FROM TRIANGULATED CATEGORIES TO CLUSTER ALGEBRAS II

PHILIPPE CALDERO AND BERNHARD KELLER

Abstract. In the acyclic case, we establish a one-to-one correspondence between the tilting objects of the cluster category and the clusters of the associated cluster algebra. This correspondence enables us to solve conjectures on cluster algebras. We prove a multiplicativity theorem, a denominator theorem, and some conjectures on properties of the mutation graph. As in the previous article, the proofs rely on the Calabi-Yau property of the cluster category.

1. Introduction

Cluster algebras are commutative algebras, introduced in [11] by S. Fomin and A. Zelevinsky. Originally, they were constructed to obtain a better understanding of the positivity and multiplicativity properties of Lusztig’s dual (semi)canonical basis of the algebra of coordinate functions on homogeneous spaces. Cluster algebras are generated by the so-called cluster variables gathered into sets of fixed cardinality called clusters. In the framework of the present paper, the cluster variables are obtained by a recursive process from an antisymmetric square matrix B.

Denote by Q the quiver associated to the matrix B. Assume that Q is connected. A theorem of Fomin and Zelevinsky asserts that the number of cluster variables of the corresponding cluster algebra A_{Q} is finite if and only if Q is mutation-equivalent to a quiver whose underlying graph is a simply laced Dynkin diagram. In this case, it is known that the combinatorics of the clusters are governed by the generalized associahedron.

Let Q be any finite quiver without oriented cycles and let k be an algebraically closed field. The cluster category C = C_{Q} was introduced in [8] for type A_{n} and in [6] in the general case. This construction was motivated by the combinatorial similarities of C_{Q} with the cluster algebra A_{Q}. The cluster category is the category of orbits under an autoequivalence of the bounded derived category D^{b} of the category of finite dimensional kQ-modules. By [18], the category C_{Q} is a triangulated category. Let us denote its shift functor by S and write Ext_{C}^{1}(M, N) for Hom_{C}(M, SN) for any objects M, N of C. By construction, the cluster category is Calabi-Yau of CY-dimension 2; in other terms, the functor Ext^{1} is symmetric in the following sense:

Ext_{C}^{1}(M, N) ≃ D Ext_{C}^{1}(N, M).

In a series of articles [6], [3], [4], the authors study the tilting theory of the cluster category. More precisely, they describe the combinatorics of the cluster tilting objects of the category C, i.e. the objects without self-extensions and with a maximal number of nonisomorphic indecomposable summands. In [4], the authors define a map β between the set of clusters of A_{Q} and the set of tilting objects of the category C_{Q}. A natural question arises: does β provide a one-to-one correspondence between both sets?

In the articles [7] and [10], it is proved that in the finite case, i.e. the Dynkin case, the cluster algebra can be recovered from the corresponding cluster category as the so-called exceptional Hall algebra of the cluster category. More precisely, in [7], the authors give an explicit correspondence M ↦ X_{M} between indecomposable objects of C_{Q} and cluster variables of A_{Q}. In [10], we provide a multiplication rule for the algebra A_{Q} in terms of the triangulated category C_{Q}.

Date: version du 12/10/2005, modifiée le 12/04/2006.

An ingenious application of the methods of [10] can be found in [14], where the authors give a multiplication formula for elements of Lusztig’s dual semicanonical basis. Here, the cluster category is replaced by the category of finite-dimensional modules over the preprojective algebra and the rôle of the cluster algebra is played by the coordinate algebra of the maximal unipotent subgroup in the corresponding semisimple algebraic group.

The aim of the present article is to generalize some of the results of [7], [10] to the case where Q is any finite quiver without oriented cycles. Building on the important results obtained in [4] we strengthen here the connections between the cluster category and the cluster algebra by giving an explicit expression for the correspondence β and proving that β is one-to-one. The key ingredient of the proof is a natural analogue of the map M ↦ X_{M} of [7]. With the help of a multiplicativity result, we show that M ↦ X_{M} defines a bijection between the indecomposable objects without self-extensions of C_{Q} and the cluster variables of A_{Q}.

This correspondence between cluster algebras and cluster categories gives positive answers to some of the conjectures which S. Fomin and A. Zelevinsky formulated in [13]. We prove connectedness properties of some mutation graphs, cf. section 5.3. As a byproduct, we obtain a cluster-categorical interpretation of the passage to a submatrix of the exchange matrix. This strengthens a key result of [4] and may be of independent interest.

Another consequence of the bijectivity of β is that each seed is determined by its cluster. As we have learned recently, this result is obtained independently in [5].

The paper is organized as follows: In the first part, we recall well-known facts on the cluster category. For any object M of the cluster category, we define the Laurent polynomial X_{M} as in [7]. With the techniques of [10], we prove an ‘exchange relation’ for the X_{M}. To be more precise, we prove that if M and N are indecomposable objects of the category C = C_{Q} such that Ext_{C}^{1}(M, N) = k, then

X_{M}X_{N} = X_{B} + X_{B′},

where B and B′ are the unique objects (up to isomorphism) such that there exist non split triangles

N → B → M → SN, M → B′ → N → SM.

This formula is an analogue of the ‘exchange relation’ between cluster variables. With the help of a comparison theorem of [4], we prove by induction that for any indecomposable exceptional object M, X_{M} is a cluster variable and that its (monomial) denominator is given by the dimension vector \underline{dim} (M). From this denominator property, we deduce that the map M ↦ X_{M} is injective when restricted to the set of indecomposable objects of C_{Q} without self-extensions. The connectedness of the tilting graph proved in [4] then implies that the map M ↦ X_{M} is a one-to-one correspondence between the set of tilting objects of C_{Q} and the set of clusters of A_{Q}.

We then deduce some applications of this correspondence to conjectures of [13].

Acknowledgements: The first author is indebted to Thomas Brüstle and Ralf Schiffler for useful conversations. He also wishes to thank Andrei Zelevinsky for his kind hospitality and for pointing out to him the conjectures of [13]. The authors thank Andrew Hubery for pointing out a gap in a previous version of this article.

2. The cluster category and the cluster variable formula

2.1. Let H be a finite dimensional hereditary algebra over an algebraically closed field k. We denote by H -mod the category of finitely generated H-modules. We choose representatives S_{i}, 1 ≤ i ≤ n, of the isoclasses of the simple H-modules and denote by I_{i} the injective hull and by P_{i} the projective cover of S_{i}.

The Grothendieck group of H -mod is the group G_{0}(H -mod) generated by the isoclasses of modules in H -mod and subject to the relations X = M + N obtained from exact sequences 0 → M → X → N → 0 in H -mod. We denote by [M] the class of a module M in G_{0}(H -mod). We put α_{i} = [S_{i}]. The Grothendieck group is free abelian on the α_{i}. The dimension vector \underline{dim} (M) of a module M is by definition the vector of the coordinates of [M] in this basis.

We define the Euler form by < M, N >= dim Hom(M, N)−dim Ext^{1}(M, N), for any M, N in H -mod. Since H is hereditary, this form is well-defined on the Grothendieck group.

Let τ be the Auslander-Reiten functor of H -mod. This functor verifies the Auslander-Reiten formula:

D Hom(N, τM) = Ext^{1}(M, N),

where D is the functor Hom_{k}(?, k).

2.2. For any H-module M, and any e in G_{0}(H -mod), we denote by Gr_{e}(M) the Grassmannian of submodules of M with dimension vector e:

Gr_{e}(M) = \{N, N ∈ H -mod, N ⊂ M, \underline{dim} (N) = e\}.

It is a closed subvariety of the classical Grassmannian of the vector space M. Let χ_{c} be the Euler-Poincaré characteristic of the etale cohomology with proper support defined by

χ_{c}(X) = ∑_{i=0}^{∞}(−1)^{i} dim H_{c}^{i}(X, \overline{ℚ}_{l}).

Let ℚ[x_{i}^{±1}, 1 ≤ i ≤ n] be the ℚ-algebra of Laurent polynomials in the variables x_{i}. As in [7], for any module M, we set

X_{M} = ∑_{e} χ_{c}(Gr_{e}(M)) ∏_{i} x_{i}^{−<e,α_{i}>−<α_{i},m−e>} ∈ ℚ[x_{i}^{±1}, 1 ≤ i ≤ n],

where m := \underline{dim} (M). Note that, as M is finite dimensional, there only exists a finite number of non zero terms in this sum. Remark that X_{M} only depends on the isoclass of the module M. As in [7] one shows that

χ_{c}(Gr_{g}(M ⊕ N)) = ∑_{e+f=g} χ_{c}(Gr_{e}(M))χ_{c}(Gr_{f}(N)).

Hence, the bilinearity of the Euler form implies that

X_{M⊕N} = X_{M}X_{N}.

2.3. As H is hereditary and finite dimensional, there exists a finite quiver Q without oriented cycles such that H is Morita equivalent to the path algebra kQ of Q. Let Q_{0} be the set of vertices and Q_{1} the set arrows of Q. Let n be the number of vertices of Q.

