Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 5/latex
\setcounter{section}{5}
\zwischenueberschrift{Affine schemes}
A scheme $U$ is called \stichwort {affine} {} if it is isomorphic to the spectrum of some commutative ring $R$. If the scheme is of finite type over a field
\zusatzklammer {or a ring} {} {}
$K$
\zusatzklammer {if we have a variety} {} {,}
then this is equivalent to saying that there exist global functions
\mavergleichskettedisp
{\vergleichskette
{ g_1 , \ldots , g_m
}
{ \in} { \Gamma(U, {\mathcal O}_U)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
such that the mapping
\maabbeledisp {} { U } { { {\mathbb A}_{ K }^{ m } }
} { x } { { \left( g_1(x) , \ldots , g_m (x) \right) }
} {,}
is a closed embedding. The relation to cohomology is given by the following well-known theorem of Serre \cite[Theorem III.3.7]{hartshorne}.
\inputfakt{Noethersches Schema/Affin/Kohomologisches Kriterium/en/Fakt}{Theorem}{}
{
\faktsituation {Let $U$ denote a noetherian scheme.}
\faktuebergang {Then the following properties are equivalent.}
\faktfolgerung {\aufzaehlungdrei{$U$ is an affine scheme.
}{For every quasicoherent sheaf ${\mathcal F}$ on $U$ and all
\mavergleichskette
{\vergleichskette
{ i
}
{ \geq }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have
\mavergleichskette
{\vergleichskette
{ H^{i}( U, {\mathcal F} )
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}{For every coherent ideal sheaf ${\mathcal I}$ on $U$ we have
\mavergleichskette
{\vergleichskette
{ H^{1}( U, {\mathcal I} )
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}}
\faktzusatz {}
\faktzusatz {}
}
It is in general a difficult question whether a given scheme $U$ is affine. For example, suppose that
\mavergleichskette
{\vergleichskette
{ X
}
{ = }{ \operatorname{Spec} { \left( R \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is an affine scheme and
\mavergleichskettedisp
{\vergleichskette
{ U
}
{ =} { D({\mathfrak a})
}
{ \subseteq} { X
}
{ } {
}
{ } {
}
}
{}{}{}
is an open subset
\zusatzklammer {such schemes are called \stichwort {quasiaffine} {}} {} {}
defined by an ideal
\mavergleichskette
{\vergleichskette
{ {\mathfrak a}
}
{ \subseteq }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
When is $U$ itself affine? The cohomological criterion above simplifies to the condition that
\mavergleichskette
{\vergleichskette
{ H^{i}(U, {\mathcal O}_{ X } )
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for
\mavergleichskette
{\vergleichskette
{ i
}
{ \geq }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Of course, if
\mavergleichskette
{\vergleichskette
{ {\mathfrak a}
}
{ = }{ (f)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is a principal ideal
\zusatzklammer {or up to radical a principal ideal} {} {,}
then
\mavergleichskettedisp
{\vergleichskette
{ U
}
{ =} {D(f)
}
{ \cong} { \operatorname{Spec} { \left( R_f \right) }
}
{ } {
}
{ } {
}
}
{}{}{}
is affine. On the other hand, if
\mathl{(R, {\mathfrak m})}{} is a local ring of dimension $\geq 2$, then
\mavergleichskettedisp
{\vergleichskette
{ D( {\mathfrak m} )
}
{ \subset} { \operatorname{Spec} { \left( R \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is not affine, since
\mavergleichskettedisp
{\vergleichskette
{ H^{d-1}(U , {\mathcal O}_{ X } )
}
{ =} { H^d_{ {\mathfrak m} }(R)
}
{ \neq} {0
}
{ } {
}
{ } {
}
}
{}{}{}
by the relation between sheaf cohomology and local cohomology and a Theorem of Grothendieck \cite[Theorem 3.5.7]{brunsherzog}.
