The Stacks project

Lemma 75.10.2. Let $S$ be a scheme. Let $(U \subset X, V \to X)$ be an elementary distinguished square of algebraic spaces over $S$.

  1. For every sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}$ we have a short exact sequence

    \[ 0 \to \mathcal{F} \to j_{U, *}\mathcal{F}|_ U \oplus j_{V, *}\mathcal{F}|_ V \to j_{U \times _ X V, *}\mathcal{F}|_{U \times _ X V} \to 0 \]
  2. For any object $E$ of $D(\mathcal{O}_ X)$ we have a distinguished triangle

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

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

Proof. Let $W$ be an object of $X_{\acute{e}tale}$. We claim the sequence

\[ 0 \to \mathcal{F}(W) \to \mathcal{F}(W \times _ X U) \oplus \mathcal{F}(W \times _ X V) \to \mathcal{F}(W \times _ X U \times _ X V) \]

is exact and that an element of the last group can locally on $W$ be lifted to the middle one. By Lemma 75.9.2 the pair $(W \times _ X U \subset W, V \times _ X W \to W)$ is an elementary distinguished square. Thus we may assume $W = X$ and it suffices to prove the same thing for

\[ 0 \to \mathcal{F}(X) \to \mathcal{F}(U) \oplus \mathcal{F}(V) \to \mathcal{F}(U \times _ X V) \]

We have seen that

\[ 0 \to j_{U \times _ X V!}\mathcal{O}_{U \times _ X V} \to j_{U!}\mathcal{O}_ U \oplus j_{V!}\mathcal{O}_ V \to \mathcal{O}_ X \to 0 \]

is a exact sequence of $\mathcal{O}_ X$-modules in Lemma 75.10.1 and applying the right exact functor $\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(- , \mathcal{F})$ gives the sequence above. This also means that the obstruction to lifting $s \in \mathcal{F}(U \times _ X V)$ to an element of $\mathcal{F}(U) \oplus \mathcal{F}(V)$ lies in $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{O}_ X, \mathcal{F}) = H^1(X, \mathcal{F})$. By locality of cohomology (Cohomology on Sites, Lemma 21.7.3) this obstruction vanishes ├ętale locally on $X$ and the proof of (1) is complete.

Proof of (2). 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. Then $\mathcal{I}^\bullet |U$ is a K-injective complex (Cohomology on Sites, Lemma 21.20.1). Hence $Rj_{U, *}E|_ U$ is represented by $j_{U, *}\mathcal{I}^\bullet |_ U$. Similarly for $V$ and $U \times _ X V$. 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

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

This sequence is exact by (1). $\square$

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