The bounded derived category D^{b} = D^{b}(H) of H -mod is a triangulated category. We denote its shift functor M ↦ M[1] by S. The category D^{b} is a Krull-Schmidt category and, up to canonical triangle equivalence, it only depends on the underlying graph of Q, see [15]. We identify the category H -mod with the full subcategory of D^{b} formed by the complexes whose homology is concentrated in degree 0. We simply call ‘modules’ the objects in this subcategory. The indecomposable objects of D^{b} are the shifts S^{i}M, i ∈ ℤ, of the indecomposable objects of H -mod. We still denote by τ the AR-functor of D^{b}; it is known that τ is an autoequivalence characterized by the Auslander-Reiten formula.

Let F be the autoequivalence τ^{−1}S of D^{b}. The AR-formula implies that

Ext_{D^{b}}^{1}(M, N) = Hom_{D^{b}}(M, SN) = D Ext_{D^{b}}^{1}(F N, M),

for any objects M, N of D^{b}. Let C = C(H) be the orbit category D^{b}/F: the objects of C are the objects of D^{b} and the morphisms of C are given by

Hom_{C}(M, N) = ⊕_{i∈ℤ} Hom_{D^{b}}(M, F^{i}N).

The category C is the so-called cluster category, introduced and studied in depth in [6]. Let π be the canonical functor from D^{b} to C. We will often omit the functor π from the notations. Statements (i) and (ii) of the following theorem were proved in [18], statements (iii) and (iv) in [6]:

Theorem 1. (i) The category C is triangulated and (ii) the functor π : D → C is a triangle functor. (iii) The category C is a Krull-Schmidt category and (iv) For any indecomposable object without self-extensions M of C, we have End_{C}(M) =

k.

The shift functor of the triangulated category C will still be denoted by S. For any objects M, N of C, the formulas above imply that there exists an (almost canonical) duality

ϕ : Ext_{C}^{1}(M, N) × Ext_{C}^{1}(N, M) → k.

The set of indecomposable objects of C is given by

Ind(C) = Ind(H -mod) ∐{SP_{i}, 1 ≤ i ≤ n}.

Note that SP_{i} = S^{−1}τ^{−1}SP_{i} = S^{−1}I_{i}.

We extend the definition of X_{M} to any object M of the category C by setting X_{SP_{i}} = x_{i}, 1 ≤ i ≤ n, and requiring X_{M⊕N} = X_{M}X_{N} for all objects M, N of C.

The AR-formula and the fact that τ passes to the Grothendieck group of the derived category of H -mod allow us to rewrite X_{M} for a module M as

(2.1)  X_{M} = ∑_{e} χ_{c}(Gr_{e}(M)) x^{τ(e)−\underline{dim} M+e} ,

where we have set

x^{v} = ∏_{i=1}^{n} x_{i}^{<\underline{dim} S_{i},v>},

for any v in ℤ^{n}. Remark that this notation gives

X_{SP_{i}} = x^{\underline{dim} I_{i}}.

2.4. Each object M of C can be uniquely decomposed in the following way:

M = M_{0} ⊕ SP_{M} = M_{0} ⊕ S^{−1}I_{M},

where M_{0} is the image under π of a module in D^{b}, and where P_{M}, respectively I_{M}, is a uniquely determined projective, respectively injective, module. We will say that an object M of C is a module if M = M_{0}, and that M is the shift of a projective module if M = SP_{M}.

From [6], we recall the Proposition 1. For any indecomposable modules M and N in C, we have

Ext_{C}^{1}(M, N) = Ext_{H}^{1}(M, N) ⊕ D Ext_{H}^{1}(N, M).

The module M_{0} can be recovered using the functor

H^{0} = Hom_{C}(H_{H},?) : C → H -mod .

Indeed, we have

H^{0}(M) = H^{0}(M_{0}) ⊕ H^{0}(SP_{M}) = Hom_{H -mod}(H_{H}, M_{0}) ⊕ Hom_{C}(⊕_{i}P_{i}, SP_{M}) = M_{0} ,

as the last factor is zero. The functor H^{0} is a homological functor, i.e. it maps triangles in C to long exact sequences of H-modules.

3. A multiplication formula

3.1. The aim of the section is to prove the following theorem:

Theorem 2. Let M and N be indecomposable objects of the category C such that Ext_{C}^{1}(M, N) is one-dimensional. Then we have

X_{M}X_{N} = X_{B} + X_{B′},

where B and B′ are the unique objects (up to isomorphism) such that there exist non split triangles

N → B → M → SN, M → B′ → N → SM.

Note that when H is the path algebra of a Dynkin quiver, the theorem is a particular case of the cluster multiplication formula of [10]. Actually, we will see that the method of [10] generalizes nicely to the framework of the theorem.

Thanks to the hypotheses of the theorem and the symmetry of Ext^{1}, we just need to consider the two following cases

1. N = SP_{i} for an i ∈ Q_{0} and M is an indecomposable module.

2. M and N are indecomposable modules. Indeed, the isomorphisms M = SP_{j} and N = SP_{i} would imply

Ext_{C}^{1}(M, N) = Ext_{C}^{1}(P_{j}, P_{i}) = 0.

3.2. We now prove the theorem in the first case. Suppose N = SP_{i}, and let M be an indecomposable module such that Ext_{C}^{1}(SP_{i}, M) = k. Using theorem 1 and the AR-formula, we obtain

\underline{dim} (M)_{i} = dim Hom_{H}(P_{i}, M) = dim Hom_{C}(P_{i}, M) = dim Ext_{C}^{1}(SP_{i}, M) = 1.

Hence, up to a multiplicative scalar, there exists a unique non zero morphism ζ : M → I_{i} and a non zero morphism ζ′ : P_{i} → M.

Lemma 1. Let M′ be a submodule of M. Then either M′ ⊂ ker ζ or im ζ′ ⊂ M′.

Proof. By the formula above, the space M′_{i} is of dimension 0 or 1. We claim that 1. dim(M′) = 0 if and only if M′ ⊂ ker ζ,

2. dim(M′) = 1 if and only if im ζ′ ⊂ M′.

The lemma follows from the claim. Let’s prove part 1. The second part is similar and left to the reader. The module im ζ is non zero and so it contains the simple socle S_{i} of I_{i}. Hence, dim(ker ζ)_{i} = 0, which gives the ‘if’ part. Conversely, if dim(M′)_{i} = 0, then ζ(M′) ∩ S_{i} = 0, hence ζ(M′) = 0 as S_{i} is the socle of I_{i}. □

Applying the functor H^{0} the non split triangle

(3.1)  \xymatrix{SP_{i} \ar[r]^{ι} & B \ar[r]^{π} & M \ar[r]^{ζ} & S^{2}P_{i} = I_{i}}

we obtain a long exact sequences of H-modules

\xymatrix{0 \ar[r] & H^{0}B \ar[r]^{H^{0}π} & M \ar[r]^{H^{0}ζ} & I_{i} \ar[r]^{H^{1}ι} & H^{0}τB \ar[r]^{H^{1}π} & H^{0}τM} ,

Now, H^{0}τB = τH^{0}B ⊕ I_{B}, and the first factor is non injective. As the quotient of an injective module is still injective, we have im(H^{1}ι) ⊂ I_{B}. Moreover, as H^{0}τM is non injective, we have I_{B} ⊂ ker(H^{1}π). Hence, we have equality and so the following exact sequence holds

(3.2)  \xymatrix{0 \ar[r] & H^{0}B \ar[r]^{H^{0}π} & M \ar[r]^{H^{0}ζ} & I_{i} \ar[r]^{H^{1}ι} & I_{B} \ar[r] & 0} .

Note that the morphism H^{0}ζ = ζ is non zero.

In the same way, applying the functor H^{0}, the non split triangle

(3.3)  \xymatrix{P_{i} \ar[r]^{ζ′} & M \ar[r]^{π′} & B′ \ar[r]^{ι′} & SP_{i},}

we obtain

(3.4)  \xymatrix{0 \ar[r] & P_{B′} \ar[r] & P_{i} \ar[r]^{H^{0}ζ′} & M \ar[r]^{H^{0}π′} & H^{0}B′ \ar[r] & 0} .

Note that the morphism H^{0}ζ′ = ζ′ is non zero.

Now, the lemma implies that for any submodule M′ of M, M′ is a either submodule of im H^{0}π or contains ker H^{0}π′. Hence, there is a natural bijection between Gr_{e} M and Gr_{e}(H^{0}B) ∐ Gr_{e−k}(H^{0}B′), where

(3.5)  k := \underline{dim} ker H^{0}π′ = \underline{dim} P_{i} − \underline{dim} P_{B′}.

We want to prove the multiplication formula, which in this case is

x^{\underline{dim} I_{i}} ∑_{e} χ_{c}(Gr_{e} M)x^{τ(e)−\underline{dim} M+e} =

x^{\underline{dim} I_{B}} ∑_{e} χ_{c}(Gr_{e} H^{0}B)x^{τ(e)−\underline{dim} H^{0}B+e} + x^{\underline{dim} I_{B′}} ∑_{e} χ_{c}(Gr_{e} H^{0}B′)x^{τ(e)−\underline{dim} H^{0}B′+e}.

So, it remains to prove that

\underline{dim} I_{i} + τ(e) − \underline{dim} M + e = \underline{dim} I_{B} + τ(e) − \underline{dim} H^{0}B + e,

and

\underline{dim} I_{i} + τ(e) − \underline{dim} M + e = \underline{dim} I_{B′} + τ(e − k) − \underline{dim} H^{0}B′ + e − k.