\zwischenueberschrift{Affineness and superheight}
One can show that for an open affine subset
\mathl{U \subseteq X}{} the closed complement
\mathl{Y=X \setminus U}{} must be of pure codimension one
\zusatzklammer {$U$ must be the complement of the support of an effective divisor} {} {.}
In a regular or
\zusatzklammer {locally $\Q$} {} {-}
factorial domain the complement of every divisor is affine, since the divisor can be described
\zusatzklammer {at least locally geometrically} {} {}
by one equation. But it is easy to give examples to show that this is not true for normal threedimensional domains. The following example is a standardexample for this phenomenon and is in fact given by a forcing algebra.
\inputexample{}
{
Let $K$ be a field and consider the ring
\mavergleichskettedisp
{\vergleichskette
{ R
}
{ =} { K[x,y,u,v]/(xu-yv)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The ideal
\mavergleichskette
{\vergleichskette
{ {\mathfrak p}
}
{ = }{ (x,y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is a prime ideal in $R$ of height one. Hence the open subset
\mavergleichskette
{\vergleichskette
{ U
}
{ = }{ D(x,y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is the complement of an irreducible hypersurface. However, $U$ is not affine. For this we consider the closed subscheme
\mavergleichskettedisp
{\vergleichskette
{ {\mathbb A}^{2}_{K}
}
{ \cong} { Z
}
{ =} { V(u,v)
}
{ \subseteq} { \operatorname{Spec} { \left( R \right) }
}
{ } {
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ Z \cap U
}
{ \subseteq }{ U
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
If $U$ were affine, then also the closed subscheme
\mavergleichskette
{\vergleichskette
{ Z \cap U
}
{ \cong }{ {\mathbb A}^{2}_{K} \setminus \{(0,0)\}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
would be affine, but this is not true, since the complement of the punctured plane has codimension $2$.
}
The argument employed in this example rests on the following definition and the next theorem.
\inputdefinition
{}
{
Let $R$ be a noetherian commutative ring and let
\mavergleichskette
{\vergleichskette
{ I
}
{ \subseteq }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ideal. The
\zusatzklammer {noetherian} {} {}
\stichwort {superheight} {}
is the supremum
\mathdisp {\operatorname{ sup}_{ } ^{ } { \left( \operatorname{ ht}_{ } ^{ } { \left( IS \right) } :\, S \text{ is a notherian } R-\text{algebra} \right) }} { . }
}
\inputfakt{Quasiaffines Schema/Affin/Superhöhe 1 und endlich erzeugter Schnittring/en/Fakt}{Theorem}{}
{
\faktsituation {Let $R$ be a noetherian commutative ring and let
\mavergleichskette
{\vergleichskette
{ I
}
{ \subseteq }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ideal and
\mavergleichskette
{\vergleichskette
{ U
}
{ = }{ D(I)
}
{ \subseteq }{ X
}
{ = }{ \operatorname{Spec} { \left( R \right) }
}
{ }{
}
}
{}{}{.}}
\faktuebergang {Then the following are equivalent.}
\faktfolgerung {\aufzaehlungzwei {$U$ is an affine scheme.
} {$I$ has superheight $\leq 1$ and
\mathl{\Gamma(U, {\mathcal O}_X)}{} is a finitely generated $R$-algebra.
}}
\faktzusatz {}
\faktzusatz {}
} It is not true at all that the ring of global sections of an open subset $U$ of the spectrum $X$ of a noetherian ring is of finite type over this ring. This is not even true if $X$ is an affine variety. This problem is directly related to Hilbert's fourteenth problem, which has a negative answer. We will later present examples where $U$ has superheight one, yet is not affine, hence its ring of global sections is not finitely generated.
If $R$ is a two-dimensional local ring with parameters
\mathl{f,g}{} and if $B$ is the forcing algebra for some ${\mathfrak m}$-primary ideal, then the ring of global sections of the torsor is just
\mavergleichskettedisp
{\vergleichskette
{ \Gamma(D( {\mathfrak m}B) , {\mathcal O}_B )
}
{ =} { B_f \cap B_g
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
In the following two examples we use results from tight closure theory to establish
\zusatzklammer {non} {} {-}affineness properties of certain torsors.
