The Stacks project

Lemma 73.10.2. 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 73.9.2 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 73.10.1 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). $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08GW. Beware of the difference between the letter 'O' and the digit '0'.