The Stacks project

Remark 91.10.5. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a ringed topos. A first order thickening $i : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}')$ is said to be trivial if there exists a morphism of ringed topoi $\pi : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ which is a left inverse to $i$. The choice of such a morphism $\pi$ is called a trivialization of the first order thickening. Given $\pi$ we obtain a splitting

as sheaves of algebras on $\mathcal{C}$ by using $\pi ^\sharp$ to split the surjection $\mathcal{O}' \to \mathcal{O}$. Conversely, such a splitting determines a morphism $\pi$. The category of trivialized first order thickenings of $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ is equivalent to the category of $\mathcal{O}$-modules.