Kurs:Vector bundles, forcing algebras and local cohomology (Medellin 2012)/Lecture 2/latex
\setcounter{section}{2}
\zwischenueberschrift{Forcing algebras and closure operations}
Let $R$ denote a commutative ring and let
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be an ideal. Let
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and 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 and
\maabbdisp {\varphi} { \operatorname{Spec} { \left( B \right) } } { \operatorname{Spec} { \left( R \right) }
} {}
the corresponding spectrum morphism. How are properties of $\varphi$
\zusatzklammer {or of the $R$-algebra $B$} {} {}
related to certain ideal closure operations?
We start with some examples. The element $f$ belongs to the ideal $I$ if and only if we can write
\mavergleichskette
{\vergleichskette
{ f
}
{ = }{ r_1f_1 + \cdots + r_nf_n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mavergleichskette
{\vergleichskette
{ r_i
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
By the universal property of the forcing algebra this means that there exists an $R$-algebra-homomorphism
\maabbdisp {} {B} {R
} {,}
hence
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds if and only if $\varphi$ admits a scheme section. This is also equivalent to
\maabbdisp {} {R} {B
} {}
admitting an $R$-module section or $B$ being a pure $R$-algebra
\zusatzklammer {so for forcing algebras properties might be equivalent which are not equivalent for arbitrary algebras} {} {.}
\stichwort {The radical of an ideal} {}
Now we look at the \stichwort {radical} {} of the ideal $I$,
\mavergleichskettedisp
{\vergleichskette
{ \operatorname{rad} { \left( I \right) }
}
{ =} { { \left\{ f \in R \mid f^k \in I \text{ for some }k \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The importance of the radical comes mainly from Hilbert's Nullstellensatz, saying that for algebras of finite type over an algebraically closed field there is a natural bijection between radical ideals and closed algebraic zero-sets. So geometrically one can see from an ideal only its radical. As this is quite a coarse closure operation we should expect that this corresponds to a quite coarse property of the morphism $\varphi$ as well. Indeed, it is true that
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ \operatorname{rad} { \left( I \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
if and only if $\varphi$ is surjective. This is true since the radical of an ideal is the intersection of all prime ideals in which it is contained. Hence an element $f$ belongs to the radical if and only if for all residue class homomorphisms
\maabbdisp {\theta} {R} {\kappa( {\mathfrak p} )
} {}
where $I$ is sent to $0$, also $f$ is sent to $0$. But this means for the forcing equation that whenever the equation degenerates to $0$, then also the inhomogeneous part becomes zero, and so there will always be a solution to the inhomogeneous equation.
Exercise: Define the radical of a submodule inside a module.
\stichwort {Integral closure of an ideal} {}
Another closure operation is \stichwort {integral closure} {} \cite{hunekeswansonintegral}. It is defined by
\mavergleichskettedisp
{\vergleichskette
{ \overline{ I}
}
{ =} { { \left\{ f \in R \mid f^k +a_{1}f^{k-1} + \cdots + a_{k-1} f + a_k = 0 \text{ for some } k \text{ and } a_i \in I^i \right\} }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
This notion is important for describing the normalization of the blow up of the ideal $I$. Another characterization
\zusatzklammer {assume that $R$ is noetherian} {} {}
is that there exists a
\mavergleichskette
{\vergleichskette
{ z
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
not contained in any minimal prime ideal of $R$, such that
\mavergleichskette
{\vergleichskette
{ zf^n
}
{ \in }{ I^n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds for all $n$. Another equivalent property
\zusatzgs {the valuative criterion} {}
is that for all ring homomorphisms
\maabbdisp {\theta} {R} {D
} {}
to a discrete valuation domain $D$ the containment
\mavergleichskette
{\vergleichskette
{ \theta(f)
}
{ \in }{ \theta(I)D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
holds.
