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. $\square$
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