Lemma 29.21.11 (02FV)—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 29.21.11 (cite)

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Lemma 29.21.11. Let $f : X \to Y$ be a morphism of schemes over $S$.

If $X$ is locally of finite presentation over $S$ and $Y$ is locally of finite type over $S$, then $f$ is locally of finite presentation.
If $X$ is of finite presentation over $S$ and $Y$ is quasi-separated and locally of finite type over $S$, then $f$ is of finite presentation.

Proof. Proof of (1). Via Lemma 29.21.2 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is of finite presentation and $A \to B$ is of finite type, then $B \to C$ is of finite presentation. See Algebra, Lemma 10.6.2.

Part (2) follows from (1) and Schemes, Lemmas 26.21.13 and 26.21.14. $\square$

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