Since Q is acyclic, so isQ, hence kQ is finite-dimensional and hereditary. Thus, we have the cluster category CQ which is a triangulated 2-CY realization of the matrix B −I r I r 0 .
In particular, CQ ⊃ add kQ is a principal gluing for C Q ⊃ add T . Thus, Proposition 6.2, Theorem 6.3, Theorem 6.5 and Proposition 6.6 hold for acyclic cluster algebras with principal coefficients. Let P i , 1 ≤ i ≤ 2r, be the non isomorphic indecomposable projective right modules of kQ. Let P = add(P r+1 ⊕ . . . ⊕ P 2r ). We have a triangle equivalence Recall that there is a partial order on Z r defined by α ≤ β iff α(i) ≤ β(i), for 1 ≤ i ≤ r, where α, β ∈ Z r . Proposition 6.10. Let B be a 2r ×r integer matrix, whose principal part is antisymmetric and acyclic and whose complementary part is the identity matrix. Let σ be a sequence k 1 , . . ., k m with 1 ≤ k i ≤ r. Denote by B σ the matrix . Let E σ = (e 1 , e 2 , . . . , e r ) be the complementary part of B σ , where e i ∈ Z r , 1 ≤ i ≤ r. Then for each i, we have e i ≤ 0 or e i ≥ 0.
Proof. Suppose that there is some k such that e k 0 and e k 0. For simplicity, assume that k = 1, i.e. there are r < i, j ≤ 2r such that b σ i1 > 0 and b σ j1 < 0. Let Q be the quiver corresponding to the principal part of B andQ as constructed above. By the argument above, there is a cluster tilting object T ′ of C kQ such that B(T ′ ) 0 = B σ . We have arrows P i → T ′ 1 and T ′ 1 → P j , where T ′ 1 is the indecomposable direct summand of T ′ corresponding to the first column of B σ . Now if we consider the mutation in direction 1 of T ′ , we will have an arrow P i → P j in Q µ 1 (T ′ ) . But this is impossible, since for r < l ≤ 2r, the P l are simple pairwise non isomorphic modules so we have Hom C kQ (P i , P j ) = Hom kQ (P i , P j ) = 0.