Lemma 29.15.4. The base change of a morphism which is locally of finite type is locally of finite type. The same is true for morphisms of finite type.

**Proof.**
In the proof of Lemma 29.15.2 we saw that being of finite type is a local property of ring maps. Hence the first statement of the lemma follows from Lemma 29.14.6 combined with the fact that being of finite type is a property of ring maps that is stable under base change, see Algebra, Lemma 10.14.2. By the above and the fact that a base change of a quasi-compact morphism is quasi-compact, see Schemes, Lemma 26.19.3 we see that the base change of a morphism of finite type is a morphism of finite type.
$\square$

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