Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 6/latex
\setcounter{section}{6}
In the remaining lectures we will continue with the question when are the torsors given by a forcing algebras over a two-dimensional ring affine? We will look at the graded situation to be able to work on the corresponding projective curve.
In particular we want to address the following questions \aufzaehlungvier{Is there a procedure to decide whether the torsor is affine? }{Is it non-affine if and only if there exists a geometric reason for it not to be affine \zusatzklammer {because the superheight is too large} {} {?} }{How does the affineness vary in an arithmetic family, when we vary the prime characteristic? }{How does the affineness vary in a geometric family, when we vary the base ring? }
In terms of tight closure, these questions are directly related to the tantalizing question of tight closure \zusatzklammer {is it the same as plus closure} {} {,} the dependence of tight closure on the characteristic and the localization problem of tight closure.
\zwischenueberschrift{Geometric interpretation in dimension two}
We will restrict now to the two-dimensional homogeneous case in order to work on the corresponding projective curve. We want to find an object over the curve which corresponds to the forcing algebra.
Let $R$ be a two-dimensional standard-graded normal domain over an algebraically closed field $K$. Let
\mavergleichskette
{\vergleichskette
{ C
}
{ = }{ \operatorname{Proj} { \left( R \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be the corresponding smooth projective curve and let
\mavergleichskettedisp
{\vergleichskette
{ I
}
{ =} { { \left( f_1 , \ldots , f_n \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
be an $R_+$-primary homogeneous ideal with generators of degrees
\mathl{d_1 , \ldots , d_n}{.} Then we get on $C$ the short exact sequence
\mathdisp {0 \longrightarrow \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } (m) \longrightarrow \bigoplus_{i=1}^n{\mathcal O}_C(m-d_i) \stackrel{f_1, \ldots ,f_n}{\longrightarrow} {\mathcal O}_C (m) \longrightarrow 0} { . }
Here
\mathl{\operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } (m)}{} is a vector bundle, called the
\betonung{syzygy bundle}{,} of rank
\mathl{n-1}{} and of degree
\mathdisp {((n-1)m - \sum_{i=1}^n d_i) \operatorname{deg} \, (C)} { . }
Thus a homogeneous element $f$ of degree $m$ defines a cohomology class
\mathl{\delta(f) \in H^1(C, \operatorname{ Syz}_{ } ^{ } { \left( f_1 , \ldots , f_n \right) }(m) )}{,} so this defines a torsor over the projective curve. We mention an alternative description of the torsor corresponding to a first cohomology class in a locally free sheaf which is better suited for the projective situation.
\inputremark
{}
{
Let ${\mathcal S}$ denote a locally free sheaf on a scheme $X$. For a cohomology class
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(X, {\mathcal S})
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
one can construct a geometric object: Because of
\mavergleichskette
{\vergleichskette
{ H^1(X ,{\mathcal S})
}
{ \cong }{ \operatorname{Ext}^1( {\mathcal O}_{ X }, {\mathcal S})
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the class defines an extension
\mathdisp {0 \longrightarrow {\mathcal S} \longrightarrow { {\mathcal S}'} \longrightarrow {\mathcal O}_{ X } \longrightarrow 0} { . }
This extension is such that under the connecting homomorphism of cohomology,
\mavergleichskette
{\vergleichskette
{ 1
}
{ \in }{ \Gamma(X, {\mathcal O}_X)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is sent to
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(X, {\mathcal S})
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
The extension yields a projective subbundle
\mavergleichskettedisp
{\vergleichskette
{ {\mathbb P}({\mathcal S}^{\vee })
}
{ \subset} { {\mathbb P}({ {\mathcal S}'}^{\vee})
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
If $V$ is the corresponding geometric vector bundle of ${\mathcal S}$, one may think of
\mathl{{\mathbb P}({\mathcal S}^{\vee})}{} as
\mathl{{\mathbb P}(V)}{} which consists for every base point
\mavergleichskette
{\vergleichskette
{ x
}
{ \in }{ X
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
of all the lines in the fiber
\mathl{V_x}{} passing through the origin. The projective subbundle
\mathl{{\mathbb P}(V)}{} has codimension one inside
\mathl{{\mathbb P}(V')}{,} for every point it is a projective space lying
\zusatzklammer {linearly} {} {}
inside a projective space of one dimension higher. The complement is then over every point an affine space. One can show that the global complement
\mavergleichskettedisp
{\vergleichskette
{ T
}
{ =} { {\mathbb P}({ {\mathcal S} '}^{\vee}) \setminus {\mathbb P}({\mathcal S}^{\vee})
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is another model for the torsor given by the cohomology class. The advantage of this viewpoint is that we may work, in particular when $X$ is projective, in an entirely projective setting.
