Necessity of a coherence hypothesis in the relative Proj construction

Question

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.

Answer 1

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.