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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