I have questions related to Hartshorne's Exercise II.5.14, set up as follows: Let $X$ be a connected normal closed subscheme of $\mathbb{P}^r_k$, where $k$ is an algebraically closed field, and write $S$ for the homogeneous coordinate ring of $X$. Also take $S' = \oplus_{n \ge 0} \Gamma(X,\mathcal{O}_X(n))$ and view it as a graded ring.
I'm trying to show $S$ is a domain and $S'$ is its integral closure. So far, I've proven $X$ is an integral scheme. My current questions:
It follows from the text (namely Exercise 3.12(b) and Corollary 5.16(a)) that $X$ can be identified with Proj $S$. If $S$ is a domain, surely Proj $S$ is integral; is the converse true?
Hartshorne's suggestion is to view $S'$ as $\Gamma(X,\mathcal{F})$ for a sheaf of rings $\mathcal{F} = \bigoplus_n \mathcal{O}_X(n)$, and in fact show $\mathcal{F}$ is a sheaf of integrally closed domains. Can anyone provide a hint on how this might be done?
To show $S$ is a domain, it suffices to show that $I_X$ is prime, which is equivalent to showing that $X$ is irreducible. Note that $X$ is reduced. Else then some local ring $\mathcal{O}_{x, X}$ contains nilpotents and then $\mathcal{O}_{x, X}$ is not integrally closed, since it is not an integral domain. If $X$ were reducible, then some point $x$ would be contained in two irreducible components, so the local ring at the point would have zero divisors. So since $X$ is normal, $X$ is irreducible and $S$ is a domain.
Consider the sheaf$$\mathscr{L} = \bigoplus_{n \ge 0} \mathcal{O}_X(n).$$Then$$\mathscr{L}_\mathfrak{p} = \bigoplus_{n \ge 0} S(n)_{(\mathfrak{p})} = \left\{ {s\over f} \in S_\mathfrak{p} : \deg s \ge \deg f\right\}.$$Any element integral over $\mathscr{L}_\mathfrak{p}$ is of course integral over $S_\mathfrak{p}$, and thus is in $S_\mathfrak{p}$ since $X$ is normal. However, nothing with total negative degree can be integral over $\mathscr{L}_\mathfrak{p}$, so $\mathscr{L}_\mathfrak{p}$ is integrally closed. Thus,$$\Gamma(X, \mathscr{L}) = \bigoplus_{n \ge 0} \Gamma(X, \mathcal{O}_X(n)) = S'$$is integrally closed. $S'$ is contained in the integral closure of $S$ by pages 122-123 of Hartshorne, so $S'$ is the integral closure of $S$.
The ring is an algebraically closed field in this exercise, so it is basically the usual case. Perhaps that sentence should say "$I_X$ is prime if and only if $X$ is irreducible and reduced," but I handle reducibility in the next sentences, so I think my solution is fine as is.