Lemma 36.11.3 (08E2)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.11: Derived category of coherent modules
Lemma 36.11.3 (cite)

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Lemma 36.11.3. Let $S$ be a Noetherian scheme. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $E$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ such that the support of $H^ i(E)$ is proper over $S$ for all $i$ . Then $Rf_*E$ is an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ S)$ .

Proof. Consider the spectral sequence

$R^ pf_*H^ q(E) \Rightarrow R^{p + q}f_*E$

see Derived Categories, Lemma 13.21.3. By assumption and Cohomology of Schemes, Lemma 30.26.10 the sheaves $R^ pf_*H^ q(E)$ are coherent. Hence $R^{p + q}f_*E$ is coherent, i.e., $Rf_*E \in D_{\textit{Coh}}(\mathcal{O}_ S)$ . Boundedness from below is trivial. Boundedness from above follows from Cohomology of Schemes, Lemma 30.4.5 or from Lemma 36.4.1. $\square$

Comments (2)

Comment #4826 by awllower on December 11, 2019 at 10:31

In $E \in D_{\textit{Coh}}(\mathcal{O}_ S)$ it should be $Rf_* E\in D_{\textit{Coh}}(\mathcal{O}_ S)$ .

Comment #5129 by Johan on June 07, 2020 at 13:06

Thanks and fixed here.

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