Throughout, suppose $S$ is a graded ring which is finitely generated by $S_{1}$ and an $S_{0}$-algebra. Let $X = \text{Proj} S$. There is the usual associated graded module given by $$ \Gamma_{\bullet}(\mathcal{O}_{X}) = \bigoplus_{d \in \mathbb{Z}} \Gamma(X, \mathcal{O}_{X}(d)). $$ My question is about what this graded module is in various cases. It is well known that if $S = A[x_{0}, x_{1}, \ldots , x_{r}]$, then $\Gamma_{\bullet}(\mathcal{O}_{X}) \simeq S$. In particular, this is true even if $A$ is not an integral domain. A standard proof of this is given in Hartshorne Proposition 5.13.

I was studying the proof of this, and noticed that the crucial step in that proof is observing that $\Gamma_{\bullet}(\mathcal{O}_{X})$ can be identified with the intersection $$ \bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}}. $$ This observation, along with the inclusion in the above math environment, depends only on the $x_{i}$ not being zero divisors.

This leads me to think that this proof admits a significant generalisation.

Let $S$ be a graded ring with $S_{0} = A$ an arbitrary commutative ring with identity and suppose $S$ is finitely generated by $S_{1}$ as an $A$-algebra. Suppose further that a family of generators $x_{0}, x_{1}, \ldots , x_{r}$ can be chosen so that none of the $x_{i}$ are zero-divisors. Is it still true that $$ \Gamma_{\bullet}(\mathcal{O}_{X}) \simeq \bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}} ? $$

An immediate followup question would be to ask whether the above question is even well-posed. In particular, if I chose a different family of generators $y_{0}, y_{1}, \ldots y_{q}$, with possible algebraic dependencies, does the same fact remain? Of course one would hope so.

Finally, what is the most general setting in which this holds. In particular, what adjectives do I need to add to the statement "$S$ is a graded ring finitely generated by $S_{1}$ as an $S_{0}$-algebra" in order for $$ \Gamma_{\bullet}(\mathcal{O}_{X}) \simeq \bigcap S_{x_{i}} \subseteq S_{x_{0}x_{1} \cdots x_{r}} $$ to hold?


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