}
\zwischenueberschrift{Semistability of vector bundles}
In the situation of a forcing algebra for homogeneous elements, this torsor $T$ can also be obtained as
\mathl{\operatorname{Proj} \, B}{,} where $B$ is the (not necessarily positively) graded forcing algebra. In particular, it follows that the containment
\mathl{f \in I^*}{} is equivalent to the property that $T$ is not an affine variety. For this properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need the concept of
\zusatzklammer {Mumford} {-} {} semistability.
\inputdefinition
{}
{
Let ${\mathcal S}$ be a vector bundle on a smooth projective curve $C$. It is called \stichwort {semistable} {,} if
\mavergleichskette
{\vergleichskette
{ \mu( {\mathcal T} )
}
{ = }{ \frac{\deg( {\mathcal T} )}{ \operatorname{rk}( {\mathcal T} )}
}
{ \leq }{ \frac{\deg( {\mathcal S} )}{ \operatorname{rk}({\mathcal S} )}
}
{ = }{ \mu({\mathcal S} )
}
{ }{
}
}
{}{}{}
for all subbundles ${\mathcal T}$.
Suppose that the base field has positive characteristic
\mavergleichskette
{\vergleichskette
{ p
}
{ > }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then ${\mathcal S}$ is called \stichwort {strongly semistable} {,} if all
\zusatzklammer {absolute} {} {}
Frobenius pull-backs
\mathl{F^{e*}( {\mathcal S} )}{} are semistable.
}
Note that a semistable vector bundle need not be strongly semistable, the following is probably the simplest example.
\inputexample{}
{
Let $C$ be the smooth Fermat quartic given by
\mathl{x^4+y^4+z^4}{} and consider on it the syzygy bundle
\mathl{\operatorname{Syz} { \left( x,y,z \right) }}{}
\zusatzklammer {which is also the restricted cotangent bundle from the projective plane} {} {.}
This bundle is semistable. Suppose that the characteristic is $3$. Then its Frobenius pull-back is
\mathl{\operatorname{Syz} { \left( x^3,y^3,z^3 \right) }}{.} The curve equation gives a global non-trivial section of this bundle of total degree $4$. But the degree of
\mathl{\operatorname{Syz} { \left( x^3,y^3,z^3 \right) } (4)}{} is negative, hence it can not be semistable anymore.
}
For a strongly semistable vector bundle ${\mathcal S}$ on $Y$ and a cohomology class
\mathl{c \in H^1(Y, {\mathcal S})}{} with corresponding torsor we obtain the following affineness criterion.
\inputfakt{Torsor über Kurve/Stark semistabil/Affinitätskriterium/en/Fakt}{Theorem}{}
{
\faktsituation {Let $C$ denote a smooth projective curve over an algebraically closed field $K$ and let ${\mathcal S}$ be a strongly semistable vector bundle over $C$ together with a cohomology class
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(C, {\mathcal S} )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then the torsor
\mathl{T(c)}{} is an affine scheme if and only if
\mavergleichskette
{\vergleichskette
{ \operatorname{ deg}_{ } ^{ } { \left( {\mathcal S} \right) }
}
{ < }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ c
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
\zusatzklammer {
\mavergleichskettek
{\vergleichskettek
{ F^e(c)
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all $e$ in positive characteristic} {} {.}}
\faktzusatz {}
\faktzusatz {}
}
This result rests on the ampleness of
\mathl{{\mathcal S}'^\vee}{} occuring in the dual exact sequence
\mathl{0 \rightarrow {\mathcal O}_Y \rightarrow {\mathcal S}'^\vee \rightarrow {\mathcal S}^\vee \rightarrow 0}{} given by $c$
\zusatzklammer {work of Hartshorne and Gieseker} {} {.} It implies for a strongly semistable syzygy bundles the following
\betonung{degree formula}{} for tight closure.
