Lemma 29.34.11 (01V7)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.34: Smooth morphisms
Lemma 29.34.11 (cite)

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Smooth morphisms are locally standard smooth.

Lemma 29.34.11. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point. Let $V \subset S$ be an affine open neighbourhood of $f(x)$ . The following are equivalent

The morphism $f$ is smooth at $x$ .
There exists an affine open $U \subset X$ , with $x \in U$ and $f(U) \subset V$ such that the induced morphism $f|_ U : U \to V$ is standard smooth.

Proof. Follows from the definitions and Algebra, Lemmas 10.137.7 and 10.137.10. $\square$

Comments (1)

Comment #835 by Johan Commelin on July 24, 2014 at 18:15

Suggested slogan: Smooth morphisms are locally standard smooth.

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