The Stacks project

Definition 18.34.1. Let $\mathcal{C}$ be a site. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. Let $k \geq 0$ be an integer. Let $\mathcal{F}$, $\mathcal{G}$ be sheaves of $\mathcal{O}_2$-modules. A differential operator $D : \mathcal{F} \to \mathcal{G}$ of order $k$ is an is an $\mathcal{O}_1$-linear map such that for all local sections $g$ of $\mathcal{O}_2$ the map $s \mapsto D(gs) - gD(s)$ is a differential operator of order $k - 1$. For the base case $k = 0$ we define a differential operator of order $0$ to be an $\mathcal{O}_2$-linear map.