Lemma 17.28.5 (08RQ)—The Stacks project

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Part 1: Preliminaries
Chapter 17: Sheaves of Modules
Section 17.28: Modules of differentials
Lemma 17.28.5 (cite)

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Lemma 17.28.5. Let $X$ be a topological space. Let $\varphi : \mathcal{O}_1 \to \mathcal{O}_2$ be a homomorphism of sheaves of rings. For $U \subset X$ open there is a canonical isomorphism

$\Omega _{\mathcal{O}_2/\mathcal{O}_1}|_ U = \Omega _{(\mathcal{O}_2|_ U)/(\mathcal{O}_1|_ U)}$

compatible with universal derivations.

Proof. Holds because $\Omega _{\mathcal{O}_2/\mathcal{O}_1}$ is the cokernel of the map (17.28.2.1). $\square$

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