The Stacks project

Lemma 114.25.9. Let $(S, \mathcal{O}_ S)$ be a ringed space and let $\mathcal{J}$ be an $\mathcal{O}_ S$-module.

  1. The set of extensions of sheaves of rings $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_ S \to 0$ where $\mathcal{J}$ is an ideal of square zero is canonically bijective to $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ S}(\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{J})$.

  2. Given a morphism of ringed spaces $f : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$, an $\mathcal{O}_ X$-module $\mathcal{G}$, an $f$-map $c : \mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels:

    1. $0 \to \mathcal{J} \to \mathcal{O}_{S'} \to \mathcal{O}_ S \to 0$ corresponding to $\alpha \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ S}(\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{J})$,

    2. $0 \to \mathcal{G} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0$ corresponding to $\beta \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathop{N\! L}\nolimits _{X/\mathbf{Z}}, \mathcal{G})$

    then there is a morphism $X' \to S'$ fitting into Deformation Theory, Equation (90.7.0.1) if and only if $\beta $ and $\alpha $ map to the same element of $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Lf^*\mathop{N\! L}\nolimits _{S/\mathbf{Z}}, \mathcal{G})$.

Proof. This follows from Deformation Theory, Lemmas 90.7.4 and 90.7.6. $\square$


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