The first formula is a direct consequence of 3.2. The second one comes from 3.4, 3.5 and the formula τ(\underline{dim} P_{j}) = −\underline{dim} I_{j}.

3.3. This subsection and the following one are devoted to the proof of the theorem in the second case. In order to simplify notations, we will write (X, Y) for Hom_{C}(X, Y).

Let M and N be two indecomposable modules such that Ext_{C}^{1}(N, M) = k. By proposition 1, we can suppose that Ext_{H}^{1}(N, M) = k and Ext_{H}^{1}(M, N) = 0. In this case, by theorem 1, there exists (up to isomorphism) a unique non split short exact sequence of H-modules

\xymatrix{0 \ar[r] & M \ar[r]^{i} & B_{+} \ar[r]^{p} & N \ar[r] & 0} ,

and two triangles in C

\xymatrix{M \ar[r]^{i} & B_{+} \ar[r]^{p} & N \ar[r] & SM} ,

\xymatrix{N \ar[r]^{i′} & B_{−} \ar[r]^{p′} & M \ar[r] & SN} .

Note that B_{+} is a ‘module’ of C but B_{−} is just an object; they both are uniquely determined up to isomorphism. We want to prove the formula

X_{B_{+}} + X_{B_{−}} = X_{M}X_{N},

and the idea is first to construct a morphism Ψ between Gr B_{+} ∐ Gr H^{0}B_{−} and Gr M ×Gr N. For any submodule B′_{+} of B_{+}, set Ψ(B′_{+}) = (i^{−1}B′_{+}, pB′_{+}), and for any submodule B′_{−} of B_{−}, set Ψ(B′_{−}) = ((H^{0}p′)B′_{−}, (H^{0}i′)^{−1}B′_{−}). As a first step, we want to prove the proposition

Proposition 2. The variety Gr M×Gr N is the disjoint union of Ψ(Gr B_{+}) and Ψ(Gr H^{0}B_{−}). Moreover, the fibers of Ψ are affine spaces.

This proposition will be proved at the end of this subsection.

Given a submodule M′ of M, a submodule N′ of N, and the corresponding embeddings i_{M} and i_{N}, we have a diagram

\xymatrix{S^{−1}M \ar[r]^{ε′} & N \ar[r]^{ε} & SM \\ S^{−1}M′ \ar[u]^{S^{−1}i_{M}} & N′ \ar[u]^{i_{N}} & SM′ \ar[u]^{Si_{M}}}

and two complexes

\xymatrix{(S^{−1}M, N′) \ar[r]^{α′} & (S^{−1}M, N) ⊕ (S^{−1}M′, N′) \ar[r]^{β′} & (S^{−1}M′, N) \\ (N′, SM) & (N, SM) ⊕ (N′, SM′) \ar[l]_{α} & (N, SM′) \ar[l]_{β}}

where

α′ = [ \begin{matrix}(i_{N′})_{∗} \\ (S^{−1}i_{M′})^{∗} \end{matrix} ] , β′ = [(S^{−1}i_{M′})^{∗}, −(i′_{N})_{∗}] , α = [(i_{N′})_{∗}, (Si_{M′})^{∗}] , β = [ \begin{matrix}(Si_{M′})^{∗} \\ −(i_{N′})_{∗} \end{matrix} ] .

The two sequences are dual to each other via the canonical duality ϕ.

The following proposition is straightforward by using basic properties of triangulated categories.

Proposition 3. The following conditions are equivalent:

(i) There exists a submodule B′_{+} ⊂ B_{+} such that the diagram

\xymatrix{0 \ar[r] & M \ar[r] & B_{+} \ar[r] & N \ar[r] & 0 \\ 0 \ar[r] & M′ \ar[u] \ar[r] & B′_{+} \ar[u] \ar[r] & N′ \ar[u] \ar[r] & 0.}

commutes. (ii) There exists a morphism η : N′ → SM′ such that the square

\xymatrix{N \ar[r]^{ε} & SM \\ N′ \ar[u]^{i_{N}} \ar[r]^{η} & SM′ \ar[u]^{Si_{M}}}

commutes.

(iii) The composed morphism

ker α ↪ (N′, SM′) ⊕ (N, SM) → (N, SM)

is non zero. (iv) The composed morphism

cok α′ ← (S^{−1}M′, N′) ⊕ (S^{−1}M, SN) ↩ (S^{−1}M, N)

is non zero.

The following proposition sheds light on the situation when the conditions of proposition 3 do not hold.

Proposition 4. The following conditions are equivalent:

(i) The composition

cok α′ ← (S^{−1}M′, N′′) ⊕ (S^{−1}M, SN) ↩ (S^{−1}M, N)

vanishes, i.e. (S^{−1}M, N) is contained in the image of α′. (ii) There exist a submodule B′_{−} ↪ H^{0}B_{−} and a commutative diagram

\xymatrix{N \ar[r]^{H^{0}i′} & H^{0}B′ \ar[r]^{H^{0}p′} & M \\ N′ \ar[u] \ar[r] & B′_{−} \ar[u] \ar[r] & M′ \ar[u] \ar[r] & 0 \\ 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

where M′ := H^{0}p′(B′_{−}) and N′ = (H^{0}i′)^{−1}(B′_{−}).

Proof. Let us show that (i) implies (ii). By the assumption, we can find a commutative square

\xymatrix{S^{−1}M \ar[r]^{ε′} \ar[dr]^{f} & N \\ S^{−1}M′ \ar[u]^{S^{−1}i_{M}} \ar[r]_{0} & N′ \ar[u]^{i_{N}}}

We complete it to a morphism between triangles:

\xymatrix{S^{−1}M \ar[r]^{ε′} \ar[dr]^{f} & N \ar[r]^{i′} & B_{−} \ar[r]^{p′} & M \\ S^{−1}M′ \ar[u] \ar[r]_{0} & N′ \ar[u]^{i_{N}} \ar[r] & N′ ⊕ M′ \ar[u] \ar[r] & M′ \ar[u]^{i_{M}}}

We take the homology:

\xymatrix{H^{0}(S^{−1}M) \ar[r]^{H^{0}(ε′)} \ar[dr]^{H^{0}(f)} & N \ar[r]^{H^{0}i′} & H^{0}B_{−} \ar[r]^{H^{0}p′} & M \\ 0 \ar[r] & N′ \ar[u]^{i_{N}} \ar[r] & N′ ⊕ M′ \ar[u] \ar[r] & M′ \ar[u]^{i_{M}} \ar[r] & 0}

We take B′_{−} as the image of N′ ⊕M′ → H^{0}B_{−}. Let us show that N′ ⊂ N is H^{0}(i′)^{−1}(B′_{−}). Indeed, clearly the image of N′ is contained in B′_{−}. Conversely, if we have x ∈ N whose image lies in B′_{−}, then the image is the image of (x′, y′) in N′ ⊕M′, and the image of x ∈ N under N → H^{0}B_{−} → M vanishes. So, the image of y′ in M vanishes. But M′ → M is mono. So y′ vanishes and we get x′ in N′ such that x in N and x′ have the same image in H^{0}B_{−}. Then x = x′ + (H^{0}(ε′)(z) for some z in S^{−1}M. But H^{0}ε′ = (H^{0}i_{N}) ∘ (H^{0}f). So (H^{0}ε′)(z) lies in fact in N′ ⊂ N and x lies in N′.

Let us show that M′ is the image of B′_{−}. Clearly, the image of B′_{−} is contained in M′. Conversely, if x′ ∈ M′, we consider the image y in B′_{−} of (0, x′) ∈ N′ ⊕ M′. Then clearly, the image of y is x′.

Let us prove that (ii) implies (i). The hypothesis yields the following diagram

\xymatrix{H^{0}(S^{−1}M) \ar[r]^{H^{0}(ε′)} \ar@{.>}[dr] & N \ar[r]^{H^{0}i′} & H^{0}B_{−} \ar[r]^{H^{0}p′} & M \ar[r]^{H^{0}(Sε′)} & H^{0}(SN) \\ H^{0}(S^{−1}M′) \ar[u] & N′ \ar[u] \ar[r] & B′_{−} \ar[u] \ar[r] & M′ \ar[u] \ar[r]_{0} & H^{0}(SN′). \ar[u] \\  & 0 \ar[u] & 0 \ar[u] & 0 \ar[u] & 0 \ar[u]}

As the composition H^{0}(S^{−1}M) → N → H^{0}B_{−} vanishes, the image of H^{0}(S^{−1}M) is contained in N′, which is the inverse image of B′_{−}.

As the composition H^{0}(B_{−}) → M → H^{0}(SN) vanishes, M′ is contained in the kernel of M → H^{0}(SN). We know that M is not injective, so, S^{−1}M = τ^{−1}M is still a module.

Moreover, we have

D Ext^{1}(N, M) = Hom_{H}(τ^{−1}M, N) = Hom_{H}(M, τN) = k.

We obtain the commutative diagrams

\xymatrix{S^{−1}M \ar[r]^{ε′} \ar@{.>}[dr]_{f} & N & M \ar[r]^{Sε′} & SN \\  & N′ \ar[u]_{i_{N}} & M′ \ar[u]^{i_{M}} \ar@{.>}[ur]_{0}}

The module M′ has no injective direct summand, because M is indecomposable and non injective. So, S^{−1}M′ is still a module. Consider

\xymatrix{S^{−1}M \ar[r]^{ε′} \ar[dr]^{f} & N \\ S^{−1}M \ar[u]_{S^{−1}i_{M}} \ar[r]_{0} & N′ \ar[u]^{i_{N}}}.

