# $\mathrm{Proj}(A)$ and when does $A_1 = \Gamma(\mathcal{O}(1))$

Let $A=\bigoplus_{k \in \mathbb{Z}} A_k$ be a Noetherian $\mathbb{Z}$-graded commutative ring. Let $X=\mathrm{Proj}(A)$ and let $\mathcal{O}(1)$ be the sheaf on $X$ associated to the graded $A$-module $A(1)$ (whose degree $n$th term is $A_{n+1}$). We have a natural map $i:A_1 \to \Gamma(X, \mathcal{O}(1))$.

1) what is an example where $i$ is not an isomorphism?

2) A book I'm reading says that if $A$ is integrally closed in its total quotient ring, then $i$ is an isomorphism, under the hypothesis there exists a finite collection of elements $a_i \in A_1$ with $X = \bigcup_i D_+(a_i)$ and $A_+:=\bigoplus_{k \geq 1}A_k$ is generated by $A_1$. What is a proof or a reference?

For (1), take $A$ to be the homogeneous coordinate ring of the rational curve $C$ in $\mathbb P^3$ parametrized by $[s^4,s^3t,st^3,t^4]$ (this is the projection of the rational quartic in $\mathbb P^4$ to a hyperplane). One can check that $C$ is contained in no hyperplane, so that $A_1$ is $4$-dimensional (spanned by the images of the coordinate hyperplanes from the homogeneous coordinate ring of $\mathbb P^3$). However, we have that $\mathcal O_A(1)\cong \mathcal O_{\mathbb P^1}(4)$ (it has to be of the form $\mathcal O_{\mathbb P^1}(k)$, and just check that a general hyperplane in $\mathbb P^3$ meets $C$ in 4 points). Since $\dim H^0(\mathbb P^1, \mathcal O_{\mathbb P^1}(4))=5$, the map $A^1\to H^0(\mathbb P^1, \mathcal O_{\mathbb P^1}(4))$ cannot be surjective.
By the way, this gave an example where the map isn't surjective, but it also need not be injective either! One issue here is that any $A'$ differing by $A$ only in finitely many graded pieces determines the same scheme $X$ and sheaf $\mathcal O_X(1)$, so (for example) one can throw away or add elements of $A_1$ without changing $H^0(X,\mathcal O_X(1))$. For a more systematic approach to examining when the kernel and cokernel of $A^1\to H^0(X, \mathcal O_{X}(1))$, one can use local cohomology (at least when $A$ is $\mathbb N$-graded). For details see (for example) the appendix to Eisenbud's Geometry of Syzygies; essentially, the idea is that by choosing a surjection $k[x_0,\ldots,x_n]\to A$ one determines an embedding $\mathrm{Proj}\,A\hookrightarrow \mathbb P^{n}$; we then have an exact sequence of graded modules $$0\to H^0_{(x_0,\dots,x_n)}(A)\to A \to \bigoplus_d H^0(X,\mathcal O_X(d))\to H^1_{(x_0,\dots,x_n}(A)\to 0.$$ In particular, the obstacle to your map being an isomorphism lies in the first graded pieces of the initial and final terms of the sequence.