The Stacks project

Remark 10.150.9. Let $R \to S \to S'$ be ring maps with $S \to S'$ formally étale (for example étale). Let $M_ i$, $i = 1, 2, 3$ be $S$-modules and let $D_ i : M_ i \to M_{i + 1}$, $i = 1, 2$ be differential operators of finite order. Then if $D'_ i : M'_ i \to M'_{i + 1}$, $i = 1, 2$ are the extensions of $D_ i$ to $M'_ i = S' \otimes _ S M_ i$ as in Lemma 10.150.8, then $D'_2 \circ D'_1$ is the extension of $D_2 \circ D_1$. In particular, if $M$ is an $S$-module, then $M' = S' \otimes _ S M$ is a module over the $S$-algebra $\text{Diff}_{S/R}(M, M)$.