Lemma 37.33.1. 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

1. $X_0$, $S_0$ are affine schemes,

2. $S_0$ of finite type over $\mathbf{Z}$,

3. $f_0$ 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. $\square$

Comment #6736 by 羽山籍真 on

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

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