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

for a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules a

*$Y$-derivation*$D : \mathcal{O} \to \mathcal{F}$ is just a $f^\sharp $-derivation, andthe

*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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