The Stacks project

74.3 Generalities

In this section we put some general results on cohomology of unbounded complexes of modules on algebraic spaces.

Lemma 74.3.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Given an ├ętale morphism $V \to Y$, set $U = V \times _ Y X$ and denote $g : U \to V$ the projection morphism. Then $(Rf_*E)|_ V = Rg_*(E|_ U)$ for $E$ in $D(\mathcal{O}_ X)$.

Proof. Represent $E$ by a K-injective complex $\mathcal{I}^\bullet $ of $\mathcal{O}_ X$-modules. Then $Rf_*(E) = f_*\mathcal{I}^\bullet $ and $Rg_*(E|_ U) = g_*(\mathcal{I}^\bullet |_ U)$ by Cohomology on Sites, Lemma 21.20.1. Hence the result follows from Properties of Spaces, Lemma 65.26.2. $\square$

Definition 74.3.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Let $T \subset |X|$ be a closed subset. We say $E$ is supported on $T$ if the cohomology sheaves $H^ i(E)$ are supported on $T$.

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