We have i_{N} ∘ f ∘ S^{−1}i_{M} = ε′ ∘ S^{−1}i_{M} = 0. As i_{N} is injective, this gives f ∘ S^{−1}M = 0, which implies (i). □

Propositions 3 and 4 imply the first part of proposition 2. The second part is a wellknown fact, cf. lemma 3.8 of [7].

3.4. We want to prove the multiplication formula for the second. It reads as follows:

∑_{e} χ_{c}(Gr_{e} M)x^{τ(e)−\underline{dim} M+e} ∑_{f} χ_{c}(Gr_{f} N)x^{τ(f)−\underline{dim} N+f} =

∑_{g} χ_{c}(Gr_{g} H^{0}B_{+})x^{τ(g)−\underline{dim} B_{+}+g} + x^{\underline{dim} I_{B_{−}}} ∑_{g} χ_{c}(Gr_{g} H^{0}B_{−})x^{τ(g)−\underline{dim} H^{0}B_{−}+g}.

By combining proposition 2 with proposition 3.6 of [7], we can compare Euler characteristics on both sides of the equality. What we need to prove now is

(3.6)  τ(e) − \underline{dim} M + e + τ(f) − \underline{dim} N + f = τ(g) − \underline{dim} B_{+} + g,

with e = \underline{dim} M′, f = \underline{dim} N′, g = \underline{dim} B′_{+}, in the setting of proposition 3 (i), and then

(3.7)  τ(e) − \underline{dim} M + e + τ(f) − \underline{dim} N + f = \underline{dim} I_{B_{−}} + τ(g) − \underline{dim} H^{0}B_{−} + g,

with e = \underline{dim} M′, f = \underline{dim} N′, g = \underline{dim} B′_{−}, in the setting of proposition 4 (ii). The formula 3.6 is clear since g = e + f in this case. In order to prove the second formula, we need to complete the diagram of proposition 4

by adding kernels and cokernels

\xymatrix{ &  & N/N′ \ar[r] & (H^{0}B_{−})/B′_{−} \ar[r] & M/M′ \ar[r] & C \ar[r] & 0 \\  & H^{0}(S^{−1}M) \ar[r] & N \ar[u] \ar[r]^{H^{0}(i)} & H^{0}B_{−} \ar[u] \ar[r]^{H^{0}(p)} & M \ar[u] \ar[r] & H^{0}(SN) \\ 0 \ar[r] & K \ar[r] & N′ \ar[u] \ar[r] & B′_{−} \ar[u] \ar[r] & M′ \ar[u]}

With the notation above, the diagram implies the equalities

τ(e) + τ(f) = τ(g) + τ(\underline{dim} K), (\underline{dim} M − e) + (\underline{dim} N − f) = (\underline{dim} H^{0}B_{−} − g) + \underline{dim} C.

So, in order to prove formula 3.7, we remains to show that

(3.8)  τ\underline{dim} K − \underline{dim} C = \underline{dim} I_{B_{−}}.

For this, we first note that we have the three triangles

\xymatrix{N \ar[r]^{H^{0}(i)} & H^{0}B_{−} \ar[r] & cok(H^{0}i) ⊕ SK \ar[r] & SN}
\xymatrix{SP_{B_{−}} \ar[r] & B_{−} \ar[r] & H^{0}B_{−} \ar[r]^{0} & I_{B_{−}}}
N → Y → M → SN

in C_{Q}. Note that H^{0}i is the composition of the morphism N → B_{−} with the projection B_{−} → H^{0}B_{−}. If we form the octahedron associated with this composition, the three triangles we have just mentioned appear among its faces, as well as a new triangle, namely

\xymatrix{SP_{B_{−}} \ar[r] & M \ar[r] & cok(H^{0}i) ⊕ SK \ar[r] & I_{B_{−}}} .

If we apply H^{∗} to this triangle, we obtain the exact sequence of H-modules

\xymatrix{0 \ar[r] & M \ar[r] & cok(H^{0}i) ⊕ H^{0}(τK) \ar[r] & I_{B_{−}} \ar[r] & H^{0}(τM)} .

Since M is an indecomposable module, τM is either an indecomposable non injective module or zero. The image of I_{B_{−}} → τM = H^{0}τM is injective (as a quotient of an injective module). Hence it is zero and we get an exact sequence

\xymatrix{0 \ar[r] & M \ar[r] & cok(H^{0}i) ⊕ H^{0}(τK) \ar[r] & I_{B_{−}} \ar[r] & 0} .

In the Grothendieck group, this yields

0 = \underline{dim} M − \underline{dim} cok(H^{0}i) − \underline{dim} H^{0}(τK) + \underline{dim} I_{B_{−}} = \underline{dim} C − \underline{dim} H^{0}(τK) + \underline{dim} I_{B_{−}}.

Now, by the third triangle, K is a quotient of H^{0}(S^{−1}M) = H^{0}(τ^{−1}M). As M is a non injective indecomposable module, H^{0}(τ^{−1}M) = τ^{−1}M, so τK is a quotient of M, and hence, τK is a module. Thus, we get formula 3.8 as desired. This ends the proof of theorem 2.

4. A denominator theorem.

4.1. Weakly positive Laurent polynomials. We recall an idea from [4]: Define an integer polynomial P in n variables x_{1}, …, x_{n} to be weakly positive if we have P (z) > 0 for each point z of ℕ^{n} which has at most one vanishing component. If P is an integer polynomial and d an n-tuple of integers, then L = P/x^{d} is a Laurent polynomial, where x^{d} is the product of the x_{i}^{d_{i}}. It is a weakly positive Laurent polynomial if we can choose P weakly positive. Clearly, in this case, no factor x_{i} divides P so that the factorization L = (1/x^{d}) · P is unique. We call x^{d} the denominator of L. For example, the Laurent polynomial L = x_{1} is weakly positive with denominator 1/x_{1}. The proof of the following lemma is elementary and left to the reader.

Lemma 2. a) If L_{1} and L_{2} are weakly positive Laurent polynomials, so is their sum L_{1} + L_{2}. Moreover, if x^{d} and x^{e} are the denominators of L_{1} and L_{2}, the denominator of L_{1} + L_{2} is x^{sup(d,e)}, where by definition sup(d, e)_{i} = sup(e_{i}, d_{i}), 1 ≤ i ≤ n.

b) Suppose that L_{1} and L_{2} are Laurent polynomials and L_{1} is weakly positive. Then L_{2} is weakly positive iff L_{1}L_{2} is weakly positive. Moreover, if x^{d} and x^{e} are the denominators of L_{1} and L_{2}, the denominator of L_{1}L_{2} is x^{d+e}.

4.2. Denominators and dimension vectors. From the multiplication formula, we obtain the following denominator property for exceptional modules. A direct proof for arbitrary modules has recently been obtained in [17].

Theorem 3. Let M be an indecomposable exceptional H-module with dimension vector \underline{dim} M = (m_{i}). Then the denominator of X_{M} as an irreducible fraction of integral polynomials in the variables x_{i} is ∏_{i} x_{i}^{m_{i}}.

Proof. Let us start with some preliminary remarks: By the explicit formula for X_{M}, its denominator as an irreducible fraction of integral polynomials is a monomial

x^{den(M)} = ∏_{i} x_{i}^{den(M)_{i}} ,

where den(M) ∈ ℤ^{n}. We claim that for each exceptional indecomposable M, the Laurent polynomial X_{M} is weakly positive. Indeed, if (T, T′) is an exchange pair of exceptional objects (cf. [6]) and

T → B → T′ → ST and T′ → B′ → T → ST′

are non split triangles, then we have

X_{T′} = \frac{X_{B} + X_{B′}}{X_{T}}

by the multiplication formula. Thus, by the lemma above, if X_{B}, X_{B′} and X_{T} are weakly positive, so is X_{T′}. The claim therefore follows from the facts that each exceptional object is a direct summand of a cluster tilting object and that the cluster tilting graph is connected, cf. [6]. The lemma also shows that for an exchange pair (T, T′), we have

den(T) + den(T′) = sup(den(B), den(B′)).

Now for an object X of C_{Q}, we define

δ(X) = \underline{dim} H^{0}(X) − ∑ m_{i}e_{i} ,

where m_{i} is the multiplicity of SP_{i} in the decomposition of X into indecomposables and e_{i} the ith vector of the canonical basis of ℤ^{n}. With this notation, we have to prove that

δ(M) = den(M)

for each indecomposable exceptional M. If P is indecomposable projective, this equality is trivial for M = SP and is shown for M = P by computing X_{P} explicitly as in [7, lemma 3.9]. To prove that the equality holds for every indecomposable exceptional, it suffices therefore, by induction, to prove that we have

δ(T) + δ(T′) = sup(δ(B), δ(B′)).

if (T, T′) is an exchange pair and B, B′ are the non split extensions of T′ by T and T by T′. This will be proved by distinguishing two cases according to whether T and T′ are modules or one of them is a shifted projective (note that they cannot both be shifted projectives since Ext_{C_{Q}}^{1}(T, T′) ≠ 0).