\inputexample{}
{
Let $K$ be a field and consider the Fermat ring
\mavergleichskettedisp
{\vergleichskette
{ R
}
{ =} { K[X,Y,Z]/ { \left( X^d+Y^d+Z^d \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
together with the ideal
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ (X,Y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ f
}
{ = }{ Z^2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For
\mavergleichskette
{\vergleichskette
{ d
}
{ \geq }{ 3
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
we have
\mavergleichskette
{\vergleichskette
{ Z^2
}
{ \notin }{ (X,Y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
This element is however in the tight closure
\mathl{(X,Y)^*}{} of the ideal in positive characteristic
\zusatzklammer {assume that the characteristic $p$ does not divide $d$} {} {}
and is therefore also in characteristic $0$ inside the tight closure and inside the solid closure. Hence the open subset
\mavergleichskettedisp
{\vergleichskette
{ D(X,Y)
}
{ \subseteq} { \operatorname{Spec} { \left( K[X,Y,Z,S,T]/ { \left( X^d+Y^d+Z^d, SX+TY-Z^2 \right) } \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is not an affine scheme. In positive characteristic, $Z^2$ is also contained in the plus closure
\mathl{(X,Y)^+}{} and therefore this open subset contains punctured surfaces
\zusatzklammer {the spectrum of the forcing algebra contains two-dimensional closed subschemes which meet the exceptional fiber
\mathl{V(X,Y)}{} in only one point; the ideal
\mathl{(X,Y)}{} has superheight two in the forcing algebra} {} {.}
In characteristic zero however,
the superheight is one because plus closure is trivial for normal domains in characteristic $0$, and therefore by
Theorem 5.4
the algebra
\mathl{\Gamma({D(X,Y),\mathcal O}_B )}{} is not finitely generated. For
\mavergleichskette
{\vergleichskette
{ K
}
{ = }{ {\mathbb C}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ d
}
{ = }{ 3
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
one can also show that
\mathl{D(X,Y)_{\mathbb C}}{} is, considered as a complex space, a Stein space.
}
\inputexample{}
{
Let $K$ be a field of positive characteristic
\mavergleichskette
{\vergleichskette
{ p
}
{ \geq }{ 7
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and consider the ring
\mavergleichskettedisp
{\vergleichskette
{ R
}
{ =} { K[X,Y,Z]/{ \left( X^5+Y^3+Z^2 \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
together with the ideal
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ (X,Y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mavergleichskette
{\vergleichskette
{ f
}
{ = }{ Z
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Since $R$ has a rational singularity, it is $F$-regular, i.e. all ideals are tightly closed. Therefore
\mavergleichskette
{\vergleichskette
{ Z
}
{ \notin }{ (X,Y)^*
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and so the torsor
\mavergleichskettedisp
{\vergleichskette
{ D(X,Y)
}
{ \subseteq} { \operatorname{Spec} { \left( K[X,Y,Z,S,T]/ { \left( X^5+Y^3+Z^2, SX+TY-Z \right) } \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is an affine scheme. In characteristic zero this can be proved by either using that $R$ is a quotient singularity or by using the natural grading
\zusatzklammer {\mathlk{\operatorname{ deg}_{ } ^{ } { \left( X \right) }=6,\, \operatorname{ deg}_{ } ^{ } { \left( Y \right) }=10,\, \operatorname{ deg}_{ } ^{ } { \left( Z \right) }=15}{}} {} {}
where the corresponding cohomology class
\mathl{{ \frac{ Z }{ XY } }}{} gets degree $-1$ and then applying the geometric criteria on the corresponding projective curve
\zusatzklammer {rather the corresponding curve of the standard-homogenization
\mavergleichskettek
{\vergleichskettek
{ U^{30}+ V^{30} +W^{30}
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {.}
}