The Stacks project

Lemma 91.7.4. Let $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces. Let $\mathcal{G}$ be a $\mathcal{O}_ X$-module. The set of isomorphism classes of extensions of $f^{-1}\mathcal{O}_ S$-algebras

\[ 0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0 \]

where $\mathcal{G}$ is an ideal of square zero1 is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/S}, \mathcal{G})$.

Proof. To prove this we apply the previous results to the case where ( is given by the diagram

\[ \xymatrix{ 0 \ar[r] & \mathcal{G} \ar[r] & {?} \ar[r] & \mathcal{O}_ X \ar[r] & 0 \\ 0 \ar[r] & 0 \ar[u] \ar[r] & \mathcal{O}_ S \ar[u] \ar[r]^{\text{id}} & \mathcal{O}_ S \ar[u] \ar[r] & 0 } \]

Thus our lemma follows from Lemma 91.7.3 and the fact that there exists a solution, namely $\mathcal{G} \oplus \mathcal{O}_ X$. (See remark below for a direct construction of the bijection.) $\square$

[1] In other words, the set of isomorphism classes of first order thickenings $i : X \to X'$ over $S$ endowed with an isomorphism $\mathcal{G} \to \mathop{\mathrm{Ker}}(i^\sharp )$ of $\mathcal{O}_ X$-modules.

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