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?


1 Answer 1


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.

Note that you can produce a lot of examples this way: any projective variety embedded by a noncomplete linear series (i.e., by some proper subspace of global sections) will give rise to such an example.

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.

For (2), exercise II.5.14 of Hartshorne's Algebraic Geometry sketches a proof of this fact; if you need further help on any of the steps let me know and I can fill it in here.


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