The Stacks project

Remark 91.10.6. Let $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ be a trivial first order thickening of ringed topoi and let $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a trivialization. Then given any triple $(\mathcal{F}, \mathcal{K}, c)$ consisting of a pair of $\mathcal{O}$-modules and a map $c : \mathcal{I} \otimes _\mathcal {O} \mathcal{F} \to \mathcal{K}$ we may set

and use the splitting (91.10.5.1) associated to $\pi$ and the map $c$ to define the $\mathcal{O}'$-module structure and obtain an extension (91.10.0.1). 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 (91.10.0.1) we can use $\pi ^\sharp : \mathcal{O} \to \mathcal{O}'$ to think of $\mathcal{F}'$ as an $\mathcal{O}$-module extension, hence a class $\xi _{\mathcal{F}'}$ in $\mathop{\mathrm{Ext}}\nolimits ^1_\mathcal {O}(\mathcal{F}, \mathcal{K})$. Lemma 91.10.3 assures that $\mathcal{F}' \mapsto \xi _{\mathcal{F}'}$ induces a bijection

Moreover, the trivial extension $\mathcal{F}'_{c, triv}$ maps to the zero class.