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

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?

## 1 Answer

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 $$\Gamma(X,\mathcal{O}_X)$$.