# Necessity of a coherence hypothesis in the relative Proj construction

In Hartshorne the hypotheses for the relative Proj construction are given as follows:

Let $$X$$ be a noetherian scheme and $$L$$ be quasi-coherent sheaf of $$O_X$$-modules which has a structure of sheaf of graded $$O_X$$-algebras. Thus $$L = \oplus L_d$$ where $$d \ge 0$$. Assume furthermore that $$L_0 =O_X$$, that $$L_1$$ is a coherent module and that $$L$$ is locally generated by $$L_1$$ as an $$O_X$$-algebra.

My question is do we need this hypothesis for the construction of the relative Proj? I think only quasi-coherence of $$L$$ is needed.

Question: "My question is do we need this hypothesis for the construction of the relative Proj? I think only quasi-coherence of L is needed."

Answer: You are correct. You find a proof in

Grothendieck, Alexander; Dieudonné, Jean A. Éléments de géométrie algébrique. I. (English) Zbl 0203.23301 Die Grundlehren der mathematischen Wissenschaften. 166. Berlin-Heidelberg-New York: Springer-Verlag. IX, 466 p. (1971).

Proposition 9.7.7. They prove that for any scheme $$S$$ and any quasi coherent $$\mathcal{O}_S$$-module $$E$$, it follows $$G:=Grass_d(E)$$ is a separated $$S$$-scheme. If $$E$$ is of finite type it follows $$G$$ is of finite type. In particular it follows $$\mathbb{P}(E^*)$$ is a separated $$S$$-scheme for any such $$E$$. I believe the techniques of proof can be adapted to your situation of a general sheaf $$L:=\oplus_{d \geq 0}L_d$$ generated by $$L_1$$ as $$L_0$$-module.

Note: There is an equlity

$$Hom_{\mathcal{O}_S-mod}(L_1, L) \cong Hom_{\mathcal{O}_S-alg}(Sym_{\mathcal{O}_S}^*(L_1),L)$$

and you get an exact sequence

$$0 \rightarrow I \rightarrow Sym_{\mathcal{O}_S}^*(L_1) \rightarrow L \rightarrow 0$$

and $$Sym_{\mathcal{O}_S}^*(L_1)/I \cong L$$. Hence there is a closed subscheme

$$V(I) \subseteq \mathbb{P}(L_1):=Proj(Sym_{\mathcal{O}_S}^*(L_1))$$

and $$\mathbb{P}(L) \cong V(I) \subseteq \mathbb{P}(L_1)$$

and since $$\mathbb{P}(L_1)$$ is a scheme it follows $$\mathbb{P}(L)$$ is a scheme.