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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.

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Jesko -- if you are willing to accept complexes of sheaves rather than just sheaves, then in characteristic 0 one may take the sum over all smooth strata of the Euler characteristics of the complexes of differential forms. –  algori Jul 14 '11 at 8:42
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Well, in a smooth flat 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
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Over $\mathbb{C}$ one can use hodge theory to write $H^k(X,\mathbb{C})=\oplus_{p+q=k}H^q(X,\Omega_X^p)$, where $\Omega_X^p$ is the sheaf of $p$-forms on $X$. –  Daniel Loughran Jul 14 '11 at 12:25
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Also the top Chern class of the tangent sheaf gives the topological Euler characteristic (of a smooth, projective, complex variety). –  Jason Starr Jul 14 '11 at 15:14
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@Jesko: smooth strata means this: write $X = X_0 \cup X_1 \cup \cdots \cup X_n$ where $X_0$ is the non-singular locus of $X$, $X_1$ is the non-singular locus of $X \setminus X_0$, etc. –  Steven Sam Jul 14 '11 at 18:03
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