Definition 17.28.10. Let $(f, f^\sharp ) : (X, \mathcal{O}_ X) \to (S, \mathcal{O}_ S)$ be a morphism of ringed spaces.

Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. An

*$S$-derivation*into $\mathcal{F}$ is a $f^{-1}\mathcal{O}_ S$-derivation, or more precisely a $f^\sharp $-derivation in the sense of Definition 17.28.1. We denote $\text{Der}_ S(\mathcal{O}_ X, \mathcal{F})$ the set of $S$-derivations into $\mathcal{F}$.The

*sheaf of differentials $\Omega _{X/S}$ of $X$ over $S$*is the module of differentials $\Omega _{\mathcal{O}_ X/f^{-1}\mathcal{O}_ S}$ endowed with its universal $S$-derivation $\text{d}_{X/S} : \mathcal{O}_ X \to \Omega _{X/S}$.

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