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

such that

$X_0$, $S_0$ are affine schemes,

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

$f_0$ is of finite type.

Lemma 37.33.1. 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

$X_0$, $S_0$ 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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