Lemma 18.34.5 (09CV)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 18: Modules on Sites
Section 18.34: Finite order differential operators
Lemma 18.34.5 (cite)

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Lemma 18.34.5. Let $\mathcal{C}$ be a site. Let $\mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of presheaves of rings. Let $\mathcal{F}$ be a presheaf of $\mathcal{O}_2$ -modules. Then $\mathcal{P}^ k_{\mathcal{O}_2^\# /\mathcal{O}_1^\# }(\mathcal{F}^\# )$ is the sheaf associated to the presheaf $U \mapsto P^ k_{\mathcal{O}_2(U)/\mathcal{O}_1(U)}(\mathcal{F}(U))$ .

Proof. This can be proved in exactly the same way as is done for the sheaf of differentials in Lemma 18.33.4. Perhaps a more pleasing approach is to use the universal property of Lemma 18.34.3 directly to see the equality. We omit the details. $\square$

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