Acyclic Cluster Algebras Revisited

Abstract

We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and sufficient combinatorial criterion for a collection of vectors to be the c-vectors of some cluster in the cluster algebra associated to a given skew-symmetrizable matrix. Our approach also yields a simple proof of the known result that the c-vectors of an acyclic cluster algebra are sign-coherent, from which Nakanishi and Zelevinsky have showed that it is possible to deduce in an elementary way several important facts about cluster algebras.

Appendix: Derived Categories of Hereditary Categories

This paper uses the language of derived categories, because it is the simplest and most natural language in which to present our results. However, we fear that this might frighten away some readers, who feel that nothing which mentions the word “derived” can be elementary. We therefore seek to explain why, in this case, the derived category is not an object to be feared.

Let A be a ring (not necessarily commutative) and let be the category of finitely generated A-modules. We will write and for Hom and Ext of A-modules, so that undecorated $\operatorname{Hom}$ and $\operatorname{Ext}$ can stand for the Hom and Ext in the derived category, as they do throughout this paper. A complex of A-modules is a doubly-infinite sequence ⋯←C ₋₁←C ₀←C ₁←C ₂←⋯ of A-modules and A-module maps, such that the composition C _i ←C _i+1←C _i+2 is 0 for all i. All our complexes will be bounded, meaning that all but finitely many C _i are zero; we usually will not mention this explicitly. For a complex C _•, we write H _i (C _•) for the homology group Ker(C _i−1←C _i )/Im(C _i ←C _i+1).

Objects of the derived category are bounded complexes, but many different bounded complexes can be isomorphic to each other in the derived category and, as usual in category theory, there will be little reason to distinguish isomorphic objects. For a general derived category, if complexes B _• and C _• are isomorphic, then we can deduce that H _i (B _•)≅H _i (C _•), but the converse does not hold.

However, now suppose that the ring A is what is called hereditary, meaning that vanishes for all j≥2 and all A-modules M and N. Then we have

Theorem A.2

([14, Sect. I.5.2])

If A is hereditary, then the complexes B _• and C _• are isomorphic in the derived category if and only if H _i (B _•)≅H _i (C _•) for all i.

Remark A.3

Happel has a standing assumption that k is algebraically closed in the section we cite. As Happel says, this assumption is “not really needed”, and the careful reader should have little difficulty removing it.

In particular, C _• is isomorphic to the complex which has H _i (C _•) in position i, and where all the maps are zero. If you like, whenever we speak of an object of the derived category, you can use this trick to simply think of a sequence of modules, taking all the maps between them to be zero. We will generally only be interested in indecomposable objects in the derived category. If we view an indecomposable object as a sequence of modules in this way, exactly one of the modules in the sequence will be non-zero.

We introduce the following notations: For an A-module M, the object M[i] is the complex which is M in position i, and 0 in every other position. More generally, for any complex C _•, the complex C[i]_• has C[i] _j =C _j−i , with correspondingly shifted maps. We define direct sums of complexes in the obvious way, so ⨁M _i [i] is the complex which is M _i in position i, with all the maps being 0.

In a category, one wishes to know the homorphisms between objects, and how to compose them. In the derived category, for M,N objects of , we have $\operatorname{Hom}(M[a], N[b]) = 0$ if a>b and if b≥a. We sometimes adopt the notation $\operatorname{Ext}^{j}(B_{\bullet}, C_{\bullet})$ as shorthand for $\operatorname{Hom}(B_{\bullet}, C[j]_{\bullet})$, for this reason. The composition $\operatorname{Hom}(M[a], N[b]) \times \operatorname{Hom}(N[b], P[c]) \to \operatorname{Hom}(M[a], P[c])$ is the Yoneda product .

