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
\mathop{\mathrm{Ext}}\nolimits ^1_{\textit{Mod}(\mathcal{O}_ X)}(\mathcal{O}_ X, \mathcal{F}) \longrightarrow H^1(X, \mathcal{F})
which associates to the extension
0 \to \mathcal{F} \to \mathcal{E} \to \mathcal{O}_ X \to 0
the image of 1 \in \Gamma (X, \mathcal{O}_ X) in H^1(X, \mathcal{F}).
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
\xymatrix{ 0 \ar[r] & \mathcal{F} \ar[r] \ar@{=}[d] & \mathcal{E} \ar[r] \ar[d] & \mathcal{O}_ X \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 (1)
Comment #1606 by Keenan Kidwell on