Global sections of $\operatorname{Proj} S$ when $S$ is an integrally closed domain

Question

Let $S=\bigoplus_{n \geqslant 0}S_n$ be a finitely generated $\mathbb{Z}_{\geqslant 0}$-graded algebra over $\mathbb{C}$ with $S_+ \neq 0$, here $S_+:=\bigoplus_{n>0}S_n$. Let us also assume that $S$ is an integrally closed domain. We can consider $X:=\operatorname{Proj}S$. Is it true that the natural morphism $S_0 \rightarrow \Gamma(X,\mathcal{O}_X)$ is an isomorphism?

Answer 1

Yes.

Injectivity follows from the fact that $S$ is a domain: if $s\in S_0$ is in the kernel of $S_0\to\Gamma(X,\mathcal{O}_X)$ then it's also in the kernel of $S_0\to\Gamma(X,\mathcal{O}_X)\to\Gamma(D(f),\mathcal{O}_X)$ for any nonzero homogeneous $f$ of positive degree. But this map is $S_0\to S_{(f)}$ by $s\mapsto \frac{s}{1}$, so if $\frac{s}{1}=\frac{0}{1}$ then $f^ns=0$, so $s=0$.

Surjectivity is slightly more interesting. Up to replacing $S$ by $S^{(d)}$, which preserves $\Gamma(X,\mathcal{O}_X)$, finite generation, and normality, we may assume $S$ is finitely generated as an $S_0$-algebra by $S_1$. Then $S$ is the coordinate ring of some closed subscheme $X\subset\Bbb P^n_{S_0}$, and $X$ is projectively normal by definition. On the other hand, for any projectively normal closed subscheme of $\Bbb P^n_A$ and any $d\geq 0$, the map $\Gamma(\Bbb P^n_A,\mathcal{O}_{\Bbb P^n_A}(d))\to\Gamma(X,\mathcal{O}_{X}(d))$ is surjective (cf Hartshorne chapter II section 5 exercise 14, for instance). Applying this to our situation, we have that $S_0=\Gamma(\Bbb P^n_{S_0},\mathcal{O}_{\Bbb P^n_{S_0}})$ surjects on to $\Gamma(X,\mathcal{O}_X)$.