Definition 17.28.1 (01UN)—The Stacks project

Definition 17.28.1. 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. An $\mathcal{O}_1$-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 $\mathcal{F}$.

Comments (2)

Comment #1814 by Keenan Kidwell on February 03, 2016 at 00:57

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?

Comment #1831 by Johan on February 04, 2016 at 20:34

Yes, that is true.

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