The Stacks project

Lemma 29.15.3. The composition of two morphisms which are 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 29.14.5 combined with the fact that being of finite type is a property of ring maps that is stable under composition, see Algebra, Lemma 10.6.2. By the above and the fact that compositions of quasi-compact morphisms are quasi-compact, see Schemes, Lemma 26.19.4 we see that the composition of morphisms of finite type is of finite type. $\square$

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