Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 4/latex
\setcounter{section}{4}
\zwischenueberschrift{Forcing algebras and induced torsors}
As $T_U$ is a $V_U$-torsor, and as every $V$-torsor is represented by a unique cohomology class, there should be a natural cohomology class coming from the forcing data. To see this, let $R$ be a noetherian ring and
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ideal. Then on
\mavergleichskette
{\vergleichskette
{ U
}
{ = }{ D(I)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have the short exact sequence
\mathdisp {0 \longrightarrow \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } \longrightarrow {\mathcal O}^n_U \longrightarrow {\mathcal O}_U \longrightarrow 0} { . }
An element
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
defines an element
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ \Gamma(U, {\mathcal O}_U)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and hence a cohomology class
\mavergleichskette
{\vergleichskette
{ \delta(f)
}
{ \in }{ H^1(U , \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Hence $f$ defines in fact a
\mathl{\operatorname{Syz} { \left( f_1 , \ldots , f_n \right) }}{-}torsor over $U$. We will see that this torsor is induced by the forcing algebra given by
\mathl{f_1 , \ldots , f_n}{} and $f$.
\inputfaktproof
{Erzwingende Algebra/Primäres Ideal/Induzierter Torsor/en/Fakt}
{Theorem}
{}
{
\faktsituation {Let $R$ denote a noetherian ring, let
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an ideal and let
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be another element. Let
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ H^1(D(I), \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be the corresponding cohomology class and let
\mavergleichskettedisp
{\vergleichskette
{ B
}
{ =} { R[T_1 , \ldots , T_n]/ { \left( f_1T_1 + \cdots + f_nT_n -f \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
denote the forcing algebra for these data.}
\faktfolgerung {Then the scheme
\mathl{\operatorname{Spec} { \left( B \right) } {{|}}_{D(I)}}{} together with the natural action of the syzygy bundle on it is isomorphic to the torsor given by $c$.}
\faktzusatz {}
\faktzusatz {}
}
{
We compute the cohomology class
\mavergleichskette
{\vergleichskette
{ \delta(f)
}
{ \in }{ H^1(U, \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and the cohomology class given by the forcing algebra. For the first computation we look at the short exact sequence
\mathdisp {0 \longrightarrow \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } \longrightarrow {\mathcal O}^n_U \stackrel{f_1 , \ldots , f_n}{ \longrightarrow } {\mathcal O}_U \longrightarrow 0} { . }
On
\mathl{D(f_i)}{,} the element $f$ is the image of
\mathl{\left( 0 , \ldots , 0 , \, { \frac{ f }{ f_i } } , \, 0 , \ldots , 0) \right)}{}
\zusatzklammer {the non-zero entry is at the $i$th place} {} {.}
The cohomology class is therefore represented by the family of differences
\mathdisp {\left( 0 , \ldots , 0 , \, { \frac{ f }{ f_i } } , \, 0 , \ldots , 0 , \, -{ \frac{ f }{ f_j } } , \, 0 , \ldots , 0 \right) \in \Gamma(D(f_i) \cap D(f_j),\operatorname{Syz} { \left( f_1 , \ldots , f_n \right) })} { . }
On the other hand, there are isomorphisms
\maabbeledisp {} {V {{|}}_{D(f_i)} } { T {{|}}_{D(f_i)}
} {\left( s_1 , \ldots , s_n \right) } {\left( s_1 , \ldots , s_{i-1} , \, s_i + { \frac{ f }{ f_i } } , \, s_{i+1} , \ldots , s_n \right)
} {.}
The composition of two such isomorphisms on
\mathl{D(f_if_j)}{} is the identity plus the same section as before.
