# Vector bundles and tight closure (Triest 2023)/Lecture 3/latex

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\zwischenueberschrift{Plus closure}
For an ideal \mavergleichskette {\vergleichskette { I } { \subseteq }{ R } { }{ } { }{ } { }{ } } in a domain $R$ define its \definitionswort {plus closure} by \mavergleichskettedisp {\vergleichskette { I^+ } { =} { { \left\{ f \in R \mid \text{there exists a finite domain extension } R \subseteq T \text{ such that } f \in IT \right\} } } { } { } { } { } { } { } }{.} Equivalent: Let $R^+$ be the \betonung{absolute integral closure} of $R$. This is the integral closure of $R$ in an algebraic closure of the quotient field $Q(R)$ \zusatzklammer {first considered by Artin \cite{artinjoints}} {.} Then \mathdisp {f \in I^+ \text{ if and only if } f \in IR^+} { . } The plus closure commutes with localization.
We also have the inclusion \mavergleichskette {\vergleichskette { I^+ } { \subseteq }{ I^* } { }{ } { }{ } { }{ } }{.} Here the question arises:
Question: Is \mavergleichskette {\vergleichskette { I^+ } { = }{ I^* } { }{ } { }{ } { }{ } }{?}
This question is known as the \betonung{tantalizing question} in tight closure theory.
In terms of forcing algebras and their torsors, the containment inside the plus closure means that there exists a $d$-dimensional closed subscheme inside the torsor which meets the exceptional fiber \zusatzklammer {the fiber over the maximal ideal} in isolated points, and this means that the so-called superheight of the extended ideal is $d$. In this case the local cohomological dimension of the torsor must be $d$ as well, since it contains a closed subscheme with this cohomological dimension. So also the plus closure depends only on the torsor.
In characteristic zero, the plus closure behaves very differently compared with positive characteristic. If $R$ is a normal domain of characteristic $0$, then the trace map shows that the plus closure is trivial, \mavergleichskette {\vergleichskette { I^+ } { = }{ I } { }{ } { }{ } { }{ } } for every ideal $I$.
\zwischenueberschrift{Plus closure in dimension two}
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 \mathl{f_1 , \ldots , f_n} and another homogeneous element $f$ of degree $m$ yield a cohomology class \mavergleichskettedisp {\vergleichskette { c } { =} { \delta(f) } { =} { H^1(C, \operatorname{Syz} { \left(f_1 , \ldots , f_n \right) } (m)) } { } { } { } { } }{.} Let \mathl{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 \mavergleichskette {\vergleichskette { D( {\mathfrak m} ) } { \subseteq }{ \operatorname{Spec} { \left( R \right) } } { }{ } { }{ } { }{ } } \zusatzklammer {and to the property of not belonging to the tight closure} {.} Now we want to understand what the property \mavergleichskette {\vergleichskette { f } { \in }{ I^+ } { }{ } { }{ } { }{ } } means for $c$ and for \mathl{T(c)}{.} Instead of the plus closure we will work with the \stichwort {graded plus closure} \mathl{I^{+ \text{gr} }}{,} where \mavergleichskette {\vergleichskette { f } { \in }{ I^{+ \text{gr} } } { }{ } { }{ } { }{ } } holds if and only if there exists a finite graded extension \mavergleichskette {\vergleichskette { R } { \subseteq }{ S } { }{ } { }{ } { }{ } } such that \mavergleichskette {\vergleichskette { f } { \in }{ IS } { }{ } { }{ } { }{ } }{.} The existence of such an $S$ translates into the existence of a finite morphism \mathdisp {\varphi \colon C'= \operatorname{Proj} { \left( S \right) } \longrightarrow \operatorname{Proj} { \left( R \right) } = C} { } such that \mavergleichskette {\vergleichskette { \varphi^*(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} Here we may assume that $C'$ is also smooth. Therefore, we discuss the more general question when a cohomology class \mavergleichskette {\vergleichskette { c } { \in }{ H^1(C, {\mathcal S} ) } { }{ } { }{ } { }{ } }{,} where ${\mathcal S}$ is a locally free sheaf on $C$, can be annihilated by a finite morphism \mathdisp {C' \longrightarrow C} { } of smooth projective curves. The advantage of this more general approach is that we may work with short exact sequences \zusatzklammer {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.
