# Problem showing the saturation map of a graded ring f.g. in degree 1 induces an isomorphism of projective schemes and O(1) (The Rising Sea 15.6.G)

I've had problem working on this exercise from the July 31, 2023 version of Vakil's The Rising Sea:

15.6.G. Exercise. Show that the map of graded rings $$S_\bullet \to \Gamma_\bullet\widetilde{S_\bullet}$$ induces an isomorphism $$\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet} \cong \operatorname{Proj} S_\bullet$$, and under this isomorphism, the respective $$\mathscr{O}(1)$$'s are identified.

It could be easily seen that for homogeneous $$f \in S_n (n>0)$$, we have isomorphisms on $$D_+(f)$$ in these two projective schemes that glue nicely. However, I could not prove $$\Gamma_\bullet\widetilde{S_\bullet}$$ is finitely generated in degree 1, thus I could not show $$\mathscr{O}(1)$$ on $$\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet}$$ is an invertible sheaf. Nor could I prove this map of graded rings induces a map of projective schemes defined on all of $$\operatorname{Proj} \Gamma_\bullet\widetilde{S_\bullet}$$, since a general map of graded rings only induce a morphism on an open subscheme. How could I prove these two statements?

• I've made a number of typographical improvements to your post, like using $\operatorname{Proj} R$ to produce $\operatorname{Proj} R$ instead of the \mathrm and \space version you used before, and using \bullet for the subscript instead of a period. Please also note that since Vakil's text is still in preparation with new versions coming all the time, it's helpful to add a link to the version you're using in the post - the first time I went to look at the surrounding material, I got an older version which didn't even have an exercise 15.6.G. Commented Oct 26, 2023 at 23:48
• Sorry, I had an answer up but it fell short and I couldn't really find a good way to fix it. Your concern about showing that $V(\phi(S_+))=\emptyset$ is good and valid, and I can't think of a way to show that without quite a bit of work, some of which is done later (or not at all) in the text. See Hartshorne theorem II.5.19 and exercise II.5.9 for the result in the case when $S_0$ is a field, but it's not exactly easy - it requires most of the content of Serre's FAC to do this. Commented Oct 30, 2023 at 6:17