Lemma 17.29.2 (0G3R)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.29: Finite order differential operators
Lemma 17.29.2 (cite)

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Lemma 17.29.2. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a map of sheaves of rings on $X$ . Let $\mathcal{E}, \mathcal{F}, \mathcal{G}$ be sheaves of $\mathcal{O}_2$ -modules. If $D : \mathcal{E} \to \mathcal{F}$ and $D' : \mathcal{F} \to \mathcal{G}$ are differential operators of order $k$ and $k'$ , then $D' \circ D$ is a differential operator of order $k + k'$ .

Proof. Let $g$ be a local section of $\mathcal{O}_2$ . Then the map which sends a local section $x$ of $\mathcal{E}$ to

$D'(D(gx)) - gD'(D(x)) = D'(D(gx)) - D'(gD(x)) + D'(gD(x)) - gD'(D(x))$

is a sum of two compositions of differential operators of lower order. Hence the lemma follows by induction on $k + k'$ . $\square$

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