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The Stacks project

Lemma 75.26.3. Let S be a scheme. Let X be an algebraic space over S. Let E \in D(\mathcal{O}_ X) be perfect. The function

\chi _ E : |X| \longrightarrow \mathbf{Z},\quad x \longmapsto \sum (-1)^ i \beta _ i(x)

is locally constant on X.

Proof. Omitted. Hints: Follows from the case of schemes by étale localization. See Derived Categories of Schemes, Lemma 36.31.2. \square

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