Lemma 29.44.16. A finite morphism is projective.

**Proof.**
Let $f : X \to S$ be a finite morphism. Then $f_*\mathcal{O}_ X$ is a quasi-coherent $\mathcal{O}_ S$-module (Lemma 29.11.5) of finite type (by our definition of finite morphisms and Properties, Lemma 28.16.1). We claim there is a closed immersion

over $S$, which finishes the proof. Namely, we let $\sigma $ be the morphism which corresponds (via Constructions, Lemma 27.16.11) to the surjection

coming from the adjunction map $f^*f_* \to \text{id}$. Then $\sigma $ is a closed immersion by Schemes, Lemma 26.21.11 and Constructions, Lemma 27.21.4. $\square$

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