Definition 17.27.1. Let $X$ be a topological space. 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

$D(ab) = aD(b) + D(a)b$

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 $\mathcal{F}$.

Comment #1814 by Keenan Kidwell on

To be clear, this definition can equivalently be formulated as follows: an $\mathcal{O}_1$-derivation $D:\mathcal{O}_2\to\mathcal{F}$ is a map of abelian sheaves on $X$ such that, for each open subset $U$ of $X$, $D_U:\Gamma(U,\mathcal{O}_2)\to\Gamma(U,\mathcal{F})$ is a $\Gamma(U,\mathcal{O}_1)$-derivation of $\Gamma(U,\mathcal{O}_2)$ into $\Gamma(U,\mathcal{F})$ in the sense of 10.131.1, right?

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