The characterization of the integral closure in terms of forcing algebras requires some notions from topology. A continuous map
\maabbdisp {\varphi} {X} {Y
} {}
between topological spaces
\mathkor {} {X} {and} {Y} {}
is called a \stichwort {submersion} {,} if it is surjective and if $Y$ carries the image topology
\zusatzklammer {quotient topology} {} {}
under this map. This means that a subset
\mavergleichskette
{\vergleichskette
{ W
}
{ \subseteq }{ Y
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is open if and only if its preimage
\mathl{\varphi^{-1}(W)}{} is open. Since the spectrum of a ring endowed with the Zarisiki topology is a topological space, this notion can be applied to the spectrum morphism of a ring homomorphism. With this notion we can state that
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ \bar{I}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
if and only if the forcing morphism
\maabbdisp {\varphi} { \operatorname{Spec} { \left( B \right) } } { \operatorname{Spec} { \left( R \right) }
} {}
is a universal submersion
\zusatzklammer {universal means here that for any ring change
\maabb {} {R} {R'
} {}
to a noetherian ring $R'$, the resulting homomorphism
\maabb {} {R'} {B'
} {}
still has this property} {} {.}
The relation between these two notions stems from the fact that also for universal submersions there exists a criterion in terms of discrete valuation domains: A morphism of finite type between two affine noetherian schemes is a universal submersion if and only if the base change to any discrete valuation domain yields a submersion(see \cite[Remarque 2.6]{SGA1}). For a morphism
\maabbdisp {} {Z} { \operatorname{Spec} { \left( D \right) }
} {}
\zusatzklammer {$D$ a discrete valuation domain} {} {}
to be a submersion means that above the only chain of prime ideals in
\mathl{\operatorname{Spec} { \left( D \right) }}{,} namely
\mavergleichskette
{\vergleichskette
{ (0)
}
{ \subset }{ {\mathfrak m}_D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
there exists a chain of prime ideals
\mavergleichskette
{\vergleichskette
{ {\mathfrak p}'
}
{ \subseteq }{ {\mathfrak q}'
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in $Z$ lying over this chain. This pair-lifting property holds for a universal submersion
\maabbdisp {} { \operatorname{Spec} { \left( S \right) } } { \operatorname{Spec} { \left( R \right) }
} {}
for any pair of prime ideals
\mavergleichskette
{\vergleichskette
{ {\mathfrak p}
}
{ \subseteq }{ {\mathfrak q}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in
\mathl{\operatorname{Spec} { \left( R \right) }}{.} This property is stronger than lying over
\zusatzklammer {which means surjective} {} {}
but weaker than the going-down or the going-up property
\zusatzklammer {in the presence of surjectivity} {} {.}
If we are dealing only with algebras of finite type over the complex numbers ${\mathbb C}$, then we may also consider the corresponding complex spaces with their natural topology induced from the euclidean topology of ${\mathbb C}^n$. Then universal submersive with respect to the Zariski topology is the same as submersive in the complex topology \zusatzklammer {the target space needs to be normal} {} {.}
\inputexample{}
{
Let $K$ be a field and consider
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ K[X]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Since this is a principal ideal domain, the only interesting forcing algebras
\zusatzklammer {if we are only interested in the local behavior around $(X)$} {} {}
are of the form
\mathl{K[X,T]/ { \left( X^nT-X^m \right) }}{.} For
\mavergleichskette
{\vergleichskette
{ m
}
{ \geq }{ n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
this
\mathl{K[X]}{-}algebra admits a section
\zusatzklammer {corresponding to the fact that
\mavergleichskette
{\vergleichskette
{ X^m
}
{ \in }{ { \left( X^n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {,}
and if
\mavergleichskette
{\vergleichskette
{ n
}
{ \geq }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
there exists an affine line over the maximal ideal
\mathl{(X)}{.} So now assume
\mavergleichskette
{\vergleichskette
{m
}
{ < }{ n
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
If
\mavergleichskette
{\vergleichskette
{ m
}
{ \geq }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
then we have a hyperbola mapping to an affine line, with the fiber over
\mathl{(X)}{} being empty, corresponding to the fact that $1$ does not belong to the radical of
\mathl{{ \left( X^n \right) }}{} for
\mavergleichskette
{\vergleichskette
{ n
}
{ \geq }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
So assume finally
\mavergleichskette
{\vergleichskette
{ 1
}
{ \leq }{ m
}
{ < }{ n
}
{ }{
}
{ }{
}
}
{}{}{.}
Then $X^m$ belongs to the radical of
\mathl{{ \left( X^n \right) }}{,} but not to its integral closure
\zusatzklammer {which is the identical closure on a one-dimensional regular ring} {} {.}
We can write the forcing equation as
\mavergleichskette
{\vergleichskette
{ X^nT-X^m
}
{ = }{ X^m { \left( X^{n-m} T -1 \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
So the spectrum of the forcing algebra consists of a
\zusatzklammer {thickend} {} {}
line over
\mathl{(X)}{} and of a hyperbola. The forcing morphism is surjective, but it is not a submersion. For example, the preimage of
\mavergleichskette
{\vergleichskette
{ V(X)
}
{ = }{ \{ (X) \}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is a connected component hence open, but this single point is not open.
