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