Projective space $Proj$

We let $$S$$ be a graded algebra. I have a 2 questions regarding the Proj construction.

1. It seems to me that we do not know what $$O_{Proj S}Proj(S))$$ is ?
2. How is the map $$S_0 \rightarrow \Gamma(Proj S, O_{Proj(S)})$$ induced?
• 1. What happens when you try to prove that it's $S_0$? 2. Did you look at the proof of 01M3? – KReiser May 13 at 19:54
• Sorry which statemnt is 01M3? My understanding for 2. is $S_0$ yields a map to each $S_{(f)}$ hence induces a map to global section. My problem with 1. is I haven't found an explicit example that computes the global section ring. – CL. May 14 at 16:21
• My apologies, somehow I copied the wrong thing there. It should have been part 6 of 01M7. There are many examples on this website of computing the global sections of $\Bbb P^n_R=\operatorname{Proj} R[x_0,\cdots,x_n]$, such as this one. And for most reasonable rings, the answer will just be $R_0$. ... – KReiser May 14 at 22:14
• ... On the other hand, the treatment in the stacksproject is very short on assumptions - it does not assume $R$ noetherian, generated as an $R_0$ algebra by elements of degree 1, etc. So it would be instructive to see what happens and if there are any differences when one uses as few assumptions as they do. – KReiser May 14 at 22:14