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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