We have now described morphisms between complexes that have only one nonzero term. More generally, let M _•=⨁M _i [i] and N _•=⨁N _i [i] be two complexes with all maps 0, then $\operatorname{Hom}(M,N) = \bigoplus_{i,j} \operatorname{Hom}(M_{i}[i], N_{j}[j])$. Given three such complexes M _•, N _• and P _•, the composition $\operatorname{Hom}(M_{\bullet}, N_{\bullet}) \times \operatorname{Hom}(N_{\bullet}, P_{\bullet}) \to \operatorname{Hom}(M_{\bullet}, P_{\bullet})$ is the sum of the compositions of the individual terms. So, if one only looks at complexes where all maps are zero, one can view the derived category as a convenient notational device for organizing the $\operatorname{Ext}$ groups and the maps between them. In particular, when A is hereditary, we really can understand all the objects and morphisms in the derived category in this way.

Finally, we must describe the “triangles”. This means that, for every map $M_{\bullet} \stackrel{\phi}{\to} N_{\bullet}$, we must construct a complex E _• with maps N _•→E _• and E _•→M _•[1]. We call this “completing $M_{\bullet} \stackrel{\phi}{\to} N_{\bullet}$ to a triangle”. The sense in which this construction is natural is somewhat subtle, so we will gloss over this. We only use the triangle construction in the case that M _• and N _• are of the forms M[a] and N[b] for some A-modules M and N, so we will only discuss it in that case. Furthermore, we will now restrict ourselves to the case that A is hereditary. So there is a nonzero homorphism M[a]→N[b] if and only if b−a is 0 or 1. For notational simplicity we will restrict to the case a=0.

The following theorem is the result of unwinding the definition of a triangle, the relation between $\operatorname{Hom}(M, N[1])$ and extensions between N and M, and using Theorem A.2 to identify a complex with its cohomology.

Theorem A.4

Let A be hereditary and let M and N be A-modules.

Let ψ an A-module map M→N and ϕ the corresponding map M→N in the derived category. If ψ is injective then the completion of $M \stackrel{\phi}{\to } N$ to a triangle is isomorphic to C where C:=Coker(ψ). The map N→C is the tautological projection and the map C→M[1] comes from the class of 0→M→N→C→0 in $\operatorname{Ext}^{1}(C,M)$.

If ψ is surjective then the completion of $M \stackrel{\phi }{\to} N$ to a triangle is isomorphic to K[1], where K:=Ker(ψ). The map K[1]→M[1] is (−1) times the tautological inclusion and the map N→K[1] comes from the class of 0→K→M→N→0 in $\operatorname{Ext}^{1}(N,K)$.

Let ψ be a class in $\operatorname{Ext}^{1}(M,N)$ and let ϕ be the corresponding map M→N[1]. Then the completion of $M \stackrel {\phi}{\to} N[1]$ to a triangle is isomorphic to E[1], where E is the extension 0→N→E→M→0 corresponding to ϕ. The maps N[1] to E[1] and E[1]→M[1] are (−1) times the maps from the extension short exact sequence.

Remark A.5

We use the construction of completing to a triangle to define mutation of exceptional sequences. One of the surprising consequences of the theory of exceptional sequences is that all the maps we will deal with are either injective or surjective, so we do not need to know how to complete ψ:M→N to a triangle if ψ is neither injective nor surjective. For the interested reader, we explain nonetheless. Let K, I and C be the kernel, image and cokernel of ψ. The completion of $M \stackrel{\psi}{\to} N$ to a triangle is noncanonically isomorphic to C⊕K[1]. The maps K[1]→M[1] and N→C are the tautological maps, the former multiplied by −1. The maps C→M[1] and N→K[1] come from classes in $\operatorname{Ext}^{1}(C,M)$ and $\operatorname{Ext}^{1}(N,K)$. The precise classes depend on the noncanonical choice of isomorphism, but one can say that their images in $\operatorname{Ext}^{1}(C,I)$ and $\operatorname{Ext}^{1}(I,K)$ correspond to the extensions 0→I→N→C→0 and 0→K→M→I→0, respectively.