Lemma 91.11.5. Let i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') be a first order thickening of ringed topoi. Let \mathcal{F}', \mathcal{G}' be \mathcal{O}'-modules and set \mathcal{F} = i^*\mathcal{F}' and \mathcal{G} = i^*\mathcal{G}'. Let \varphi : \mathcal{F} \to \mathcal{G} be an \mathcal{O}-linear map. There exists an element
o(\varphi ) \in \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
whose vanishing is a necessary and sufficient condition for the existence of a lift of \varphi to an \mathcal{O}'-linear map \varphi ' : \mathcal{F}' \to \mathcal{G}'.
Proof.
It is clear from the proof of Lemma 91.11.1 that the vanishing of the boundary of \varphi via the map
\mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}'}(\mathcal{F}', \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}')
is a necessary and sufficient condition for the existence of a lift. We conclude as
\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}'}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
the adjointness of i_* = Ri_* and Li^* on the derived category (Cohomology on Sites, Lemma 21.19.1).
\square
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