Lemma 36.6.3 (0G7H)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 36: Derived Categories of Schemes
Section 36.6: Cohomology with support in a closed subset
Lemma 36.6.3 (cite)

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Lemma 36.6.3. Let $X$ be a scheme. Let $T \subset X$ be a closed subset such that $X \setminus T$ is a retrocompact open of $X$ . Then for a family of objects $E_ i$ , $i \in I$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ we have $R\mathcal{H}_ T(\bigoplus E_ i) = \bigoplus R\mathcal{H}_ T(E_ i)$ .

Proof. Set $U = X \setminus T$ and denote $j : U \to X$ the inclusion. By Cohomology, Lemma 20.34.6 there is a distinguished triangle

$i_*R\mathcal{H}_ T(E) \to E \to Rj_*(E|_ U) \to i_*R\mathcal{H}_ Z(E)[1]$

in $D(\mathcal{O}_ X)$ for any $E$ in $D(\mathcal{O}_ X)$ . The functor $E \mapsto Rj_*(E|_ U)$ commutes with direct sums on $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Lemma 36.4.5. It follows that the same is true for the functor $i_* \circ R\mathcal{H}_ T$ (details omitted). Since $i_* : D(i^{-1}\mathcal{O}_ X) \to D_ T(\mathcal{O}_ X)$ is an equivalence (Cohomology, Lemma 20.34.2) we conclude. $\square$

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