Lemma 29.15.8. Let $X \to Y$ be a morphism of schemes over a base scheme $S$. If $X$ is locally of finite type over $S$, then $X \to Y$ is locally of finite type.

**Proof.**
Via Lemma 29.15.2 this translates into the following algebra fact: Given ring maps $A \to B \to C$ such that $A \to C$ is of finite type, then $B \to C$ is of finite type. (See Algebra, Lemma 10.6.2).
$\square$

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