Definition 33.33.1. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. In this situation the Euler characteristic of $\mathcal{F}$ is the integer
\[ \chi (X, \mathcal{F}) = \sum \nolimits _ i (-1)^ i \dim _ k H^ i(X, \mathcal{F}). \]
For justification of the formula see below.
Comments (0)