purposes, deals with almost-split exact sequences. For the theory of almost-split sequences and the definition of the Auslander-Reiten translation $\tau$, we refer the reader to the notes of the courses [@Malicki] and [@Platzeck] in this volume.

\[theo::F-polynomials\] Let $0\to \tau V\to E \to V\to 0$ be an almost-split sequence of modules over $A=kQ/I$. Then $F_{\tau V}\cdot F_{V} = F_E + {\mathbf{y}}^{{\underline{\dim}\,}V}$.

The spirit of the proof of this theorem is similar to that of Proposition \[prop::direct-sum\]. The difference lies in the fact that the morphism $\Phi_{{\mathbf{e}}}$ is no longer surjective for all ${\mathbf{e}}$; the term ${\mathbf{y}}^{{\underline{\dim}\,}V}$ in the right hand side of the statement compensates, in some sense, this lack of surjectivity.

Examples of $F$-polynomials
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Let $Q$ be the quiver with one vertex and no arrows. Its path algebra is simply ${\mathbb{C}}$, and representations of $Q$ are just vector spaces.

Let $V$ be a $d$-dimensional vector space. Then $$F_V(y) = \sum_{i=0}^d \binom{d}{i} y^i.$$ This can be seen by observing that, for $d=1$, the $F$-polynomial is $1+y$, and then by applying Proposition \[prop::direct-sum\]. As a corollary, we get a nice proof of the known fact that the Euler-Poincaré characteristic of the (usual) Grassmannian ${\mbox{{\rm Gr}}_{i}}({\mathbb{C}}^d)$ is equal to $\binom{d}{i}$.

