# 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 $c\in H^{1}(C,{\mathcal {S}})$
$T\rightarrow C$
* There exists a finite morphism from a smooth projective curve $C'$ such that $\varphi ^{*}(c)=0$.
    $\varphi \colon C'\longrightarrow C$
* 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
    $\varphi \colon C'\longrightarrow C$
$T$
$T$
$\varphi ^{*}(c)=0$.
$\Box$
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.