Kurs:Vector bundles and their torsors (Kolkata 2011)/Lecture 4/latex
\setcounter{section}{4}
In this lecture we deal with closure operations which depend only on the torsor which the forcing algebra defines, so they only depend on the cohomology class of the forcing data inside the syzygy bundle. Our main example is tight closure, a theory developed by Hochster and Huneke, and related closure operations like solid closure and plus closure.
\zwischenueberschrift{Tight closure and solid closure}
Let $R$ be a noetherian domain of positive characteristic, let
\maabbeledisp {F} {R} {R
} {f} {f^p
} {,}
be the \definitionswort {Frobenius homomorphism}{} and
\maabbeledisp {F^e} {R} {R
} {f} {f^q
} {}
\zusatzklammer {mit
\mavergleichskettek
{\vergleichskettek
{ q
}
{ = }{ p^e
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {}
its $e$th iteration. Let $I$ be an ideal and set
\mavergleichskettedisp
{\vergleichskette
{ I^{[q]}
}
{ =} { \text{ extended ideal of } I \text{ under } F^e
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Then define the \definitionswort {tight closure}{} of $I$ to be the ideal
\mavergleichskettedisp
{\vergleichskette
{ I^*
}
{ =} { { \left\{ f \in R \mid \text{ there exists } z \neq 0
\mathdisplaybruch \text{ such that } zf^q \in I^{[q]} \text{ for all } q=p^e \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The element $f$ defines the cohomology class
\mathl{c \in H^1(D(I), \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) })}{.} Suppose that $R$ is normal and that $I$ has height at least $2$
\zusatzklammer {think of a local normal domain of dimension at least $2$ and an ${\mathfrak m}$-primary ideal $I$} {} {.}
Then the $e$th Frobenius pull-back of the cohomology class is
\mathdisp {F^{e*} (c) \in H^1(D(I), F^{e*} ( \operatorname{Syz} { \left( f_1 , \ldots , f_n \right) } )) \cong H^1(D(I), \operatorname{Syz} { \left( f_1^q , \ldots , f_n^q \right) })} { }
\zusatzklammer {\mathlk{q=p^e}{}} {} {}
and this is the cohomology class corresponding to $f^q$. By the height assumption,
\mathl{z F^e (c) = 0}{} if and only if
\mathl{zf^q \in (f_1^q , \ldots , f_n^q)}{,} and if this holds for all $e$ then
\mathl{f \in I^*}{} by definition. This shows already that tight closure under the given conditions does only depend on the cohomology class.
This is also a consequence of the following theorem of Hochster which gives a characterization of tight closure in terms of forcing algebra and local cohomology.
\inputfakt{Forcing algebras/Relation to tight closure/Local cohomology/Characterization/Fakt}{Theorem}{}
{Let $R$ be a normal excellent local domain with maximal ideal ${\mathfrak m}$ over a field of positive characteristic. Let
\mathl{f_1 , \ldots , f_n}{} generate an ${\mathfrak m}$-primary ideal $I$ and let $f$ be another element in $R$. Then
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ I^*
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
if and only if
\mavergleichskettedisp
{\vergleichskette
{ H^{\dim (R)}_{\mathfrak m} (B)
}
{ \neq} { 0
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
where
\mavergleichskette
{\vergleichskette
{ B
}
{ = }{ R[T_1 , \ldots , T_n]/ { \left( f_1T_1 + \cdots + f_nT_n +f \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denotes the forcing algebra of these elements.
}
If the dimension $d$ is at least two, then
\mathdisp {H^d_{\mathfrak m} (R) \longrightarrow H^d_{\mathfrak m} (B) \cong H^d_{\mathfrak m B} (B) \cong H^{d-1}(D({\mathfrak m B), \mathcal O_B})} { . }
This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point, i.e. the torsor given by these data. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true
\zusatzklammer {by Serre's
cohomological criterion
for affineness} {} {}
if and only if the open subset
\mathl{D(\mathfrak m B)}{} is an
\betonung{affine scheme}{}
\zusatzklammer {the spectrum of a ring} {} {.}
The right hand side of this equivalence \zusatzgs {the non-vanishing of the top-dimensional local cohomology} {} is independent of any characteristic assumption, and can be taken as the basis for the definition of another closure operation, called \stichwort {solid closure} {.} So the theorem above says that in positive characteristic tight closure and solid closure coincide. There is also a definition of tight closure for algebras over a field of characteristic $0$ by reduction to positive characteristic.
An important property of tight closure is that it is trivial for regular rings, i.e.
\mathl{I^*=I}{} for every ideal $I$. This rests upon Kunz's theorem saying that the Frobenius homomorphism for regular rings is flat. This property implies the following cohomological property of torsors.
