I have been stuck on the proof of the following statement for a while now.
Let $S$ be a graded noetherian domain which is finitely generated by $S_{1}$ as an $A$-algebra where $A = S_{0}$ is a finitely generated domain over a field $k$. Then $\Gamma(X, \mathcal{O}_{X}(n))$ is a finitely generated $A$-module for all $n$, where $X = \text{Proj }S$.
The result is a step in the proof of Theorem 5.19 in Hartshorne. There are a few things I am confused about. Throughout, define $X = \text{Proj} S$. He defines $$ S' = \bigoplus_{n \geq 0} \Gamma(X, \mathcal{O}_{X}), $$ and claims that S' contains $S$, and is contained in the intersection $$ \bigcap S_{x_{i}} $$ where the intersection is taken inside $S_{x_{0}x_{1} \cdots x_{r}}$. He then goes on to refer to "the quotient field of $S$ and "the quotient field of S' ". If $S'$ is contained inside a localisation of $S$, then shouldn't their quotient fields be the same?
He also argues that $S'$ is integral over $S$. I think I understand the proof of that fact, but then at the end he says "thus $S'$ is contained in the integral closure of $S$ in its quotient field". Assuming for a moment that I was mistaken above, and the quotient fields are not the same, what is "it" here. Based on what has been proven I thought $its$ referred to $S$, but given my other confusion I am not so sure.
This is all building up to apply a result form commutative algebra, which Hartshorne quotes as
Let $A$ be an integral domain which is finitely generated over a field $k$. Let $K$ be the quotient field of $A$, and let $L$ be a finite algebraic extension of $K$. Then the integral closure $B$ of $A$ in $L$ is a finitely generated $A$-module, and is also a finitely generated $k$-algebra.
How exactly is he applying this? Is he just taking the case where $K=L$?