Lemma 17.29.6 (0G3V)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.29: Finite order differential operators
Lemma 17.29.6 (cite)

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Lemma 17.29.6. Let $X$ be a topological space. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings on $X$. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_2$-modules. There is a canonical short exact sequence

\[ 0 \to \Omega _{\mathcal{O}_2/\mathcal{O}_1} \otimes _{\mathcal{O}_2} \mathcal{F} \to \mathcal{P}^1_{\mathcal{O}_2/\mathcal{O}_1}(\mathcal{F}) \to \mathcal{F} \to 0 \]

functorial in $\mathcal{F}$ called the sequence of principal parts.

Proof. Follows from the commutative algebra version (Algebra, Lemma 10.133.6) and Lemmas 17.28.4 and 17.29.5. $\square$

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