Regarding the relative Proj construction

Question

The relative projective space $\underline{Proj}(\mathcal{F})$ corresponding to a quasi-coherent sheaf $\mathcal{F}$ of graded $\mathcal{O}_X$-algebras is generally constructed by gluing $Proj(\mathcal{F}(U))$ where $U$ runs over all affine open of $X$. It appears to me that it would be enough to just consider an affine cover of $X$ instead of taking all affine opens. One can then check that the construction is well defined by showing that $\underline{Proj}(\mathcal{F})$ is the same (upto isomorphism) whether we take an affine cover of $X$ or any affine refinement of that cover. But all the references I have seen so far, glue the $Proj(\mathcal{F}(U))$ over all affine opens of $X$ to construct $\underline{Proj}(\mathcal{F})$. So, I am having doubts if I am correct. Please let me know if I am making any mistake.

Thanks in advance!

Answer 1

It seems to be that it would be easier to take the "all affine opens" version as the starting point — it seems like it would be easier both to write down definitions, write down theorems, and prove theorems about that version.

Note, in particular, the two points:

• "All affine opens" is an affine cover of $X$; the definition you want to consider is strictly more complicated.
• Given a cover of $X$, "all affine opens contained in a member of the given cover" is among the affine refinements you need to consider, and in this situation that doesn't seem any less complicated than "all affine opens".

It seems simpler and more economical to organize the overall work in a form like:

• Take the construction using all affine opens as the starting point
• Prove theorems¹ like $\underline{\mathrm{Proj}}(\mathcal{F}|_U) \cong \underline{\mathrm{Proj}}(\mathcal{F}) \times_X U$ for any open $U$, and $\underline{\mathrm{Proj}}(\mathcal{F}|_U) \cong \mathrm{Proj}(F(U))$ for affine open $U$
• Prove a theorem about gluing together the proj construction over the components of a cover.

Note that the last bullet point will be even more general than the definition you want to consider, because it will be a theorem for all covers, not just affine covers!

_{1: I'm assuming these statements are true}

Answer 2

Your (idea of) construction is perfectly valid. Finessing it a bit would lead to arguments that you can find here :

http://therisingsea.org/notes/TheRelativeProjConstruction.pdf

When you mention "being the same up to isomorphism" you in fact speak of the relative proj representing some functor. More details here in fully generality :

https://stacks.math.columbia.edu/tag/01NS

and here for special case :

https://stacks.math.columbia.edu/tag/01O4