Plus closure/Two-dimensional/Graded/Torsor/Introduction/Section

Let ${}K$ be a field and let ${}R$ be a normal two-dimensional standard-graded domain over ${}K$ with corresponding smooth projective curve ${}C$. A homogeneous ${}{\mathfrak {m}}$-primary ideal with homogeneous ideal generators ${}f_{1},\ldots ,f_{n}$ and another homogeneous element ${}f$ of degree ${}m$ yield a cohomology class

${}c=\delta (f)=H^{1}(C,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))\,.$

Let ${}T(c)$ be the corresponding torsor. We have seen that the affineness of this torsor over ${}C$ is equivalent to the affineness of the corresponding torsor over ${}D({\mathfrak {m}})\subseteq \operatorname {Spec} {\left(R\right)}$ (and to the property of not belonging to the tight closure). Now we want to understand what the property ${}f\in I^{+}$ means for ${}c$ and for ${}T(c)$. Instead of the plus closure we will work with the graded plus closure ${}I^{+{\text{gr}}}$, where ${}f\in I^{+{\text{gr}}}$ holds if and only if there exists a finite graded extension ${}R\subseteq S$ such that ${}f\in IS$. The existence of such an ${}S$ translates into the existence of a finite morphism

$\varphi \colon C'=\operatorname {Proj} {\left(S\right)}\longrightarrow \operatorname {Proj} {\left(R\right)}=C$

such that ${}\varphi ^{*}(c)=0$. Here we may assume that ${}C'$ is also smooth. Therefore, we discuss the more general question when a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$, where ${}{\mathcal {S}}$ is a locally free sheaf on ${}C$, can be annihilated by a finite morphism

$C'\longrightarrow C$

of smooth projective curves. The advantage of this more general approach is that we may work with short exact sequences (in particular, the sequences coming from the Harder-Narasimhan filtration) in order to reduce the problem to semistable bundles which do not necessarily come from an ideal situation.

Lemma

Let ${}C$ denote a smooth projective curve over an algebraically closed 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 the following conditions are equivalent.

1. There exists a finite morphism

   $\varphi \colon C'\longrightarrow C$

   from a smooth projective curve ${}C'$ such that ${}\varphi ^{*}(c)=0$.

2. There exists a projective curve ${}Z\subseteq T$.

Proof

If (1) holds, then the pull-back ${}\varphi ^{*}(T)=T\times _{C}C'$ is trivial (as a torsor), as it equals the torsor given by ${}\varphi ^{*}(c)=0$. Hence ${}\varphi ^{*}(T)$ is isomorphic to a vector bundle and contains in particular a copy of ${}C'$. The image ${}Z$ of this copy is a projective curve inside ${}T$.

If (2) holds, then let ${}C'$ be the normalization of ${}Z$. Since ${}Z$ dominates ${}C$, the resulting morphism

$\varphi \colon C'\longrightarrow C$

is finite. Since this morphism factors through ${}T$ and since ${}T$ annihilates the cohomology class by which it is defined, it follows that

${}\varphi ^{*}(c)=0$.

$\Box$

We want to show that the cohomological criterion for (non)-affineness of a torsor along the Harder-Narasimhan filtration of the vector bundle also holds for the existence of projective curves inside the torsor, under the condition that the projective curve is defined over a finite field. This implies that tight closure is (graded) plus closure for graded ${}{\mathfrak {m}}$-primary ideals in a two-dimensional graded domain over a finite field.