The Stacks project

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 ( $\square$

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