Euler characteristic of a coherent sheaf ($\chi (X,\mathcal{F})$)

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.

For a Cartier divisor $D$ on an $n$-dimensional projective scheme, $\chi (X,{\mathcal{O}}_X(mD))$ is a polynomial in $m$ whose leading term is $D^n{m}^n/n!$. The intersection product uses the fundamental cycle, with the generic multiplicity of each component. Thus the statement also applies to nonreduced schemes.