Lemma 29.21.3. The composition of two morphisms which are locally of finite presentation is locally of finite presentation. The same is true for morphisms of finite presentation.

Proof. In the proof of Lemma 29.21.2 we saw that being of finite presentation 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 presentation 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, quasi-separated morphisms are quasi-compact and quasi-separated, see Schemes, Lemmas 26.19.4 and 26.21.12 we see that the composition of morphisms of finite presentation is of finite presentation. $\square$

## Comments (1)

Comment #1796 by Matthieu Romagny on

Typo in the statement : "two morphisms which are locally of finite presentation..."

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