Lemma 10.133.11 (0G37)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.133: Finite order differential operators
Lemma 10.133.11 (cite)

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Lemma 10.133.11. Let $R \to A$ and $R \to B$ be ring maps. Let $M$ and $M'$ be $A$-modules. Let $D : M \to M'$ be a differential operator of order $k$ with respect to $R \to A$. Let $N$ be any $B$-module. Then the map

\[ D \otimes \text{id}_ N : M \otimes _ R N \to M' \otimes _ R N \]

is a differential operator of order $k$ with respect to $B \to A \otimes _ R B$.

Proof. It is clear that $D' = D \otimes \text{id}_ N$ is $B$-linear. By Lemma 10.133.9 it suffices to show that

\[ D' \circ a \otimes 1 - a \otimes 1 \circ D' = (D \circ a - a \circ D) \otimes \text{id}_ N \]

is a differential operator of order $k - 1$ which follows by induction on $k$. $\square$

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