Lemma 29.32.10 (01V0)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.32: Sheaf of differentials of a morphism
Lemma 29.32.10 (cite)

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Lemma 29.32.10. Let $X \to S$ be a morphism of schemes. Let $g : S' \to S$ be a morphism of schemes. Let $X' = X_{S'}$ be the base change of $X$ . Denote $g' : X' \to X$ the projection. Then the map

$(g')^*\Omega _{X/S} \to \Omega _{X'/S'}$

of Lemma 29.32.8 is an isomorphism.

Proof. This is the sheafified version of Algebra, Lemma 10.131.12. $\square$

Comments (3)

Comment #1063 by Charles Rezk on October 07, 2014 at 13:13

Suggested slogan: The sheaf of differentials is compatible with base change.

Comment #3275 by Kevin Carlson on May 01, 2018 at 05:09

Suggested slogan: Pullback commutes with differentials

Comment #3367 by Johan on May 19, 2018 at 17:19

Still not a really catchy slogan... Can't we do better?

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