Let $Y$ be a "nice" scheme. I am thinking projective varieties over an algebraically closed field, for now, but I am open to more general results.

In terms of singular homology (coefficients in $\mathbb{Z}$), I can define the Euler characteristic $\chi(Y)$. My question is: Can I express $\chi(Y)$ in terms of the Euler characteristic of certain coherent sheaves on $Y$, in terms of sheaf cohomology? Most preferably, I would like $\chi(Y)=\chi(Y,\mathcal{F})$ for some particular sheaf $\mathcal{F}$.

I am sorry if this is really trivial or widely known, my searching and asking (in the real world) has led me nowhere so far.

smoothflat family the fibres are diffeomorphic by Ehresmann theorem, so the topological Euler number is constant. But if the family is not smooth, it can definitely vary (think of a smooth plane cubic degenerating to a nodal one). – Francesco Polizzi Jul 14 '11 at 9:22