Lemma 20.5.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. There is a canonical bijection which associates to the extension the image of $1 \in \Gamma (X, \mathcal{O}_ X)$ in $H^1(X, \mathcal{F})$.
\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\textit{Mod}(\mathcal{O}_ X)}(\mathcal{O}_ X, \mathcal{F}) \longrightarrow H^1(X, \mathcal{F}) \]
\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O}_ X \to 0 \]
Proof. Let us construct the inverse of the map given in the lemma. Let $\xi \in H^1(X, \mathcal{F})$. Choose an injection $\mathcal{F} \subset \mathcal{I}$ with $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{O}_ X)$. Set $\mathcal{Q} = \mathcal{I}/\mathcal{F}$. By the long exact sequence of cohomology, we see that $\xi $ is the image of a section $\tilde\xi \in \Gamma (X, \mathcal{Q}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{O}_ X, \mathcal{Q})$. Now, we just form the pullback
see Homology, Section 12.6. $\square$
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Comment #1606 by Keenan Kidwell on