The Stacks project

Remark 91.4.9. Let $(X, \mathcal{O}_ X)$ be a ringed space. We define a sequence of morphisms of first order thickenings

of $(X, \mathcal{O}_ X)$ to be a complex if the corresponding maps between the ideal sheaves $\mathcal{I}_ i$ give a complex of $\mathcal{O}_ X$-modules $\mathcal{I}_3 \to \mathcal{I}_2 \to \mathcal{I}_1$ (i.e., the composition is zero). In this case the composition $(X'_1, \mathcal{O}_{X'_1}) \to (X_3', \mathcal{O}_{X'_3})$ factors through $(X, \mathcal{O}_ X) \to (X'_3, \mathcal{O}_{X'_3})$, i.e., the first order thickening $(X'_1, \mathcal{O}_{X'_1})$ of $(X, \mathcal{O}_ X)$ is trivial and comes with a canonical trivialization $\pi : (X'_1, \mathcal{O}_{X'_1}) \to (X, \mathcal{O}_ X)$.

We say a sequence of morphisms of first order thickenings

of $(X, \mathcal{O}_ X)$ is a short exact sequence if the corresponding maps between ideal sheaves is a short exact sequence

of $\mathcal{O}_ X$-modules.