point})=1$;

2.  $\chi({\mathbb{A}}^{\! n})=1$, where ${\mathbb{A}}^{\! n}$ is the affine space of dimension $n$;

3.  $\chi({\mathbb{P}}^n)=n+1$, where ${\mathbb{P}}^n$ is the projective space of dimension $n$;

4.  $\chi({\mathcal{U}}\times {\mathcal{V}}) = \chi({\mathcal{U}}) \cdot \chi({\mathcal{V}})$;

5.  if ${\mathcal{U}}$ is a disjoint union of two constructible subsets $C_1$ and $C_2$, then $\chi({\mathcal{U}}) = \chi(C_1)+\chi(C_2)$.

6.  if $f:{\mathcal{U}}\to {\mathcal{V}}$ is a surjective morphism of varieties (or even a surjective constructible map) such that all fibers $f^{-1}(x)$ have the same Euler characteristic, say $c$, then $\chi({\mathcal{U}}) = c\chi({\mathcal{V}})$.

See [@GLS072 Proposition 7.4.1], which itself refers to [@MacPherson74] and [@Dimca04]. On constructible maps, we refer the reader to [@Joyce2006].

The first property of $F$-polynomials deals with direct sums, or equivalently, with split exact sequences.

\[prop::direct-sum\] Let $V$ and $W$ be two modules over $A=kQ/I$. Then $F_V \cdot F_W = F_{V\oplus W}$.

We outline the proof of this proposition, as it gives the flavour of the methods used to prove the various formulas that appear in these notes. We follow [@CC06 Proposition 3.6].

[*Proof of Proposition \[prop::direct-sum\]*]{}. Consider the split exact sequence $$0\to V\stackrel{\iota}{\to} V\oplus W \stackrel{\pi}{\to} W \to 0.$$ To any submodule $B$ of $V\oplus W$ we associate the submodules $\iota^{-1}(B)$ and $\pi(B)$ of $V$ and $W$, 