Lemma 91.4.3. Let i : (X, \mathcal{O}_ X) \to (X', \mathcal{O}_{X'}) be a first order thickening of ringed spaces. Assume given \mathcal{O}_ X-modules \mathcal{F}, \mathcal{K} and an \mathcal{O}_ X-linear map c : \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K}. If there exists a sequence (91.4.0.1) with c_{\mathcal{F}'} = c then the set of isomorphism classes of these extensions is principal homogeneous under \mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K}).
Proof. Assume given extensions
0 \to \mathcal{K} \to \mathcal{F}'_1 \to \mathcal{F} \to 0 \quad \text{and}\quad 0 \to \mathcal{K} \to \mathcal{F}'_2 \to \mathcal{F} \to 0
with c_{\mathcal{F}'_1} = c_{\mathcal{F}'_2} = c. Then the difference (in the extension group, see Homology, Section 12.6) is an extension
0 \to \mathcal{K} \to \mathcal{E} \to \mathcal{F} \to 0
where \mathcal{E} is annihilated by \mathcal{I} (local computation omitted). Hence the sequence is an extension of \mathcal{O}_ X-modules, see Modules, Lemma 17.13.4. Conversely, given such an extension \mathcal{E} we can add the extension \mathcal{E} to the \mathcal{O}_{X'}-extension \mathcal{F}' without affecting the map c_{\mathcal{F}'}. Some details omitted. \square
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