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?