The Stacks project

Lemma 29.43.9. A base change of a (locally) projective morphism is (locally) projective.

Proof. This is true because the base change of a projective bundle over a scheme is a projective bundle, the pullback of a finite type $\mathcal{O}$-module is of finite type (Modules, Lemma 17.9.2) and the fact that the base change of a closed immersion is a closed immersion, see Schemes, Lemma 26.18.2. Some details omitted. $\square$

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