\faktsituation {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 \mavergleichskette {\vergleichskette { c } { \in }{ H^1(C, {\mathcal S}) } { }{ } { }{ } { }{ } } be a cohomology class with corresponding torsor \mathl{T \rightarrow C}{.}} \faktuebergang {Then the following conditions are equivalent.} \faktfolgerung {\aufzaehlungzwei {There exists a finite morphism \mathdisp {\varphi \colon C' \longrightarrow C} { } from a smooth projective curve $C'$ such that \mavergleichskette {\vergleichskette { \varphi^*(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} } {There exists a projective curve \mavergleichskette {\vergleichskette { Z } { \subseteq }{ T } { }{ } { }{ } { }{ } }{.} }} \faktzusatz } {
If (1) holds, then the pull-back \mavergleichskette {\vergleichskette { \varphi^*(T) } { = }{ T \times_C C' } { }{ } { }{ } { }{ } } is trivial \zusatzklammer {as a torsor} {,} as it equals the torsor given by \mavergleichskette {\vergleichskette {\varphi^*(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} Hence \mathl{\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 \mathdisp {\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 \mavergleichskette {\vergleichskette { \varphi^*(c) } { = }{ 0 } { }{ } { }{ } { }{ } }{.}
We want to show that the cohomological criterion for \zusatzklammer {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 \zusatzklammer {graded} plus closure for graded ${\mathfrak m}$-primary ideals in a two-dimensional graded domain over a finite field.
\zwischenueberschrift{Annihilation of cohomology classes of strongly semistable sheaves}
We deal first with the situation of a strongly semistable sheaf ${\mathcal S}$ of degree $0$. The following two results are due to Lange and Stuhler \cite{langestuhler}. We say that a locally free sheaf is \stichwort {\'{e}tale trivializable} if there exists a finite \'{e}tale morphism $\varphi \colon C' \rightarrow C$ such that \mavergleichskette {\vergleichskette { \varphi^*( {\mathcal S} ) } { \cong }{ {\mathcal O}_{ C' }^r } { }{ } { }{ } { }{ } }{.} Such bundles are directly related to linear representations of the \'{e}tale fundamental group.
\faktsituation {Let $K$ denote a finite field \zusatzklammer {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 $X$.} \faktfolgerung {Then ${\mathcal S}$ is \'{e}tale trivializable if and only if there exists some $n$ such that \mavergleichskette {\vergleichskette { F^{n*} {\mathcal S} } { \cong }{ {\mathcal S} } { }{ } { }{ } { }{ } }{.}} \faktzusatz
\faktsituation {Let $K$ denote a finite field \zusatzklammer {or the algebraic closure of a finite field} and let $X$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a strongly semistable locally free sheaf over $X$} \faktvoraussetzung {of degree $0$.} \faktfolgerung {Then there exists a finite morphism \mathdisp {\varphi \colon Y \longrightarrow X} { } such that \mathl{\varphi^*( {\mathcal S} )} is trivial.} \faktzusatz } {
We consider the family of locally free sheaves \mathbed {F^{e*}( {\mathcal S} )} {e \in \N} {.} Because these are all semistable of degree $0$, and defined over the same finite field, we must have \zusatzklammer {by the existence of the moduli space for vector bundles} a repetition, i.e. \mavergleichskettedisp {\vergleichskette { F^{e*}( {\mathcal S} ) } { \cong} { F^{e'*}( {\mathcal S} ) } { } { } { } { } { } { } } for some \mavergleichskette {\vergleichskette { e' } { > }{ e } { }{ } { }{ } { }{ } }{.} By Lemma 3.2 , the bundle \mathl{F^{e*}( {\mathcal S} )} admits an \'{e}tale trivialization $\varphi \colon Y \rightarrow X$. Hence the finite map \mathl{F^{e} \circ \varphi} trivializes the bundle.
\faktsituation {Let $K$ denote a finite field \zusatzklammer {or the algebraic closure of a finite field} and let $X$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a strongly semistable locally free sheaf over $X$ of nonnegative degree and let \mavergleichskette {\vergleichskette { c } { \in }{ H^1(X, {\mathcal S} ) } { }{ } { }{ } { }{ } } denote a cohomology class.} \faktfolgerung {Then there exists a finite morphism \mathdisp {\varphi \colon Y \longrightarrow X} { } such that \mathl{\varphi^*(c)} is trivial.} \faktzusatz } {
If the degree of ${\mathcal S}$ is positive, then a Frobenius pull-back \mathl{F^{e*}( {\mathcal S} )} has arbitrary large degree and is still semistable. By Serre duality we get that \mavergleichskette {\vergleichskette { H^1(X, F^{e*}( {\mathcal S} )) } { = }{ 0 } { }{ } { }{ } { }{ } }{.} So in this case we can annihilate the class by an iteration of the Frobenius alone.
