Do we have $\mathit\Gamma_*(\mathcal O_X)\cong S$?

Question

In Hartshorne Proposition 5.13, the author says $r\ge 1$, but I think if $r=0$, the following proposition also holds, doesn't it?

Let $A$ be a ring, let $S=A[x_0]$, and let $X=\operatorname{Proj}S$, then $\mathit\Gamma_*(\mathcal O_X)\cong S$.

$\mathit\Gamma_*(\mathcal O_{\operatorname{Proj}S})=A[x_0]_{x_0}\neq A[x_0]=S$.

Answer 1

Take a look at Proposition 5.15. It states

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}\ S$, and let $\mathcal F$ be a quasi-coherent sheaf on $X$. Then there is a natural isomorphism ${\Gamma_*(\mathcal F)}^\sim \simeq \mathcal F$.

I believe your case fits the criteria ($S_0 = A$, $S_1 = (x_0)A$), so there is a natural isomorphism $\Gamma_*(\mathcal O_X)^\sim \simeq \mathcal O_X$.

edit

Actually, and correct me if I'm wrong, but since we define $\text{Proj} \ S$ as the set of homogeneous prime ideals in $S= A[x]$ that do not contain $(x)$ we have only the prime ideals of $A$ left, so $\text{Proj} \ S = \text{Spec} \ A$.