Lemma 75.25.4 (0CTM)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 75: Derived Categories of Spaces
Section 75.25: Cohomology and base change, VI
Lemma 75.25.4 (cite)

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Lemma 75.25.4. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of finite presentation of algebraic spaces over $S$ .

Let $E \in D(\mathcal{O}_ X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Let $\mathcal{G}$ be an $\mathcal{O}_ X$ -module of finite presentation, flat over $S$ . Then $Rf_*\mathcal{G}$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.

Proof. Special cases of Lemma 75.25.1 applied with (1) $\mathcal{G}^\bullet$ equal to $\mathcal{O}_ X$ in degree $0$ and (2) $E = \mathcal{O}_ X$ and $\mathcal{G}^\bullet$ consisting of $\mathcal{G}$ sitting in degree $0$ . $\square$

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