\inputfaktproof
{Regulärer Ring/Positive Charakteristik/Erzwingende Algebra/Nicht drin/Lokale Kohomologie/en/Fakt}
{Corollary}
{}
{
\faktsituation {Let
\mathl{( R, {\mathfrak m} )}{} denote a regular local ring of dimension $d$ and of positive characteristic, let
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ${\mathfrak m}$-primary ideal and
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an element with
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Let
\mavergleichskette
{\vergleichskette
{ B
}
{ = }{ R[T_1 , \ldots , T_n]/ { \left( f_1T_1 + \cdots + f_nT_n+f \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be the corresponding forcing algebra.}
\faktfolgerung {Then the extended ideal
\mathl{{\mathfrak m} B}{} satisfies
\mavergleichskettedisp
{\vergleichskette
{H^d_{ {\mathfrak m} B }(B)
}
{ =} {H^{d-1} (D( {\mathfrak m} B), {\mathcal O}_B)
}
{ =} { 0
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {}
\faktzusatz {}
}
{
This follows from
Theorem 4.1
and
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I^*
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
In dimension two this is true in every
\zusatzklammer {even mixed} {} {}
characteristic.
\inputfakt{Regulärer Ring/Dimension 2/Erzwingende Algebra/Nicht drin/Affin/en/Fakt}{Theorem}{}
{
\faktsituation {Let
\mathl{( R, {\mathfrak m} )}{} denote a two-dimensional regular local ring, let
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ${\mathfrak m}$-primary ideal and
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an element with
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Let
\mavergleichskettedisp
{\vergleichskette
{ B
}
{ =} { R[T_1 , \ldots , T_n]/ { \left( f_1T_1 + \cdots + f_nT_n+f \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
be the corresponding forcing algebra.}
\faktfolgerung {Then for the extended ideal
\mathl{{\mathfrak m} B}{} we have
\mavergleichskettedisp
{\vergleichskette
{ H^2_{ {\mathfrak m} B }(B)
}
{ =} { H^{1} (D( {\mathfrak m} B), {\mathcal O}_B)
}
{ =} { 0
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {In particular, the open subset
\mathl{T = D( {\mathfrak m} B)}{} is an affine scheme if and only if
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktzusatz {}
}
The main point for the proof of this result is that for
\mathl{f \not\in I}{,} the natural mapping
\maabbdisp {} {H^1(U, {\mathcal O}_X) } {H^1(T, {\mathcal O}_T)
} {}
is not injective by a Matlis duality argument. Since the local cohomology of a regular ring is explicitly known, this map annihilates some cohomology class of the form
\mathl{{ \frac{ 1 }{ fg } }}{} where
\mathl{f,g}{} are parameters. But then it annihilates the complete local cohomology module and then $T$ is an affine scheme. For non-regular two-dimensional rings it is a difficult question in general to decide whether a torsor is affine or not. A satisfactory answer is only known in the normal twodimensional graded case over a field, which we will deal with in the final lecture.
In higher dimension in characteristic zero it is not true that a regular ring is solidly closed \zusatzklammer {meaning that every ideal equals its solid closure} {} {,} as was shown by the following example of Paul Roberts.
\inputexample{}
{
Let $K$ be a field of characteristic $0$ and let
\mavergleichskettedisp
{\vergleichskette
{ B
}
{ =} { K[X,Y,Z][U,V,W]/ { \left( X^3U+Y^3V+Z^3W-X^2Y^2Z^2 \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Then the ideal
\mavergleichskette
{\vergleichskette
{ {\mathfrak a}
}
{ = }{ (X,Y,Z)B
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
has the property that
\mavergleichskette
{\vergleichskette
{ H^3_{ {\mathfrak a} } (B)
}
{ \neq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
This means that in
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ K[X,Y,Z]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the element
\mathl{X^2Y^2Z^2}{} belongs to the solid closure of the ideal
\mathl{{ \left( X^3,Y^3,Z^3 \right) }}{,} and hence the three-dimensional polynomial ring is not solidly closed.
} This example was the motivation for the introduction of parasolid closure, which has all the good properties of solid closure but which is also trivial for regular rings.
\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
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
\zwischenueberschrift{Plus closure}
The above mentioned \zusatzklammer {finite} {} {} superheight condition is also related to another closure operation, the 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{artinjoins}} {} {.}
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 meet the exceptional fiber \zusatzklammer {the fiber over the maximal ideal} {} {} in one point, and this means that the 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.
\inputremark
{}
{
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 $I$. This implies also that if $R$ is a twodimensional normal local ring of characteristic $0$ and $I$ an ${\mathfrak m}$-primary ideal and
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
an element with
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
then the extendend ideal
\mathl{{\mathfrak m} B}{} inside the forcing algebra $B$ has superheight $1$. If moreover $f$ belongs to the solid closure of $I$, then
\mathl{D( {\mathfrak m}B )}{} is not affine and so by
Theorem 4.7
its ring of global sections is not finitely generated.
}
In the following two examples we use results from tight closure theory to establish
\zusatzklammer {non} {} {-}affineness properties of certain torsors. These results follow also from the results mentioned in the next lecture.
\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, due to
Remark 4.8
the superheight is one and therefore by
Theorem 4.7
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
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {.}