Lemma 48.15.3. Let Y be a ringed space. Let \mathcal{I} \subset \mathcal{O}_ Y be a sheaf of ideals. Set \mathcal{O}_ X = \mathcal{O}_ Y/\mathcal{I} and \mathcal{N} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\mathcal{I}/\mathcal{I}^2, \mathcal{O}_ X). There is a canonical isomorphism c : \mathcal{N} \to \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^1_{\mathcal{O}_ Y}(\mathcal{O}_ X, \mathcal{O}_ X) .
Proof. Consider the canonical short exact sequence
48.15.3.1
\begin{equation} \label{duality-equation-second-order-thickening} 0 \to \mathcal{I}/\mathcal{I}^2 \to \mathcal{O}_ Y/\mathcal{I}^2 \to \mathcal{O}_ X \to 0 \end{equation}
Let U \subset X be open and let s \in \mathcal{N}(U). Then we can pushout (48.15.3.1) via s to get an extension E_ s of \mathcal{O}_ X|_ U by \mathcal{O}_ X|_ U. This in turn defines a section c(s) of \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^1_{\mathcal{O}_ Y}(\mathcal{O}_ X, \mathcal{O}_ X) over U. See Cohomology, Lemma 20.42.1 and Derived Categories, Lemma 13.27.6. Conversely, given an extension
0 \to \mathcal{O}_ X|_ U \to \mathcal{E} \to \mathcal{O}_ X|_ U \to 0
of \mathcal{O}_ U-modules, we can find an open covering U = \bigcup U_ i and sections e_ i \in \mathcal{E}(U_ i) mapping to 1 \in \mathcal{O}_ X(U_ i). Then e_ i defines a map \mathcal{O}_ Y|_{U_ i} \to \mathcal{E}|_{U_ i} whose kernel contains \mathcal{I}^2. In this way we see that \mathcal{E}|_{U_ i} comes from a pushout as above. This shows that c is surjective. We omit the proof of injectivity. \square
Comments (0)