The Stacks project

Lemma 20.11.7. Let $X$ be a ringed space. Let

\[ 0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0 \]

be a short exact sequence of $\mathcal{O}_ X$-modules. Let $U \subset X$ be an open subset. If there exists a cofinal system of open coverings $\mathcal{U}$ of $U$ such that $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$, then the map $\mathcal{G}(U) \to \mathcal{H}(U)$ is surjective.

Proof. Take an element $s \in \mathcal{H}(U)$. Choose an open covering $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ such that (a) $\check{H}^1(\mathcal{U}, \mathcal{F}) = 0$ and (b) $s|_{U_ i}$ is the image of a section $s_ i \in \mathcal{G}(U_ i)$. Since we can certainly find a covering such that (b) holds it follows from the assumptions of the lemma that we can find a covering such that (a) and (b) both hold. Consider the sections

\[ s_{i_0i_1} = s_{i_1}|_{U_{i_0i_1}} - s_{i_0}|_{U_{i_0i_1}}. \]

Since $s_ i$ lifts $s$ we see that $s_{i_0i_1} \in \mathcal{F}(U_{i_0i_1})$. By the vanishing of $\check{H}^1(\mathcal{U}, \mathcal{F})$ we can find sections $t_ i \in \mathcal{F}(U_ i)$ such that

\[ s_{i_0i_1} = t_{i_1}|_{U_{i_0i_1}} - t_{i_0}|_{U_{i_0i_1}}. \]

Then clearly the sections $s_ i - t_ i$ satisfy the sheaf condition and glue to a section of $\mathcal{G}$ over $U$ which maps to $s$. Hence we win. $\square$

Comments (3)

Comment #8354 by Et on

Can you not deduce this immediately from the exact sequence of Cech cohomology and the fact that the 0'th cohomology is just the global sections for sheaves?

Comment #8355 by Et on

Edit: I see now... it is an exact sequence of sheaves and the cech functor is only a delta functor on the pre-sheaf category... Perhap it's worth starting the proof with "note that lemma 20.10.2 does not apply here" to avoid confusion for future readers.

Comment #8961 by on

This is a reasonable concern to have, but since the proof is fine I am going to leave this as is.

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