Remark 91.4.6. Let $i : (X, \mathcal{O}_ X) \to (X', \mathcal{O}_{X'})$ be a trivial first order thickening of ringed spaces and let $\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_ X)$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}_ X$-modules and a map $c : \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{K}$ we may set
and use the splitting (91.4.5.1) associated to $\pi $ and the map $c$ to define the $\mathcal{O}_{X'}$-module structure and obtain an extension (91.4.0.1). We will call $\mathcal{F}'_{c, triv}$ the trivial extension of $\mathcal{F}$ by $\mathcal{K}$ corresponding to $c$ and the trivialization $\pi $. Given any extension $\mathcal{F}'$ as in (91.4.0.1) we can use $\pi ^\sharp : \mathcal{O}_ X \to \mathcal{O}_{X'}$ to think of $\mathcal{F}'$ as an $\mathcal{O}_ X$-module extension, hence a class $\xi _{\mathcal{F}'}$ in $\mathop{\mathrm{Ext}}\nolimits ^1_{\mathcal{O}_ X}(\mathcal{F}, \mathcal{K})$. Lemma 91.4.3 assures that $\mathcal{F}' \mapsto \xi _{\mathcal{F}'}$ induces a bijection
Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.
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