Case 1: T and T′ are modules. Since we have

k = Ext_{C_{Q}}^{1}(T_{1}, T′_{1}) = Ext_{kQ}^{1}(T_{1}, T′_{1}) ⊕ D Ext_{kQ}^{1}(T′_{1}, T_{1}),

exactly one of the triangles

T → B → T′ → ST and T′ → B′ → T → ST′

comes from an exact sequence of modules. Let us assume it is the first one. Then, by applying the functor H^{0}, we get exact sequences

0 → T → B → T′ → 0 and T′ → H^{0}(B′) → T.

These show that we have

\underline{dim} B = \underline{dim} T + \underline{dim} T′ and \underline{dim} H^{0}(B′) ≤ \underline{dim} T + \underline{dim} T′ ≤ \underline{dim} B.

It follows that δ(B′) ≤ \underline{dim} B = δ(B) and

δ(T′) + δ(T) = δ(B) = sup(δ(B), δ(B′)).

Case 2: We have T = SP for an indecomposable projective P and T′ is a module. Again, we have non split triangles

T → B → T′ → ST and T′ → B′ → T → ST.

We have

k = Ext_{C_{Q}}^{1}(T, T′) = Hom_{C_{Q}}(SP, ST′) = Hom_{kQ}(P, T′)

and

k = Ext_{C_{Q}}^{1}(T′, T) = Hom_{C_{Q}}(T′, S^{2}P) = Hom_{kQ}(T′, νP) ,

where ν is the Nakayama functor. Since mod kQ is hereditary, if f : L → M is a morphism of mod kQ, then in the derived category, we have a triangle

\xymatrix{L \ar[r]^{f} & M \ar[r] & cok(f) ⊕ S ker(f) \ar[r] & SL.}

It follows that the triangles above are in fact isomorphic to the triangles

S^{−1}νP → S^{−1} cok(f) ⊕ ker(f) → T′ → νP and T′ → S ker(g) ⊕ cok(g) → SP → ST′

associated with arbitrary non zero morphisms f : T′ → νP and g : P → T′. Now cok(f) is injective as a quotient of an injective and ker(g) projective as a quotient of a projective. Moreover, if i is the vertex of Q such that P = P_{i}, then, as a submodule of P_{i}, the module ker(f) is a direct sum of indecomposables P_{j} such that j < i. Similarly, as a quotient of νP_{i}, the module cok(g) is a direct sum of indecomposables νP_{j} such that i < j (note that we consider right modules and order the vertices of Q in the natural way). It follows that we have

sup(δ(S^{−1} cok(f)), δ(S ker(g))) = 0

(note that both vectors have negative components). Thus, we have

sup(δ(B), δ(B′)) = sup(\underline{dim} ker(f) + δ(S^{−1} cok(f)), \underline{dim} cok(g) + δ(S ker(g)))
= sup(\underline{dim} ker(f), \underline{dim} cok(g)).

It remains to be proved that

(4.1)  \underline{dim} T′ + δ(T) = sup(\underline{dim} ker(f), \underline{dim} cok(g)).

We check this equality by comparing both sides at each vertex j of Q. As above, let i be the vertex of Q such that P = P_{i} so that we have δ(T) = −e_{i}. We have

Hom(P_{i}, T′) = Ext_{C_{Q}}^{1}(SP_{i}, T′) = k

so that (\underline{dim} T′)_{i} = 1. The maps f : T′ → νP and g : P → T′ induce isomorphisms in Hom(P_{i},?) since g ∘ f induces an isomorphism between the one-dimensional spaces Hom(P_{i}, P) to Hom(P_{i}, νP). It follows that (\underline{dim} ker(f))_{i} and (\underline{dim} cok(f))_{i} both vanish so that the equality (4.1) holds at j = i. Now consider the exact sequences

\xymatrix{0 \ar[r] & ker(f) \ar[r] & T′ \ar[r]^{f} & νP & and & P \ar[r]^{g} & T′ \ar[r] & cok(g) \ar[r] & 0.}

Suppose that j is not a predecessor of i. Then (\underline{dim} P)_{j} = 0 and we have (\underline{dim} T′)_{j} = (\underline{dim} cok(g))_{j} by the second sequence and (\underline{dim} T′)_{j} ≥ (\underline{dim} ker(f))_{j} by the first so that

(\underline{dim} T′)_{j} = (\underline{dim} cok(g))_{j} = sup((\underline{dim} cok(g))_{j}, (\underline{dim} (ker(g)))_{j})

and (4.1) holds at j. Similarly, if j is not a successor of i, we see that the equality (4.1) holds at j. Since Q has no oriented cycles, each vertex j ≠ i of Q is a non-successor or a non-predecessor of i. Thus, the proof of (4.1) is complete. □

4.3. As a corollary of the denominator theorem, we will prove an injectivity property of the map M ↦ X_{M}.

We recall first a few facts on quiver representations.

A representation of Q over a field F is a Q_{0}-graded F -vector space V = ⊕_{i∈Q_{0}}V_{i} together with an element x = (x_{h})_{h∈Q_{1}} in E_{V} := ∏_{h∈Q_{1}} Hom(V_{s(h)}, V_{t(h)}), where s(h) is the source and t(h) the target of the arrow h. The group G_{V} := ∏_{i∈Q_{0}} GL(V_{i}) acts on E_{V} by (g_{i}).(x_{h}) = (g_{t(h)}x_{h}g_{s(h)}^{−1}). A representation (M, x) over a field F can be functorialy considered as an F Q-module and the dimension vector of this module is \underline{dim} M = (dim M_{i}).

Clearly, the isoclasses of finite-dimensional F Q-modules are naturally identified with G_{V} -orbits of representations of Q.

Corollary 1. If M and M′ are non isomorphic indecomposable modules without selfextensions, then X_{M} ≠ X_{M′}.

Proof. It is well known that in the identification above, an isoclass of kQ-module with no self-extension corresponds to an orbit which is dense in its representation space E_{V}. Therefore, if M and M′ are non isomorphic modules without self-extensions, the corresponding orbits cannot be in the same representation space. Hence, M and M′ cannot have the same dimension vector.

By the theorem above, we conclude that X_{M} ≠ X_{M′}.

□

5. Application to a class of cluster algebras

5.1. We recall some terminology on cluster algebras. The reader can find more precise and complete information in [12].

Let n be a positive integer. We fix the ambient field F = ℚ(x_{1}, …, x_{n}), where the x_{i}’s are indeterminates. Let x be a free generating set of F over ℚ and let B = (b_{ij}) be an n × n antisymmetric matrix with coefficients in ℤ. Such a pair (x, B) is called a seed.

Let (u, B) be a seed and let u_{j}, 1 ≤ j ≤ n, be in u. We define a new seed as follows. Let u′_{j} be the element of F defined by the exchange relation:

(5.1)  u_{j}u′_{j} = ∏_{b_{ij}>0} u_{i}^{b_{ij}} + ∏_{b_{ij}<0} u_{i}^{−b_{ij}}.

Set u′ = u ∪ {u′_{j}}\{u_{j}}. Let B′ be the n × n matrix given by

b′_{ik} = \begin{cases}−b_{ik} & if i = j or k = j \\ b_{ik} + \frac{1}{2}( |b_{ij}| b_{jk} + b_{ij} |b_{jk}|) & otherwise. \end{cases}

By a result of Fomin and Zelevinsky, (u′, B′) = μ_{j}(u, B) is a seed. It is called the mutation of the seed (u, B) in the direction u_{j} (or j). We consider all the seeds obtained by iterated mutations. The free generating sets occurring in the seeds are called clusters, and the variables they contain are called cluster variables. By definition, the cluster algebra A(x, B) associated to the seed (x, B) is the ℤ-subalgebra of F generated by the set of cluster variables. The graph whose vertices are the seeds and whose edges are the mutations between two seeds is called the mutation graph of the cluster algebra.

The Laurent phenomenon, see [11], asserts that the cluster variables are Laurent polynomials with integer coefficients in the x_{i}, 1 ≤ i ≤ n. So, we have A(x, B) ⊂ ℤ[x_{1}^{±1}, …, x_{n}^{±1}].

Note that an antisymmetric matrix B defines a quiver Q = Q_{B} with vertices corresponding to its rows (or columns) and which has b_{ij} arrows from the vertex i to the vertex j whenever b_{ij} ≥ 0. The quivers Q thus obtained are precisely the finite quivers without oriented cycles of length 1 or 2. For such quivers Q, we denote by B_{Q} the corresponding antisymmetric matrix. The cluster algebra associated to the seed (x, B) will be also denoted by A(Q). In the sequel, we will be concerned with cluster algebras associated to a quiver Q without oriented cycles.

5.2. We fix a quiver Q without oriented cycles and we set H = kQ. We consider the cluster category C = C_{H} associated to the quiver Q, cf. [6]. An object T of C is called exceptional if it has no self-extensions, i.e. if Ext^{1}(T, T) = 0. An exceptional object is called cluster-tilting or simply tilting (although this is an abuse of language) if it has n non isomorphic indecomposable direct summands, where n is the number of vertices of Q. In the sequel, we will often identify a tilting object with the datum of its indecomposable summands. An exceptional object is called almost tilting if it has n − 1 non isomorphic indecomposable direct summands. It was shown in [6] that any almost tilting object \overline{T} can be completed to precisely two non isomorphic tilting objects T and T^{∗}.

