The Stacks project

Lemma 20.32.4. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Given an open subspace $V \subset Y$, set $U = f^{-1}(V)$ and denote $g : U \to V$ the induced 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 Lemma 20.32.1. Since it is clear that $(f_*\mathcal{F})|_ V = g_*(\mathcal{F}|_ U)$ for any sheaf $\mathcal{F}$ on $X$ the result follows. $\square$

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