\inputexample{}
{
Let
\mathl{(R, {\mathfrak m} )}{} denote a two-dimensional normal local noetherian domain and let
\mathkor {} {f} {and} {g} {}
be two parameters in $R$. On
\mavergleichskette
{\vergleichskette
{ U
}
{ = }{ D( {\mathfrak m} )
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have the short exact sequence
\mathdisp {0 \longrightarrow {\mathcal O}_U \cong \operatorname{Syz} { \left( f,g \right) } \longrightarrow {\mathcal O}_U^2 \stackrel{f,g}{\longrightarrow } {\mathcal O}_U \longrightarrow 0} { }
and its corresponding long exact sequence of cohomology,
\mathdisp {0 \longrightarrow R \longrightarrow R^2 \stackrel{f,g}{\longrightarrow } R \stackrel{\delta}{\longrightarrow} H^1(U, {\mathcal O}_{ X } ) \longrightarrow \ldots} { . }
The connecting homomorphism $\delta$ sends an element
\mavergleichskette
{\vergleichskette
{ h
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
to
\mathl{{ \frac{ h }{ fg } }}{.} The torsor given by such a cohomology class
\mavergleichskette
{\vergleichskette
{ c
}
{ = }{{ \frac{ h }{ fg } }
}
{ \in }{ H^1 { \left( U, {\mathcal O}_X \right) }
}
{ }{
}
{ }{
}
}
{}{}{}
can be realized by the forcing algebra
\mathdisp {R[T_1,T_2]/ { \left( fT_1 +gT_2 -h \right) }} { . }
Note that different forcing algebras may give the same torsor, because the torsor depends only on the spectrum of the forcing algebra restricted to the punctured spectrum of $R$. For example, the cohomology class
\mavergleichskette
{\vergleichskette
{ { \frac{ 1 }{ fg } }
}
{ = }{ { \frac{ fg }{ f^2g^2 } }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
defines one torsor, but the two fractions yield the two forcing algebras
\mathkor {} {R[T_1,T_2]/ { \left( fT_1+gT_2-1 \right) }} {and} {R[T_1,T_2]/ { \left( f^2 T_1+g^2 T_2-fg \right) }} {,}
which are quite different. The fiber over the maximal ideal of the first one is empty, whereas the fiber over the maximal ideal of the second one is a plane.
If $R$ is regular, say
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ K[X,Y]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
\zusatzklammer {or the localization of this at \mathlk{(X,Y)}{} or the corresponding power series ring} {} {}
then the first cohomology classes are $K$-linear combinations of
\mathbed {{ \frac{ 1 }{ x^iy^j } }} {}
{i,j \geq 1} {}
{} {} {} {.}
They are realized by the forcing algebras
\mathdisp {K[X,Y,T_1,T_2]/ { \left( X^iT_1+Y^jT_2 -1 \right) }} { . }
Since the fiber over the maximal ideal is empty, the spectrum of the forcing algebra equals the torsor. Or, the other way round, the torsor is itself an affine scheme.
}
The closure operations we have considered in the second lecture can be characterized by some property of the forcing algebra. However, they can not be characterized by a property of the corresponding torsor alone. For example, for
\mathl{R=K[X,Y]}{,} we may write
\mavergleichskettedisp
{\vergleichskette
{ { \frac{ 1 }{ XY } }
}
{ =} { { \frac{ X }{ X^2Y } }
}
{ =} { { \frac{ XY }{ X^2Y^2 } }
}
{ =} { { \frac{ X^2Y^2 }{ X^3Y^3 } }
}
{ } {
}
}
{}{}{,}
so the torsors given by the forcing algebras
\mathdisp {R[T_1,T_2]/( XT_1+YT_2 +1), \, R[T_1,T_2]/( X^2T_1+YT_2 + X) , \, R[T_1,T_2]/( X^2T_1+Y^2T_2 + XY) \text{ and } R[T_1,T_2]/( X^3T_1+Y^3 T_2 +X^2Y^2)} { }
are all the same
\zusatzklammer {the restriction over \mathlk{D(X,Y)}{}} {} {,}
but their global properties are quite different. We have a non-surjection, a surjective non submersion, a submersion which does not admit
\zusatzklammer {for \mathlk{K={\mathbb C}}{}} {} {}
a continuous section and a map which admits a continuous section.