Plus closure/Two-dimensional/Graded/Torsor/General case/Section

We look now at an arbitrary locally free sheaf ${}{\mathcal {S}}$ on ${}C$, a smooth projective curve over a finite field. We want to show that the same numerical criterion (formulated in terms of the Harder-Narasimhan filtration) for non-affineness of a torsor holds also for the finite annihilation of the corresponding cohomomology class (or the existence of a projective curve inside the torsor).

Theorem

Let ${}K$ denote a finite field (or the algebraic closure of a finite field) and let ${}X$ be a smooth projective curve over ${}K$. Let ${}{\mathcal {S}}$ be a locally free sheaf over ${}X$ and let ${}c\in H^{1}(X,{\mathcal {S}})$ denote a cohomology class. Let ${}{\mathcal {S}}_{1}\subset \ldots \subset {\mathcal {S}}_{t}$ be a strong Harder-Narasimhan filtration of ${}F^{e*}({\mathcal {S}})$. We choose ${}i$ such that ${}{\mathcal {S}}_{i}/{\mathcal {S}}_{i-1}$ has degree ${}\geq 0$ and that ${}{\mathcal {S}}_{i+1}/{\mathcal {S}}_{i}$ has degree ${}<0$. We set

${}{\mathcal {Q}}=F^{e*}({\mathcal {S}})/{\mathcal {S}}_{i}$. Then the following are equivalent.

1. The class ${}c$ can be annihilated by a finite morphism.
2. Some Frobenius power of the image of ${}F^{e*}(c)$ inside ${}H^{1}(X,{\mathcal {Q}})$ is ${}0$.

Proof

Suppose that (1) holds. Then the torsor is not affine and hence by fact also (2) holds.

So suppose that (2) is true. By applying a certain power of the Frobenius, we may assume that the image of the cohomology class in ${}{\mathcal {Q}}$ is ${}0$. Hence the class stems from a cohomology class ${}c_{i}\in H^{1}(X,{\mathcal {S}}_{i})$. We look at the short exact sequence

$0\longrightarrow {\mathcal {S}}_{i-1}\longrightarrow {\mathcal {S}}_{i}\longrightarrow {\mathcal {S}}_{i}/{\mathcal {S}}_{i-1}\longrightarrow 0,$

where the sheaf on the right hand side has a nonnegative degree. Therefore the image of ${}c_{i}$ in ${}H^{1}(X,{\mathcal {S}}_{i}/{\mathcal {S}}_{i-1})$ can be annihilated by a finite morphism due to fact. Hence, after applying a finite morphism, we may assume that ${}c_{i}$ stems from a cohomology class ${}c_{i-1}\in H^{1}(X,{\mathcal {S}}_{i-1})$. Going on inductively we see that ${}c$ can be annihilated by a finite morphism.

$\Box$

Theorem

Let ${}C$ denote a smooth projective curve over the algebraic closure of a finite field ${}K$, let ${}{\mathcal {S}}$ be a locally free sheaf on ${}C$ and let ${}c\in H^{1}(C,{\mathcal {S}})$

be a cohomology class with corresponding torsor ${}T\rightarrow C$. Then ${}T$ is affine if and only if it does not contain any projective curve.

Proof

Due to fact and fact, for both properties the same numerical criterion does hold.

$\Box$

These results imply the following theorem in the setting of a two-dimensional graded ring.

Theorem

Let ${}R$ be a standard-graded, two-dimensional normal domain over (the algebraic closure of) a finite field. Let ${}I$ be an ${}R_{+}$-primary graded ideal. Then

${}I^{*}=I^{+}\,.$

This is also true for non-primary graded ideals and also for submodules in finitely generated graded submodules. Moreover, G. Dietz has shown that one can get rid also of the graded assumption (of the ideal or module, but not of the ring).