Lemma 29.32.3 (01US)—The Stacks project

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Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.3 (cite)

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Lemma 29.32.3. Let $f : X \to S$ be a morphism of schemes. Let $U \subset X$ , $V \subset S$ be open subschemes such that $f(U) \subset V$ . Then there is a unique isomorphism $\Omega _{X/S}|_ U = \Omega _{U/V}$ of $\mathcal{O}_ U$ -modules such that $\text{d}_{X/S}|_ U = \text{d}_{U/V}$ .

Proof. This is a special case of Modules, Lemma 17.28.5 if we use the canonical identification $f^{-1}\mathcal{O}_ S|_ U = (f|_ U)^{-1}\mathcal{O}_ V$ . $\square$

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