The Stacks project

Definition 18.33.10. Let $X = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ and $Y = (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be ringed topoi. Let $(f, f^\sharp ) : X \to Y$ be a morphism of ringed topoi. In this situation

  1. for a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules a $Y$-derivation $D : \mathcal{O} \to \mathcal{F}$ is just a $f^\sharp $-derivation, and

  2. the sheaf of differentials $\Omega _{X/Y}$ of $X$ over $Y$ is the module of differentials of $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, see Definition 18.33.3.

Thus $\Omega _{X/Y}$ comes equipped with a universal $Y$-derivation $\text{d}_{X/Y} : \mathcal{O} \longrightarrow \Omega _{X/Y}$. We sometimes write $\Omega _{X/Y} = \Omega _ f$.

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