Lemma 36.30.4 (0B91)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.30: Cohomology and base change, VI
Lemma 36.30.4 (cite)

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Lemma 36.30.4. Let $S$ be a scheme. Let $f : X \to S$ be a proper morphism of finite presentation.

Let $E \in D(\mathcal{O}_ X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$ 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 $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. Special cases of Lemma 36.30.1 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 $0$ . $\square$

Comments (2)

Comment #4352 by Remy on August 20, 2019 at 11:07

In (2), $f$ does not need to be flat.

Comment #4499 by Johan on September 02, 2019 at 12:48

Yes, indeed. Thanks. Fixed here.

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