For any tilting object T of C, let Q_{T} be the quiver associated to the algebra End_{C}(T). To be explicit, fix an ordering of the indecomposable summands T_{1}, …, T_{n} of T and let A be the endomorphism algebra of the sum of the T_{i}. Let e_{i} ∈ A the idempotent corresponding to T_{i}. Then the vertices of Q_{T} are 1, …, n, and the number of arrows from i to j is equal to dim e_{j}((rad A)/(rad A)^{2})e_{i}. A pair (T, Q_{T}) is called a cluster seed.

For 1 ≤ i ≤ n, we define, following [4], the mutation of the cluster seed (T, Q_{T}) in direction i by

δ_{i}(T, Q_{T}) := (T^{∗}, Q_{T^{∗}}),

where T and T^{∗} are the two completions of the almost tilting object

\overline{T} = T_{1} ⊕ … ⊕ T_{i−1} ⊕ T_{i+1} ⊕ … ⊕ T_{n}.

Note that there exists an indecomposable object T_{i}^{∗}, unique up to isomorphism, such that

T^{∗} = T_{1} ⊕ … ⊕ T_{i−1} ⊕ T_{i}^{∗} ⊕ T_{i+1} ⊕ … ⊕ T_{n},

which provides a natural ordering of the indecomposable summands of T^{∗}.

The following theorem is the main result of this article. The first assertion is a refinement of Conjecture 9.1 of [6] and the second assertion strengthens the main result of [4].

Theorem 4. Let Q be a quiver with n vertices and no oriented cycles, and let H = kQ be the hereditary algebra associated to Q. Then

(i) The correspondence M ↦ X_{M} provides a bijection between the set of indecomposable objects without self-extensions of C_{H} and the set of cluster variables of A(Q).

(ii) The correspondence {T_{1}, …, T_{n}} ↦ {X_{T_{1}}, …, X_{T_{n}}} provides a bijection compatible with mutations between the set of tilting objects of C_{H} and the set of clusters of A(Q).

Proof. By construction, any cluster variable belongs to a cluster. As the map M ↦ X_{M} is injective on the set of indecomposable objects without self-extensions by corollary 1, it is enough to prove (ii).

Let us prove (ii). Suppose that T = T_{1} ⊕ … ⊕ T_{n} is a tilting object of C and let T^{∗} be its mutation in direction i. Then Ext^{1}(T_{i}, T_{i}^{∗}) is one-dimensional by [6]. Hence, by theorem 2, we have

(5.2)  X_{T_{i}}X_{T_{i}^{∗}} = ∏_{j} X_{T_{j}}^{a_{ij}} + ∏_{j} X_{T_{j}}^{c_{ij}},

where a_{ij} and c_{ij} are integers defined by the following non split triangles (unique up to isomorphism)

T_{i} → ⊕a_{ij}T_{j} → T_{i}^{∗} → ST_{i}
T_{i}^{∗} → ⊕c_{ij}T_{j} → T_{i} → ST_{i}^{∗}.

By a theorem 6.2 b) of [4], the quiver Q_{T} is determined by these triangles: for any i and j, there are a_{ij} arrows from i to j and c_{ij} arrows from j to i. Moreover, if there exists an arrow from i to j , then there is no arrow from j to i, by proposition 3.2 of [4].

We now define, as in [4], a correspondence β between tilting seeds and cluster seeds. First note that the shift of H, is a tilting object and that (SH, Q) is a tilting seed. For a given word i_{1} … i_{t}, we can define

(5.3)  β(SH, Q) = (x, B_{Q}),
(5.4)  β(δ_{i_{t}} … δ_{i_{1}}(SH, Q)) = μ_{i_{t}} … μ_{i_{1}}(x, B_{Q}).

Set (T, Q_{T}) := δ_{i_{t}} … δ_{i_{1}}(SH, Q)). By [4], the quiver obtained from Q by the sequence of tilting mutations in the directions i_{1}, …, i_{t} is equal to the quiver obtained from Q by the sequence of cluster mutations in the directions i_{1}, …, i_{t}. Hence, by comparing the cluster exchange relation 5.1 and the tilting exchange relation 5.2, we obtain by induction that

β(δ_{i_{t}} … δ_{i_{1}}(SH, Q)) = ({X_{T_{1}}, …, X_{T_{n}}}, B_{Q_{T}}).

In particular, β(δ_{i_{t}} … δ_{i_{1}}(SH, Q)) does not depend on the choice of the word i_{1} … i_{t}.

By proposition 3.5 of [6], the mutation graph on the set of tilting seeds is connected. Hence, equalities 5.3 and 5.4 define a map β from the complete set of tilting seeds to the set of cluster seeds. The surjectivity of β follows from the fact that its image is stable under mutation. The injectivity of β follows from corollary 1. □

5.3. This section is devoted to the proof of some of the conjectures formulated by S. Fomin and A. Zelevinsky in [13]. The first corollary is a straightforward consequence of theorem 4. It corresponds to [13, Conjecture 4.14 (2)] in the acyclic case.

Corollary 2. Let Q be an finite quiver without oriented cycles. Then a cluster seed (u, B) of A(Q) only depends on u.

This corollary is [13, Conjecture 4.14 (3)] in the acyclic case.

Corollary 3. For any cluster variable x, the set of seeds whose clusters contain x form a connected subgraph of the exchange graph.

Proof. Indeed, the cluster variable x corresponds to an exceptional indecomposable object T_{1} of C_{Q}. Without restriction of generality, we assume that T_{1} is non projective. The seeds containing x are in bijection with the completions of T_{1}, i.e. the sets {T_{2}, …, T_{n}} of indecomposables such that the sum of the T_{i} is cluster tilting. Two seeds are joined by an edge of the exchange graph iff the corresponding sets of exceptional indecomposables are obtained from one each other by a mutation. By [6], this occurs iff they differ by precisely two indecomposables T_{i} and T_{i}^{∗} and these satisfy

dim Ext^{1}(T_{i}, T_{i}^{∗}) = 1.

This makes it clear that theorem 5 below yields a bijection compatible with mutations

{T_{2}, …, T_{n}} ↦ {P T_{2}, …, P T_{n}}

between the completions of T_{1} and the basic tilting sets of C_{Q′}, where Q′ is the quiver of the endomorphism ring of a projective generator of the category H′ ⊂ mod kQ of modules L with

Hom(M, L) = 0 = Ext^{1}(M, L).

Thus, by theorem 4 (ii), the subgraph of the exchange graph of Q formed by the seeds containing x is isomorphic to the exchange graph of Q′, which is connected by definition.

□

A consequence of theorem 4 is also the proof of [13, Conjecture 4.14 (4)] in the general case.

Corollary 4. The set of seeds whose matrix is acyclic form a connected subgraph (possibly empty) of the exchange graph.

Proof. A seed with an acyclic matrix corresponds to a cluster tilting object T whose endomorphism algebra A = End_{C_{Q}}(T) has a quiver without oriented cycles. Thus, the algebra A is both, cluster-tilted and of finite global dimension. By corollary 2.1 of [19], it is hereditary. So the category C_{A} is well-defined and the equivalence between the derived categories of A and Q induces a triangle equivalence C_{A} →^{∼} C_{Q} which takes A to T. Such an equivalence induces an isomorphism

Γ_{A} → Γ_{B}

of the Auslander-Reiten quivers of the two cluster categories. We refer to [6] for the description of the Auslander-Reiten quivers. Since A is hereditary, the quiver of its indecomposable projectives forms a slice of the component Γ_{A}^{pr} of Γ_{A} containing the projectives (recall that a slice is a full connected subquiver whose vertices are a system of representatives of the τ-orbits in the component). The isomorphism must take Γ_{A}^{pr} to Γ_{B}^{pr} since this is the only components isomorphic to the repetition ℤR of a finite quiver R. It is clear that any slice of Γ_{B}^{pr} can be transformed to the slice of the projectives by finitely many reflections at sources or sinks. □

5.4. Cluster tilting objects containing a given summand. Here, we refine a technique pioneered in section 2 of [4]: Let H be a finite-dimensional hereditary algebra and H the category of finite-dimensional right H-modules. Let M ∈ H be a non projective indecomposable with Ext^{1}(M, M) = 0. Then End(M) is a (possibly non commutative) field. Let H′ be the full subcategory on the modules L such that

Hom(M, L) = 0 and Ext^{1}(M, L) = 0.

We know from [16] and [15] that H′ is a hereditary abelian category with enough projectives and that a projective generator G of H′ is obtained by choosing an exact sequence

0 → H → G → M^{r} → 0

which induces an isomorphism

Hom(M, M^{r}) →^{∼} Ext^{1}(M, H).

Let C_{H} and C_{H′} be the cluster categories associated with H and H′. The following theorem is an elaboration on theorem 2.13 of [4].

Theorem 5. Let C(H, M) be the full additive subcategory of C_{H} whose objects are the sums of indecomposables L of C_{H} such that Ext^{1}(M, L) = 0. There is a canonical equivalence of k-linear categories

P : \xymatrix{C(H, M)/(M) \ar[r]^{∼} & C_{H′}} ,

where (M) denotes the ideal of morphisms factoring through a sum of copies of M. Moreover, we have

Ext^{1}(L_{1}, L_{2}) ≅ Ext^{1}(P L_{1}, P L_{2})

for all L_{1}, L_{2} ∈ C(H, M).

