Lemma 91.5.1. 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. 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. The set of lifts of \varphi to an \mathcal{O}_{X'}-linear map \varphi ' : \mathcal{F}' \to \mathcal{G}' is, if nonempty, a principal homogeneous space under \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{I}\mathcal{G}').
Proof. This is a special case of Lemma 91.4.1 but we also give a direct proof. We have short exact sequences of modules
0 \to \mathcal{I} \to \mathcal{O}_{X'} \to \mathcal{O}_ X \to 0 \quad \text{and}\quad 0 \to \mathcal{I}\mathcal{G}' \to \mathcal{G}' \to \mathcal{G} \to 0
and similarly for \mathcal{F}'. Since \mathcal{I} has square zero the \mathcal{O}_{X'}-module structure on \mathcal{I} and \mathcal{I}\mathcal{G}' comes from a unique \mathcal{O}_ X-module structure. It follows that
\mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{I}\mathcal{G}') \quad \text{and}\quad \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})
The lemma now follows from the exact sequence
0 \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{I}\mathcal{G}') \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G}') \to \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_{X'}}(\mathcal{F}', \mathcal{G})
see Homology, Lemma 12.5.8. \square
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