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.

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