Let $S=\bigoplus_{d\geq 0}S_d$ denote a graded algebra over a field $k$ (say S integral, finitely generated, and $k$ algebraically closed). Under which conditions does $\mathcal{O}(Proj(S))\simeq S_0$ hold?
I think this question is related to properties of the morphism $Proj(S)\to Spec(S_0)$ (because if it has connected fibres, the isomorphism holds).
My question is motivated by Geometric Invariant Theory. There, a quotient $X^{sst}(E)\to Y$ is given as $Y=Proj(\bigoplus_{d\geq 0} H^0(X,E^d)^G)$, where $H^0(X,E^0)^G=\mathcal{O}(X)^G$. If $X$ is normal, and the unstable locus is of codimension two, we get $\mathcal{O}(Y)\simeq \mathcal{O}(X^{sst}(E))\simeq \mathcal{O}(X)^G.$ But it seems more natural to consider this question in an abstract formulation (?).