Definition 10.131.1. Let $R \to S$ be a ring map. Let $M$, $N$ be $S$-modules. Let $k \geq 0$ be an integer. We inductively define a differential operator $D : M \to N$ of order $k$ to be an $R$-linear map such that for all $g \in S$ the map $m \mapsto D(gm) - gD(m)$ 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 $S$-linear map.
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