Note that C(H, M) is not a triangulated subcategory and not even stable under the shift functor. The theorem merely claims that as a k-linear category, C_{H′} is a ‘subquotient’ of C_{H}. To construct the equivalence P, we choose a ‘fundamental domain’ for the action of the autoequivalence F = τ^{−1}S on D.

Let P be the full subcategory of the projectives of H and H^{+} the full additive subcategory of D = D^{b}(H) each of whose indecomposables lies in H or SP. Let π : D → C_{H} be the projection functor. We know from [6] that π induces a bijection from the set of isoclasses of indecomposables of H^{+} to that of C_{H} and that we have

Ext^{1}(π(L_{1}), π(L_{2})) →^{∼} Ext^{1}(L_{1}, L_{2}) ⊕ D Ext^{1}(L_{2}, L_{1})

for any two indecomposables of H^{+}. Moreover, the category C_{H} is equivalent to the category whose objects are those of H^{+} and whose morphisms are given by

Hom(L_{1}, L_{2}) ⊕ Hom(L_{1}, F L_{2})

with the natural composition. Therefore, theorem 5 follows from

Theorem 6. There is a canonical bijection L ↦ L′ from the set of isoclasses of indecomposables L of H^{+} with

L ≅ M̸ , Ext^{1}(L, M) = 0 and Ext^{1}(M, L) = 0 (∗)

to the set of isoclasses of indecomposables of H^{′+}. Moreover, for any two objects L_{1}, L_{2} of H^{+} satisfying (∗), there is a canonical isomorphism

Ext^{1}(L_{1}, L_{2}) →^{∼} Ext^{1}(L′_{1}, L′_{2})

and there are canonical isomorphisms

Hom(L_{1}, L_{2})/(M) →^{∼} Hom(L′_{1}, L′_{2}) and Hom(L_{1}, F L_{2})/(M) →^{∼} Hom(L′_{1}, F′L′_{2})

compatible with compositions.

Before giving the proof, let us illustrate the statement on the following example: We consider the path algebra H = kQ of a linearly oriented quiver Q of type A_{6}. Below, we have drawn the Auslander-Reiten quiver of its derived category. The vertices corresponding to indecomposables concentrated in degree 0 lie between the two hatched lines. We use the symbols

□ for the 14 indecomposables L of H^{+} not isomorphic to M and which satisfy

Ext^{1}(L, M) = 0 = Ext^{1}(M, L) ,

• for the 5 indecomposable projectives of H′, . for indecomposable non projective objects of H′, ♢ for shifted copies SP of projectives of H′.

Notice the two rectangular zones starting from S^{−1}M respectively ending in SM where no □ occurs. If L ↦ L′ denotes the map of the theorem, we have a triangle

L_{U} → L → L′ → SL_{U}

where L_{U} is a sum of copies of M and L_{U} → L induces a bijection Hom(M, L_{U}) →^{∼} Hom(M, L). Thus we have L = L′ for all L with Hom(M, L) = 0. The corresponding triangles for the others are visible in the diagram below.

[figure: □ •; □ ♢; ♢; □ •; □.; □; ♢; S^{−1}M; □.; □; SM; M; □ •; □; ♢; □ •; □ •; □.; □; .; □ ♢]

Several of the arguments needed in the proof are contained in section 2 of [4]. For the convenience of the reader, we nevertheless include them below.

Proof. Let U ⊂ D be the full triangulated subcategory generated by M. Since Ext^{1}(M, M) vanishes and Hom(M, M) is a field, its objects are the sums of shifted copies of M. Let V be the full subcategory of D whose objects are the L ∈ D such that Hom(U, L) = 0 for all U ∈ U. Then U, V form a semiorthogonal decomposition [1] of D, i.e. for each object X of D, there is a triangle

X_{U} → X → X^{V} → SX_{U}

with X_{U} ∈ U and X^{V} ∈ V. This triangle is unique up to unique isomorphism; the functor X → X_{U} is right adjoint to the inclusion of U and the functor X ↦ X^{V} is left adjoint to the inclusion of V. We have H′ = H ∩ V and the inclusion H′ ⊂ V extends canonically to an equivalence D^{b}(H′) → V. In particular, each object of V is a direct sum of shifts of objects of H′. We have U ∩ H = M, the full subcategory on the direct sums of copies of M. The inclusion H′ ⊂ H commutes with kernels, cokernels and preserves Ext^{1}-groups. We will show that L ↦ L′ = L^{V} yields the bijection announced in the assertion.

Let L be indecomposable in H^{+} such that (∗) holds. Let us first show that Hom(S^{i}M, L) vanishes if i ≠ 0. Indeed, if L belongs to H, then this group clearly vanishes if i ≠ 0, −1 and if i = −1, it vanishes because Ext^{1}(M, L) = 0. If L = SP for a projective P ∈ H, then Hom(S^{i}M, L) = Hom(S^{i}M, SP) clearly vanishes for i ≠ 0, 1 and it vanishes for i = 1 because M is a non projective indecomposable.

Now let us show that L^{V} is indecomposable: Consider the canonical triangle

L_{U} → L → L^{V} → SL_{U}.

Since Hom(S^{i}M, L) vanishes for i ≠ 0, we have L_{U} ∈ M. Therefore, in the associated exact sequence

Hom(L, L) → Hom(L, L^{V}) → Hom(L, SL_{U})

the third term vanishes. Thus the composition

Hom(L, L) → Hom(L, L^{V}) →^{∼} Hom(L^{V}, L^{V})

is surjective and End(L^{V}) is local as a quotient of the local ring End(L).

Let us show that L′ belongs to H^{′+}. Since L_{U} belongs to M, the canonical morphism f : L_{U} → L is a morphism of H and therefore its cone L^{V} in D is isomorphic to cok(f) ⊕ S ker(f). Since L^{V} is indecomposable, one of the two summands vanishes. If ker(f) vanishes, then L^{V} belongs to H′ ⊂ H^{′+}. If cok(f) vanishes, we have to show that ker(f) is projective in H′. Now indeed the short exact sequence

0 → ker(f) → L_{U} → L → 0

induces a surjection

Ext_{H}^{1}(L_{U}, U) → Ext_{H}^{1}(ker(f), U) → 0

for each U ∈ H. The left hand term vanishes since L_{U} is a sum of copies of M and the right hand term is isomorphic to Ext_{H′}^{1}(ker(f), U) because the inclusion H′ ⊂ H preserves extension groups. Thus, ker(f) is projective in H′.

From what we have shown, we conclude that the map L → L′ is well-defined. Let us show that it is injective. For this, we show that the morphism L^{V} → SL_{U} occurring in the canonical triangle is a minimal left SM-approximation. Then L is determined up to isomorphism as the shifted cone over this morphism. To show that L^{V} → SL_{U} is a minimal left approximation, consider the canonical triangle

L_{U} → L → L^{V} → SL_{U}

and the induced sequence

Hom(L, SM) ← Hom(L^{V}, SM) ← Hom(SL_{U}, SM).

Since Hom(L, SM) vanishes by assumption, we do get a surjection Hom(L^{V}, SM) ← Hom(SL_{U}, SM). If it is not minimal, then there is a retraction r : SL_{U} → SM whose composition with L^{V} → SL_{U} vanishes. Then r extends to a retraction r̃: SL → SM. This is impossible since L is indecomposable and not isomorphic to M.

Let us show now that L ↦ L′ is surjective. Let N be indecomposable in H^{′+}. Let N → SM′ be a minimal SM-approximation and form the triangle

M′ → L → N → SM′.

Let us show that L is indecomposable. Since M′ ∈ U, we have L^{V} →^{∼} N^{V} and since N ∈ V, we have L_{U} →^{∼} M′. If L is decomposable, say L = L_{1} ⊕ L_{2}, then we get

L_{1}^{V} ⊕ L_{2}^{V} →^{∼} N

and, say, L_{1}^{V} vanishes. Then L_{1} belongs to U and thus M′ →^{∼} L_{1} ⊕ (L_{2})_{U}. Since N → SM′ is a minimal SM-approximation, we have L_{1} = 0. So L is indecomposable.

Let us show that L belongs to H^{+}. It is clear from the above triangle that L has homology at most in degrees 0 and 1. Since L is indecomposable, its homology is concentrated in one degree. If the homology is concentrated in degree 0, then L belongs to H ⊂ H^{+}. Suppose that L has its homology concentrated in degree 1. Then we must have N = SQ for some indecomposable projective Q of H′. We know that if P_{H} is a projective generator for H, then P_{H}^{V} is is a projective generator for H′. Thus, there is a projective P of H and a section s : Q → P^{U} which identifies Q with a direct factor of P^{U}. Since N → SM′ is an SM-approximation, the composition

\xymatrix{N \ar[r]^{Ss} & SP^{U} \ar[r] & SP_{M}}

extends to SM′ so that we obtain a morphism of triangles

\xymatrix{M′ \ar[r] \ar[d] & L \ar[r] \ar[d] & N \ar[r] \ar[d]^{Ss} & SM′ \ar[d] \\ SP_{U} \ar[r] & SP \ar[r] & SP^{U} \ar[r] & SP_{M}.}

The morphism L → SP is non zero since its composition with SP → SP^{U} equals the composition of the non zero morphism L → N with the section Ss. So we obtain a non zero morphism S^{−1}L → P in H. Since S^{−1}L is indecomposable and P is projective, S^{−1}L is projective and we have L ∈ H^{+}.

