Let $X\to S$ a morphism of Noetherian schemes. Assume that $\mathcal F$ is a coherent sheaf on $X$ with the following property: the support of $\mathcal F$ is proper over a subscheme of $S$ of dimension $0$. Then I've read in Kollar's book "Rational curves on algebraic varieties (chap VI appendix 2)", that in this case it is possibile to define the Euler-Poincare characteristic $\chi_S(\mathcal F)$. This is quite weird because I've seen the Euler-Poincare characteristic only for algebraic varieties.
What is $\chi_S$? I suppose that is the alternating sum of the lengths of cohomologies where somehow we use the properties of Artinian modules. Can you please explain the construction in details?
You are right. If $g: X \to S$ is proper and the $R^ig_*\mathcal{F}$ have $0$-dimensional support, say at $s$, then one can take the length of the $R^ig_*\mathcal{F}$ and their alternating sum.