The Stacks project

Lemma 115.26.10. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$ be a ringed topos and let $\mathcal{J}$ be an $\mathcal{O}_\mathcal {B}$-module.

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

  2. Given a morphism of ringed topoi $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{B}), \mathcal{O}_\mathcal {B})$, an $\mathcal{O}$-module $\mathcal{G}$, an $f^{-1}\mathcal{O}_\mathcal {B}$-module map $c : f^{-1}\mathcal{J} \to \mathcal{G}$, and given extensions of sheaves of rings with square zero kernels:

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

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

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

Proof. This follows from Deformation Theory, Lemmas 91.13.4 and 91.13.6. $\square$

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