Lemma 91.5.4. Let i : (X, \mathcal{O}_ X) \to (X', \mathcal{O}_{X'}) be a first order thickening of ringed spaces. Let \mathcal{F}', \mathcal{G}' be \mathcal{O}_{X'}-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}_ X-linear map. There exists an element
o(\varphi ) \in \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(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}_{X'}-linear map \varphi ' : \mathcal{F}' \to \mathcal{G}'.
Proof.
It is clear from the proof of Lemma 91.5.1 that the vanishing of the boundary of \varphi via the map
\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_{X'}}(\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}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(Li^*\mathcal{F}', \mathcal{I}\mathcal{G}')
the adjointness of i_* = Ri_* and Li^* on the derived category (Cohomology, Lemma 20.28.1).
\square
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