So suppose that the degree is $0$. Then there exists by Theorem 3.3
a finite morphism which trivializes the bundle. So we may assume that \mavergleichskette {\vergleichskette { {\mathcal S} } { \cong }{ {\mathcal O}_{ X }^r } { }{ } { }{ } { }{ } }{.} Then the cohomology class has several components \mavergleichskette {\vergleichskette { c_i } { \in }{ H^1( X, {\mathcal O}_{ X } ) } { }{ } { }{ } { }{ } } and it is enough to annihilate them separately by finite morphisms. But this is possible by the parameter theorem of K. Smith \cite{smithparameter} \zusatzklammer {or directly using Frobenius and Artin-Schreier extensions} {.}
\zwischenueberschrift{The general case}
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 \zusatzklammer {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 \zusatzklammer {or the existence of a projective curve inside the torsor} {.}
\faktsituation {Let $K$ denote a finite field \zusatzklammer {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 \mavergleichskette {\vergleichskette { c } { \in }{ H^1(X, {\mathcal S} ) } { }{ } { }{ } { }{ } } denote a cohomology class. Let \mavergleichskette {\vergleichskette { {\mathcal S}_1 } { \subset }{ \ldots } { \subset }{ {\mathcal S}_t } { }{ } { }{ } } be a strong Harder-Narasimhan filtration of \mathl{F^{e*} ( {\mathcal S} )}{.} We choose $i$ such that \mathl{{\mathcal S}_{i} / {\mathcal S}_{i-1}} has degree $\geq 0$ and that \mathl{{\mathcal S}_{i+1} / {\mathcal S}_{i}} has degree $< 0$. We set \mavergleichskette {\vergleichskette { {\mathcal Q} } { = }{ F^{e*}({\mathcal S})/ {\mathcal S}_i } { }{ } { }{ } { }{ } }{.}} \faktuebergang {Then the following are equivalent.} \faktfolgerung {\aufzaehlungzwei {The class $c$ can be annihilated by a finite morphism. } {Some Frobenius power of the image of \mathl{F^{e*}(c)} inside \mathl{H^1(X, {\mathcal Q} )} is $0$. }} \faktzusatz } {
Suppose that (1) holds. Then the torsor is not affine and hence by Theorem 2.12
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 \mavergleichskette {\vergleichskette { c_i } { \in }{ H^1(X, {\mathcal S}_i) } { }{ } { }{ } { }{ } }{.} We look at the short exact sequence \mathdisp {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 \mathl{H^1(X , {\mathcal S}_{i} / {\mathcal S}_{i-1})} can be annihilated by a finite morphism due to Theorem 3.4 . Hence, after applying a finite morphism, we may assume that $c_i$ stems from a cohomology class \mavergleichskette {\vergleichskette { 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.
\faktsituation {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 \mavergleichskette {\vergleichskette { c } { \in }{ H^1(C, {\mathcal S}) } { }{ } { }{ } { }{ } } be a cohomology class with corresponding torsor \mathl{T \rightarrow C}{.}} \faktfolgerung {Then $T$ is affine if and only if it does not contain any projective curve.} \faktzusatz } {
Due to Theorem 2.12
and Theorem 3.5 , for both properties the same numerical criterion does hold.
These results imply the following theorem in the setting of a two-dimensional graded ring.
\faktsituation {Let $R$ be a standard-graded, two-dimensional normal domain over \zusatzklammer {the algebraic closure of} a finite field. Let $I$ be an $R_+$-primary graded ideal.} \faktfolgerung {Then \mavergleichskettedisp {\vergleichskette { I^* } { =} { I^+ } { } { } { } { } { } { } }{.}} \faktzusatz
This is also true for non-primary graded ideals and also for submodules in finitely generated graded submodules. Moreover, G. Dietz \cite{dietztight} has shown that one can get rid also of the graded assumption \zusatzklammer {of the ideal or module, but not of the ring} {.}