\inputfakt{tight closure/degree bound for inclusion/strongly semistable/curve case/Fakt}{Theorem}{}
{Suppose that
\mathl{\operatorname{Syz} { \left( f_1 , \ldots , f_n \right) }}{} is strongly semistable. Then
\mathdisp {R_m \subseteq I^* \text{ for } m \geq \frac{\sum d_i}{n-1} \text{ and (for almost all prime numbers) } R_m \cap I^* \subseteq I \text{ for } m < \frac{\sum d_i}{n-1}} { . }
}
We indicate the proof of the inclusion result. The degree condition implies that
\mathl{c= \delta(f) \in H^1(Y, {\mathcal S})}{} is such that
\mathl{{\mathcal S} =\operatorname{ Syz}_{ } ^{ } { \left( f_1 , \ldots , f_n \right) } (m)}{} has nonnegative degree. Then also all Frobenius pull-backs
\mathl{F^*(\mathcal S)}{} have nonnegative degree. Let $\mathcal L$ be a line bundle on $Y$ such that its degree is larger than the degree of
\mathl{\omega_Y^{-1}}{,} the dual of the canonical sheaf. Let
\mathl{z \in H^0(Y, {\mathcal L})}{} be a non-zero element. Then
\mathl{z F^{e*}(c) \in H^1(Y, F^{e*}({\mathcal S}) \otimes {\mathcal L})}{,} and by Serre duality we have
\mathdisp {H^1(Y, F^{e*} ({\mathcal S}) \otimes {\mathcal L}) \cong H^0 ( F^{e*}({\mathcal S}^{*} ) \otimes {\mathcal L}^{-1} \otimes \omega_Y)^{\vee}} { . }
On the right hand side we have a semistable sheaf of negative degree, which can not have a nontrivial section. Hence
\mathl{zF^{e*} = 0}{} and therefore $f$ belongs to the tight closure.
\zwischenueberschrift{Harder-Narasimhan filtration}
In general, there exists an exact criterion depending on
\mathl{c}{} and the
\betonung{strong Harder-Narasimhan filtration}{} of ${\mathcal S}$. For this we give the definition of the Harder-Narasimhan filtration.
\inputdefinition
{}
{
Let ${\mathcal S}$ be a vector bundle on a smooth projective curve $C$ over an algebraically closed field $K$. Then the
\zusatzklammer {uniquely determined} {} {}
filtration
\mavergleichskettedisp
{\vergleichskette
{ 0
}
{ =} { {\mathcal S}_0
}
{ \subset} { {\mathcal S}_1
}
{ \subset \ldots \subset} { {\mathcal S}_{t-1}
}
{ \subset} { {\mathcal S}_t
}
}
{
\vergleichskettefortsetzung
{ =} { {\mathcal S}
}
{ } {}
{ } {}
{ } {}
}{}{}
of subbundles such that all quotient bundles
\mathl{{\mathcal S}_k /{\mathcal S}_{k-1}}{} are semistable with decreasing slopes
\mavergleichskette
{\vergleichskette
{ \mu_k
}
{ = }{ \mu ({\mathcal S}_k /{\mathcal S}_{k-1})
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
is called the \stichwort {Harder-Narasimhan filtration} {} of ${\mathcal S}$.
}
The Harder-Narasimhan filtration exists uniquely
\zusatzklammer {by a Theorem of Harder and Narasimhan} {} {.}
A Harder-Narasimhan filtration is called strong if all the quotients
\mathl{{\mathcal S}_{i}/{\mathcal S}_{i-1}}{} are strongly semistable. A Harder-Narasimhan filtration is not strong in general, however, by a Theorem of A. Langer, there exists some Frobenius pull-back $F^{e*} ( {\mathcal S} )$ such that its Harder-Narasimhan filtration is strong.
\inputfakt{Torsor über Kurve/Starke Harder-Narasimhan Filtration/Affinitätskriterium/en/Fakt}{Theorem}{}
{
\faktsituation {Let $C$ denote a smooth projective curve over an algebraically closed field $K$ and let $E$ be a vector bundle over $C$ together with a cohomology class
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(C,{\mathcal S})
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Let
\mavergleichskettedisp
{\vergleichskette
{ E_1
}
{ \subset} { E_2
}
{ \subset \ldots \subset} { E_{t-1}
}
{ \subset} { E_{t}
}
{ =} { F^{e*}( E)
}
}
{}{}{}
be a strong Harder-Narasimhan filtration. We choose $i$ such that
\mathl{E_{i}/E_{i-1}}{} has degree $\geq 0$ and that
\mathl{E_{i+1}/E_{i}}{} has degree $< 0$. We set
\mavergleichskette
{\vergleichskette
{ {\mathcal Q}
}
{ = }{ F^{e*}(E)/ E_i
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then the following are equivalent.
\aufzaehlungzwei {The torsor
\mathl{T(c)}{} is not an affine scheme.
} {Some Frobenius power of the image of
\mathl{F^{e*}(c)}{} inside
\mathl{H^1(X, {\mathcal Q} )}{} is $0$.
}}
\faktzusatz {}
\faktzusatz {}
}