Lemma 20.33.2 (08BV)—The Stacks project

Lemma 20.33.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $X = U \cup V$ be the union of two open subspaces. For any object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

\[ E \to Rj_{U, *}E|_ U \oplus Rj_{V, *}E|_ V \to Rj_{U \cap V, *}E|_{U \cap V} \to E[1] \]

in $D(\mathcal{O}_ X)$.

Proof. Choose a K-injective complex $\mathcal{I}^\bullet $ representing $E$ whose terms $\mathcal{I}^ n$ are injective objects of $\textit{Mod}(\mathcal{O}_ X)$, see Injectives, Theorem 19.12.6. We have seen that $\mathcal{I}^\bullet |U$ is a K-injective complex as well (Lemma 20.32.1). Hence $Rj_{U, *}E|_ U$ is represented by $j_{U, *}\mathcal{I}^\bullet |_ U$. Similarly for $V$ and $U \cap V$. Hence the distinguished triangle of the lemma is the distinguished triangle associated (by Derived Categories, Section 13.12 and especially Lemma 13.12.1) to the short exact sequence of complexes

\[ 0 \to \mathcal{I}^\bullet \to j_{U, *}\mathcal{I}^\bullet |_ U \oplus j_{V, *}\mathcal{I}^\bullet |_ V \to j_{U \cap V, *}\mathcal{I}^\bullet |_{U \cap V} \to 0. \]

This sequence is exact because for any $W \subset X$ open and any $n$ the sequence

\[ 0 \to \mathcal{I}^ n(W) \to \mathcal{I}^ n(W \cap U) \oplus \mathcal{I}^ n(W \cap V) \to \mathcal{I}^ n(W \cap U \cap V) \to 0 \]

is exact (see proof of Lemma 20.8.2). $\square$

The Stacks project

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