Lemma 32.8.3. Notation and assumptions as in Situation 32.8.1. If

$f$ is a finite morphism, and

$f_0$ is locally of finite type,

then there exists an $i \geq 0$ such that $f_ i$ is finite.

Lemma 32.8.3. Notation and assumptions as in Situation 32.8.1. If

$f$ is a finite morphism, and

$f_0$ is locally of finite type,

then there exists an $i \geq 0$ such that $f_ i$ is finite.

**Proof.**
A finite morphism is affine, see Morphisms, Definition 29.44.1. Hence by Lemma 32.8.2 above after increasing $0$ we may assume that $f_0$ is affine. By writing $Y_0$ as a finite union of affines we reduce to proving the result when $X_0$ and $Y_0$ are affine and map into a common affine $W \subset S_0$. The corresponding algebra statement follows from Algebra, Lemma 10.168.3.
$\square$

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