Lemma 75.3.1. Let S be a scheme. Let f : X \to Y be a morphism of algebraic spaces over S. Given an étale 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).
75.3 Generalities
In this section we put some general results on cohomology of unbounded complexes of modules on algebraic spaces.
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 66.26.2. \square
Definition 75.3.2. 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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