$1+y_1+y_1y_2+y_1^2y_2 = (1+y_1y_2)(1+y_1)$.

3.  An $F$-polynomial may have negative coefficients. An example for a quiver with two vertices and four arrows is given in [@DWZ09 Example 3.6].

Cluster categories {#sect::clustercat}
==================

Derived categories
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Derived categories were introduced by J.-L. Verdier in [@Verdier77] [@Verdier67]. Their general theory is discussed in numerous books and papers; let us cite [@Hartshorne66] [@Happel] [@Kashiwara-Schapira94] [@Weibel94] and [@Keller07].

In this section, we only give a brief outline of the theory of derived categories, focusing on aspects that suit the purpose of these notes. Here, $k$ is an arbitrary field.

### Generalities

Let ${\mathcal{A}}$ be an abelian category (for example, the category of modules over a finite-dimensional $k$-algebra). In particular, every morphism in ${\mathcal{A}}$ has a kernel and a cokernel.

A *complex* of objects of ${\mathcal{A}}$ is a sequence of morphisms $$C = \ldots \stackrel{d_{i-2}}{\to} C_{i-1} \stackrel{d_{i-1}}{\to} C_i \stackrel{d_{i}}{\to} C_{i+1} \stackrel{d_{i+1}}{\to} \ldots$$ such that $d_{i+1}\circ d_i = 0$ for all integers $i$.

Let $C$ and $C'$ be two complexes. A *morphism of complexes* $f:C\to C'$ is an infinite tuple $f=(f_i)_{i \in {\mathbb{Z}}}$ such that for all integers $i$, $f_i:C_i\to C'_i$ is a morphism, and the square $$\xymatrix{ C_i\ar[r]^{d_i}\ar[d]^{f_i} & C_{i+1}\ar[d]^{f_{i+1}} \\
            C'_i\ar[r]^{d'_i} & C'_{i+1}
 }$$ commutes, 