Lemma 37.34.1 (05FB)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.34: Limit arguments
Lemma 37.34.1 (cite)

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Lemma 37.34.1. Let $f : X \to S$ be a morphism of affine schemes, which is of finite presentation. Then there exists a cartesian diagram

$\xymatrix{ X_0 \ar[d]_{f_0} & X \ar[l]^ g \ar[d]^ f \\ S_0 & S \ar[l] }$

such that

$X_0$ , $S_0$ are affine schemes,
$S_0$ of finite type over $\mathbf{Z}$ ,
$f_0$ is of finite type.

Proof. Write $S = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ . As $f$ is of finite presentation we see that $B$ is of finite presentation as an $A$ -algebra, see Morphisms, Lemma 29.21.2. Thus the lemma follows from Algebra, Lemma 10.127.18. $\square$

Comments (2)

Comment #6736 by 羽山籍真 on November 20, 2021 at 07:37

In (3), finite should imply "of finite type"; also, $f_0$ is merely of finite type, no?

Comment #6926 by Johan on January 19, 2022 at 14:15

Thanks and fixed here.

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