Lemma 29.21.9 (01TX)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.21: Morphisms of finite presentation
Lemma 29.21.9 (cite)

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Over a locally Noetherian base, finite type is finite presentation.

Lemma 29.21.9. Let $f : X \to S$ be a morphism.

If $S$ is locally Noetherian and $f$ locally of finite type then $f$ is locally of finite presentation.
If $S$ is locally Noetherian and $f$ of finite type then $f$ is of finite presentation.

Proof. The first statement follows from the fact that a ring of finite type over a Noetherian ring is of finite presentation, see Algebra, Lemma 10.31.4. Suppose that $f$ is of finite type and $S$ is locally Noetherian. Then $f$ is quasi-compact and locally of finite presentation by (1). Hence it suffices to prove that $f$ is quasi-separated. This follows from Lemma 29.15.7 (and Lemma 29.21.8). $\square$

Comments (1)

Comment #1113 by Simon Pepin Lehalleur on October 30, 2014 at 19:42

Suggested slogan: Over a locally noetherian base, "finite type" = "finite presentation"

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