Lemma 10.150.8 (0H94)—The Stacks project

Lemma 10.150.8. Let $R \to S \to S'$ be ring maps with $S \to S'$ formally étale (for example étale). Let $M$ be an $S$-module. Set $M' = S' \otimes _ R M$. Then we have

\[ S' \otimes _ S P^ k_{S/R}(M) = P^ k_{S'/R}(M') \]

It follows that for any $S$-module $N$ and any finite order differential operator $D : M \to N$ there exists a unique extension $D' : M' \to S' \otimes _ S N$ of $D$ to a differential operator (of the same or lesser order).

Proof. Let $J$ and $J'$ be as in the statement of Lemma 10.150.7. Then we have

\begin{align*} S' \otimes _ S P^ k_{S/R}(M) & = S' \otimes _ S \left((S \otimes _ R M)/J^{k + 1}(S \otimes _ R M)\right) \\ & = S' \otimes _ S \left((S \otimes _ R S)/J^{k + 1}\right) \otimes _ S M \\ & = \left(S' \otimes _ R S'/(J')^{k + 1}\right) \otimes _ S M \\ & = \left(S' \otimes _ R S'/(J')^{k + 1}\right) \otimes _{S'} M' \\ & = S' \otimes _ R M'/(J')^{k + 1}(S' \otimes _ R M') \\ & = P^ k_{S'/R'}(M') \end{align*}

The first and the last equalities are from Lemma 10.133.9. The third equality is Lemma 10.150.7. The final assertion holds because if $D$ corresponds to the linear map $\gamma : P^ k_{S/R}(M) \to N$, then we can let $D' : M' \to S' \otimes _ R N$ correspond to the linear map $1 \otimes \gamma : S' \otimes _ R P^ k_{S/R}(M) \to S' \otimes _ R N$. $\square$

The Stacks project

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