Vector bundles and tight closure (Triest 2023)/Lecture 3/latex
\setcounter{section}{3}
\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.
\inputfaktbeweis
{Projective curve/Vector bundle/Cohomology class/Finite annihilation and torsor/Fact}
{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}{.}}
\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.
\inputfakt
{Finite field/Smooth projective curve/Vector bundel/Etale trivializable and Frobenius periodicity/Fact}
{Lemma}
{}
{
\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 {}
\inputfaktbeweis
{Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree 0/Trivializable/Fact}
{Theorem}
{}
{
\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.
\inputfaktbeweis
{Finite field/Smooth projective curve/Vector bundle/Strongly semistable degree nonnegative/Finite annihilation/Fact}
{Theorem}
{}
{
\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} {} {.}
\inputfaktbeweis
{Finite field/Smooth projective curve/Vector bundle/Finite annihilation/Harder-Narasimhan-criterion/Fact}
{Theorem}
{}
{
\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.
\inputfaktbeweis
{Finite field/Smooth projective curve/Vector bundle/Affineness and nonexistence of curves/Fact}
{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 {}
}
{
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.
\inputfakt
{Tight closure/Plus closure/Two-dimensional/Standard-graded/Fact}
{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 {}
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} {} {.}