The Stacks project

Lemma 36.31.2. Let $X$ be a scheme. Let $E \in D(\mathcal{O}_ X)$ be perfect. The function

\[ \chi _ E : X \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^ i \dim _{\kappa (x)} H^ i(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \kappa (x)) \]

is locally constant on $X$.

Proof. By Cohomology, Lemma 20.47.3 we see that we can, locally on $X$, represent $E$ by a finite complex $\mathcal{E}^\bullet $ of finite free $\mathcal{O}_ X$-modules. On such an open the function $\chi _ E$ is constant with value $\sum (-1)^ i \text{rank}(\mathcal{E}^ i)$. $\square$

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