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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