The Stacks project

Definition 18.33.3. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. The module of differentials of the ring map $\varphi $ is the object representing the functor $\mathcal{F} \mapsto \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F})$ which exists by Lemma 18.33.2. It is denoted $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$, and the universal $\varphi $-derivation is denoted $\text{d} : \mathcal{O}_2 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1}$.

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