Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 9/latex
\setcounter{section}{9}
\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) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
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
\maabbdisp {\varphi} {C'= \operatorname{Proj} { \left( S \right) } } { \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
\maabbdisp {} {C'} {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.
\inputfaktproof
{Projektive Kurve/Vektorbündel/Kohomologieklasse/Endliche Annulierung und Kurven im Torsor/Fakt/en}
{Lemma}
{}
{
\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}{.} Then the following conditions are equivalent.}
\faktfolgerung {\aufzaehlungzwei {There exists a finite morphism
\maabbdisp {\varphi} {C'} {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 {}
\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
\maabbdisp {\varphi} {C'} {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
\maabb {\varphi} {C'} {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.
\inputfakt{Endlicher Körper/Glatte projektive Kurve/Vektorbündel/Etale trivialisierbar und Frobenius Periodizität/Fakt/en}{Lemma}{}
{
\faktsituation {Let $K$ denote a
\definitionsverweis {finite field}{}{}
\zusatzklammer {or the algebraic closure of a finite field} {} {}
and let $C$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a locally free sheaf over $C$.}
\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 {}
\faktzusatz {}
}
\inputfaktproof
{Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Stark semistabil Grad 0/Trivialisierbar/Fakt/en}
{Theorem}
{}
{
\faktsituation {Let $K$ denote a
\definitionsverweis {finite field}{}{}
\zusatzklammer {or the algebraic closure of a finite field} {} {}
and let $C$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a strongly semistable locally free sheaf over $C$}
\faktvoraussetzung {of degree $0$.}
\faktfolgerung {Then there exists a finite morphism
\maabbdisp {\varphi} {C' } {C
} {} such that
\mathl{\varphi^*({\mathcal S})}{} is trivial.}
\faktzusatz {}
\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 9.2
the bundle
\mathl{F^{e*}({\mathcal S})}{} admits an \'{e}tale trivialization
\maabb {\varphi} {C'} {C
} {.}
Hence the finite map
\mathl{F^{e} \circ \varphi}{} trivializes the bundle.
\inputfaktproof
{Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Stark semistabil Grad nichtnegativ/Endliche Annulation/Fakt/en}
{Theorem}
{}
{
\faktsituation {Let $K$ denote a
\definitionsverweis {finite field}{}{}
\zusatzklammer {or the algebraic closure of a finite field} {} {}
and let $C$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a strongly semistable locally free sheaf over $C$}
\faktvoraussetzung {of nonnegative degree and let
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(C, {\mathcal S} )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote a cohomology class.}
\faktfolgerung {Then there exists a finite morphism
\maabbdisp {\varphi} {C' } {C
} {}
such that
\mathl{\varphi^*(c)}{} is trivial.}
\faktzusatz {}
\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(C, 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 9.3
a finite morphism which trivializes the bundle. So we may assume that
\mavergleichskette
{\vergleichskette
{ {\mathcal S}
}
{ \cong }{ {\mathcal O}_{ C }^r
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then the cohomology class has several components
\mavergleichskette
{\vergleichskette
{ c_i
}
{ \in }{ H^1( C, {\mathcal O}_{ C } )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
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} {} {.}
\inputfaktproof
{Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Endliche Annulation/Harder-Narasimhan-Kriterium/Fakt/en}
{Theorem}
{}
{
\faktsituation {Let $K$ denote a
\definitionsverweis {finite field}{}{}
\zusatzklammer {or the algebraic closure of a finite field} {} {}
and let $C$ be a smooth projective curve over $K$. Let ${\mathcal S}$ be a locally free sheaf over $C$ and let
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(C, {\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
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then the following are equivalent.
\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(C, {\mathcal Q} )}{} is $0$.
}}
\faktzusatz {}
\faktzusatz {}
}
{
Suppose that (1) holds. Then the torsor is not affine and hence by Theorem 8.7 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(C, {\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 of the right hand side has a nonnegative degree. Therefore the image of $c_i$ in
\mathl{H^1(C, {\mathcal S}_{i}/ {\mathcal S}_{i-1})}{} can be annihilated by a finite morphism due to
Theorem 9.4.
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(C,{\mathcal S}_{i-1} )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Going on inductively we see that $c$ can be annihilated by a finite morphism.
\inputfaktproof
{Endlicher Körper/Projektive glatte Kurve/Vektorbündel/Affinität und Nichtexistenz von Kurven/Fakt/en}
{Theorem}
{}
{
\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 {}
\faktzusatz {}
}
{
Due to Theorem 8.7 and Theorem 9.5, for both properties the same numerical criterion does hold.
These results imply the following theorem in the setting of a two-dimensional graded ring.
\inputfakt{Tight closure/Plus closure/2 dim, standard-graded/Brenner/Fakt/en}{Theorem}{}
{
\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 {}
\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} {} {.} Pdf-version