Euler characteristic of a coherent sheaf
= Euler characteristic of a coherent sheaf
{title2=$\chi(X,\mathcal F)$}
On a proper scheme over a field, the <Euler characteristic of a coherent sheaf> is $\chi(X,\mathcal F)=\sum_i(-1)^i\dim H^i(X,\mathcal F)$. The sum is finite by <Grothendieck vanishing>, and <long exact sequences in sheaf cohomology> make it additive in <short exact sequences of sheaves>.