# How to see $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$ as a ring?

Let $S$ be a finitely generated graded $A$-algebra. For each homogeneous $f\in S_+$, we have a scheme structure $D(f)\cong \operatorname{Spec} S_{(f)}$ where $S_{(f)}$ denotes the zeroth piece of the graded localization ring $S_f$. These glue well and give rise to a scheme structure on $\operatorname{S}$.

Now my question is how to understand the ring $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$? In particular, if $S = k[x_1,\cdots, x_n]$, then what is the ring $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$?

As an exercise, you might try to prove in your specific example what you already know from the classical formulation, namely, that the global sections of the structure sheaf on projective space over a field is just $k$. (The only global regular functions on projective space are constant.)
• I wanted to find a natural structure morphism $A\to \Gamma(\mathscr{O}_S,\operatorname{Proj} S)$, so perhaps I can show that the maps $A\to \Gamma(\mathscr{O}_S,D(f))$ are compatible thus lifts to a morphism $A\to \Gamma(\mathscr{O}_S,\operatorname{Proj} S)$. In this way I could create the map while avoiding the specific structure of $\Gamma(\mathscr{O}_S,\operatorname{Proj} S)$. And that gives me a morphism $\operatorname{Proj}S\to \operatorname{Spec} A$. Is it correct? – mez Mar 1 '15 at 21:00