Finally, let us show that L satisfies the condition (∗). If L was isomorphic to M, we would have N = L^{V} = 0 contrary to our hypothesis that N is indecomposable.

The triangle

M′ → L → N → SM′

yields an exact sequence

Hom(M′, SM) ← Hom(L, SM) ← Hom(N, SM) ← Hom(SM′, SM).

Here the leftmost term vanishes since Ext^{1}(M, M) = 0 and the rightmost map is surjective since N → SM′ is a left SM-approximation. Thus we have Ext^{1}(L, M) = 0. The triangle also yields the sequence

Hom(S^{−1}M, M′) → Hom(S^{−1}M, L) → Hom(S^{−1}M, N).

The left hand term vanishes since Ext^{1}(L_{1}, M) = 0 and the right hand term vanishes since N belongs to V. Thus we have Ext^{1}(M, L) = 0.

Now let L_{1}, L_{2} be indecomposables of H^{+} satisfying condition (∗). Consider the triangle

(L_{2})_{U} → L_{2} → L_{2}^{V} → S(L_{2})_{U}.

It induces an exact sequence

Hom(S^{−1}L_{1}, (L_{2})_{U}) → Hom(S^{−1}L_{1}, L_{2}) → Hom(S^{−1}L_{1}, L_{2}^{V}) → Hom(S^{−1}L_{1}, S(L_{2})_{U}).

The leftmost term vanishes since Ext^{1}(L_{1}, M) = 0 and the rightmost term vanishes since Ext^{2}(L_{1}, M) = 0. Thus we have

Hom(L_{1}, SL_{2}) →^{∼} Hom(L_{1}, SL_{2}^{V}) →^{∼} Hom(L_{1}^{V}, SL_{2}^{V}) ,

which proves the assertion on the extension groups. The above triangle also induces an exact sequence

Hom(L_{1}, (L_{2})_{U}) → Hom(L_{1}, L_{2}) → Hom(L_{1}, L_{2}^{V}) → Hom(L_{1}, S(L_{2})_{U}).

The last term vanishes since Ext^{1}(M, M) = 0. Thus the kernel of the map

Hom(L_{1}, L_{2}) → Hom(L_{1}, L_{2}^{V}) →^{∼} Hom(L_{1}^{V}, L_{2}^{V})

is formed by the morphisms factoring through sums of M. Put F = τ^{−1}S. Consider the triangle

(F L_{2})_{U} → F L_{2} → (F L_{2})^{V} → S(F L_{2})_{U}.

Note that the functor F does not take V to itself. We have

Hom(S^{i}M, F L_{2}) →^{∼} D Hom(L_{2}, S^{i}τ^{2}M).

This can be non zero only if i equals 0 or 1. Thus (F L_{2})_{U} is a sum of copies of M and SM. Therefore, in the exact sequence

Hom(L_{1}, (F L_{2})_{U}) → Hom(L_{1}, F L_{2}) → Hom(L_{1}, (F L_{2})^{V}) → Hom(L_{1}, S(F L_{2})_{U})

the last term vanishes and

Hom(L_{1}, (F L_{2})^{V}) →^{∼} Hom(L_{1}^{V}, (F L_{2})^{V})

identifies with the quotient of Hom(L_{1}, F L_{2}) by the subspace of morphisms factoring through a sum of copies of M and SM. Since Hom(L_{1}, SM) vanishes, this is also the subspace of morphisms factoring through a sum of copies of M. To finish the proof, it remains to be noticed that under the canonical equivalence D^{b}(H′) →^{∼} V, if L_{2}^{V} corresponds to L′_{2}, then the object (F L_{2})^{V} does correspond to τ_{H′}^{−1}SL′_{2}, by lemma 2.14 of [4] or section 8.1 of [18].

□

6. Example on the Kronecker quiver

6.1. We consider the Kronecker quiver Q obtained by choosing the following orientation of the diagram Ã_{1}:

(6.1)  \xymatrix{1 \ar[r]^{α} \ar[r]_{β} & 2} .

As an illustration of theorems 2 and 4, we give an interpretation of some results of [20] in terms of the cluster category of the Kronecker quiver. We consider covariant representations. So if S_{1}, S_{2} are the simple representations, then

dim Hom(S_{i}, S_{i}) = 1, dim Ext^{1}(S_{1}, S_{2}) = 2, dim Ext^{1}(S_{2}, S_{1}) = 0.

Over a field k, the (finite-dimensional) indecomposable representations of the Kronecker quiver are classified as follows:

The postprojective indecomposable modules U^{n}, n ≥ 0, the preinjective indecomposable modules V^{n}, n ≥ 0, and the family of indecomposables modules W^{n} = W^{n}(x), n > 0, of the (regular) tube parametrized by x ∈ ℙ^{1}(k). They are given by

U^{n} : \xymatrix{k^{n} \ar[r]^{α} \ar[r]_{β} & k^{n+1}} with α = [ \begin{matrix}I_{n} \\ 0 \end{matrix} ] , β = [ \begin{matrix}0 \\ I_{n} \end{matrix} ]

V^{n} : \xymatrix{k^{n+1} \ar[r]^{α} \ar[r]_{β} & k^{n}} with α = [ I_{n} 0 ] , β = [ 0 I_{n} ]

W^{n}((1 : λ)) : \xymatrix{k^{n} \ar[r]^{α} \ar[r]_{β} & k^{n}} with α = I_{n}, β = ( [figure: λ; 1; 0; 0; 0; 0; 1; 0; 0; λ] ) ,

for λ in k, and

W^{n}((0 : 1)) : \xymatrix{k^{n} \ar[r]^{α} \ar[r]_{β} & k^{n}} with α = ( [figure: 0; 1; 0; 0; 0; 0; 1; 0; 0; 0] ) , β = I_{n}.

Let us calculate the cluster variables of the cluster algebra A(Q) ⊂ ℤ[x_{1}^{±1}, x_{2}^{±1}] corresponding to the Kronecker quiver. By theorem 4, they are given by X_{U^{n}}, X_{V^{n}}, n ≥ 0. By duality, X_{V^{n}} is obtained from X_{U^{n}} by exchanging x_{1} and x_{2}. So, we just have to calculate X_{U^{n}}.

Let (y_{n})_{n∈ℕ} be the sequence given by y_{0} = x_{2}, y_{1} = x_{1}, and y_{n+2} = X_{U^{n}}, n ≥ 0. Set P_{1} = U^{1}, P_{2} = U^{0}, for the indecomposable projective modules, then SP_{2} ⊕ SP_{1}, is the ”seed” tilting object and SP_{1} ⊕ U^{0} is also a tilting object of the cluster category since the first component of \underline{dim} (U^{0}) is zero. Moreover, for any n ≥ 0, U^{n} ⊕ U^{n+1} is easily seen to be a tilting kQ-module by applying recursively the inverse AR-functor to the tilting module P_{2} ⊕ P_{1} and the object SP_{1} ⊕ U^{0} . By applying theorem 4, we obtain that the y_{n}’s are cluster variables and that

μ_{2}({y_{2n}, y_{2n+1}}) = {y_{2n+2}, y_{2n+1}}, μ_{1}({y_{2n−1}, y_{2n}}) = {y_{2n+1}, y_{2n}}.

In particular, the exchange relations imply that the sequence (y_{n})_{n∈ℕ} is given by

(6.2)  y_{0} = x_{2}, y_{1} = x_{1}, y_{n−1}y_{n+1} = y_{n}^{2} + 1.

Note that in the module category Ext^{1}(W^{1}, P_{i}) = k, for i = 1, 2, which implies

Ext_{C_{Q}}^{1}(W^{1}, P_{i}) = k ,

because the P′_{i}s are projective. Applying the (AR)-autoequivalence τ in the cluster category, we obtain Ext_{C_{Q}}^{1}(W^{1}, U^{n}) = k, n ≥ 0. In the module category, we have (up to isomorphism) a unique non split exact sequence

0 → U^{n} → U^{n+1} → W^{1} → 0.

This yields a triangle in the cluster category

U^{n} → U^{n+1} → W^{1} → SU^{n}.

But, as SU^{n} = τU^{n} = U^{n−2} in the cluster category, shifting the triangle gives

W^{1} → U^{n−1} → U^{n} → SW^{1}.

Now, let w_{1} := X_{W^{1}} = \frac{1+x_{1}^{2}+x_{2}^{2}}{x_{1}x_{2}} , then theorem 2 implies

w_{1}y_{n} = y_{n+1} + y_{n−1}.

Note that this formula simplifies the initial induction 6.2. It was obtained in a direct way in [20]. We can calculate the generating series of (y_{n})_{n∈ℕ}

∑_{n≥0} y_{n}t^{n} = \frac{1 − y_{−1}t}{1 − w_{1}t + t^{2}} ,

where y_{−1} := X_{V^{0}} = \frac{1+x_{2}^{2}}{x_{1}} .

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Institut Camille Jordan, Université Claude Bernard Lyon I, 69622 Villeurbanne Cedex, France

E-mail address: caldero@math.univ-lyon1.fr

UFR de Mathématiques, Institut de Mathématiques, Case 7012, Université Paris 7 - Denis Diderot, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail address: keller@math.jussieu.fr