Lemma 21.5.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules on $\mathcal{C}$. There is a canonical bijection

\[ \mathop{\mathrm{Ext}}\nolimits ^1_{\textit{Mod}(\mathcal{O})}(\mathcal{O}, \mathcal{F}) \longrightarrow H^1(\mathcal{C}, \mathcal{F}) \]

which associates to the extension

\[ 0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O} \to 0 \]

the image of $1 \in \Gamma (\mathcal{C}, \mathcal{O})$ in $H^1(\mathcal{C}, \mathcal{F})$.

**Proof.**
Let us construct the inverse of the map given in the lemma. Let $\xi \in H^1(\mathcal{C}, \mathcal{F})$. Choose an injection $\mathcal{F} \subset \mathcal{I}$ with $\mathcal{I}$ injective in $\textit{Mod}(\mathcal{O})$. Set $\mathcal{Q} = \mathcal{I}/\mathcal{F}$. By the long exact sequence of cohomology, we see that $\xi $ is the image of a section $\tilde\xi \in \Gamma (\mathcal{C}, \mathcal{Q}) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{O}, \mathcal{Q})$. Now, we just form the pullback

\[ \xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar@{=}[d] & \mathcal{E} \ar[r] \ar[d] & \mathcal{O} \ar[r] \ar[d]^{\tilde\xi } & 0 \\ 0 \ar[r] & \mathcal{F} \ar[r] & \mathcal{I} \ar[r] & \mathcal{Q} \ar[r] & 0 } \]

see Homology, Section 12.6.
$\square$

## Comments (2)

Comment #3264 by Rene on

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