The Stacks project

Remark 91.4.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. A first order thickening $i : (X, \mathcal{O}_ X) \to (X', \mathcal{O}_{X'})$ is said to be trivial if there exists a morphism of ringed spaces $\pi : (X', \mathcal{O}_{X'}) \to (X, \mathcal{O}_ X)$ 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 $X$ by using $\pi ^\sharp $ to split the surjection $\mathcal{O}_{X'} \to \mathcal{O}_ X$. Conversely, such a splitting determines a morphism $\pi $. The category of trivialized first order thickenings of $(X, \mathcal{O}_ X)$ is equivalent to the category of $\mathcal{O}_ X$-modules.