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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