Definition 18.33.1. Let \mathcal{C} be a site. Let \varphi : \mathcal{O}_1 \to \mathcal{O}_2 be a homomorphism of sheaves of rings. Let \mathcal{F} be an \mathcal{O}_2-module. A \mathcal{O}_1-derivation or more precisely a \varphi -derivation into \mathcal{F} is a map D : \mathcal{O}_2 \to \mathcal{F} which is additive, annihilates the image of \mathcal{O}_1 \to \mathcal{O}_2, and satisfies the Leibniz rule
for all a, b local sections of \mathcal{O}_2 (wherever they are both defined). We denote \text{Der}_{\mathcal{O}_1}(\mathcal{O}_2, \mathcal{F}) the set of \varphi -derivations into \mathcal{F}.
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