The Stacks project

Lemma 75.26.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent (for example perfect). For any $i \in \mathbf{Z}$ consider the function

\[ \beta _ i : |X| \longrightarrow \{ 0, 1, 2, \ldots \} \]

defined above. Then we have

  1. formation of $\beta _ i$ commutes with arbitrary base change,

  2. the functions $\beta _ i$ are upper semi-continuous, and

  3. the level sets of $\beta _ i$ are étale locally constructible.

Proof. Choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. Then $L\varphi ^*E$ is a pseudo-coherent complex on the scheme $U$ (use Lemma 75.13.2) and we can apply the result for schemes, see Derived Categories of Schemes, Lemma 36.31.1. The meaning of part (3) is that the inverse image of the level sets to $U$ are locally constructible, see Properties of Spaces, Definition 66.8.2. $\square$

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