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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