}
\inputexample{}
{
Let $K$ be a field and let
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ K[X,Y]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be the polynomial ring in two variables. We consider the ideal
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( X^2,Y \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and the element $X$. This element belongs to the radical of this ideal, hence the forcing morphism
\maabbdisp {} { \operatorname{Spec} { \left( K[X,Y,T_1,T_2]/ { \left( X^2T_1 +YT_2+X \right) } \right) } } { \operatorname{Spec} { \left( K[X,Y] \right) }
} {}
is surjective. We claim that it is not a submersion. For this we look at the reduction modulo $Y$. In
\mavergleichskette
{\vergleichskette
{ K[X,Y]/(Y) }
{ \cong }{ K[X]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
the ideal $I$ becomes
\mathl{{ \left( X^2 \right) }}{} which does not contain $X$. Hence by the valuative criterion for integral closure, $X$ does not belong to the integral closure of the ideal. One can also say that the chain
\mavergleichskette
{\vergleichskette
{ V(X,Y)
}
{ \subset }{ V(Y)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
in the affine plane does not have a lift
\zusatzklammer {as a chain} {} {}
to the spectrum of the forcing algebra.
For the ideal
\mavergleichskettedisp
{\vergleichskette
{ I
}
{ =} { { \left( X^2,Y^2 \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
and the element $XY$ the situation looks different. Let
\maabbdisp {\theta} { K[X,Y]} { D
} {}
be a ring homomorphism to a discrete valuation domain $D$. If $X$ or $Y$ is mapped to $0$, then also $XY$ is mapped to $0$ and hence belongs to the extended ideal. So assume that
\mathkor {} {\theta(X)=u \pi^r} {and} {\theta(Y)=v \pi^s} {,}
where $\pi$ is a local parameter of $D$ and $u$ and $v$ are units. Then
\mavergleichskette
{\vergleichskette
{ \theta(XY)
}
{ = }{ uv \pi^{r+s}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and the exponent is at least the minimum of
\mathkor {} {2r} {and} {2s} {,}
hence
\mavergleichskettedisp
{\vergleichskette
{ \theta(XY)
}
{ \in} { { \left( \pi^{2r}, \pi^{2s} \right) }
}
{ =} { { \left( \theta { \left( X^2 \right) } , \theta { \left( Y^2 \right) } \right) } D
}
{ } {
}
{ } {
}
}
{}{}{.}
So $XY$ belongs to the integral closure of
\mathl{{ \left( X^2,Y^2 \right) }}{} and the forcing morphism
\maabbdisp {} { \operatorname{Spec} { \left( K[X,Y,T_1,T_2]/ { \left( X^2T_1 +Y^2T_2+XY \right) } \right) } } { \operatorname{Spec} { \left( K[X,Y] \right) }
} {}
is a universal submersion.
}
\stichwort {Continuous closure} {}
Suppose now that
\mavergleichskette
{\vergleichskette
{ R
}
{ = }{ {\mathbb C}[X_1 , \ldots , X_k]
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Then every polynomial
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
can be considered as a continuous function
\maabbeledisp {f} {{\mathbb C}^k} {{\mathbb C}
} { { \left( x_1 , \ldots , x_k \right) } } { f { \left( x_1 , \ldots , x_k \right) }
} {,}
in the complex topology. If
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ { \left( f_1 , \ldots , f_n \right) }
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is an ideal and
\mavergleichskette
{\vergleichskette
{ f
}
{ \in }{ R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is an element, we say that $f$ belongs to the \stichwort {continuous closure} {} of $I$, if there exist continuous functions
\maabbdisp {g_1 , \ldots , g_n} {{\mathbb C}^k} {{\mathbb C}
} {}
such that
\mavergleichskettedisp
{\vergleichskette
{ f
}
{ =} { \sum_{i = 1}^n g_if_i
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
\zusatzklammer {as an identity of functions} {} {.}
The same definition works for ${\mathbb C}$-algebras of finite type, see \cite{brennercontinuous}, \cite{epsteinhochstercontinuous}, \cite{kollarcontinuous}.
It is not at all clear at once that there may exist polynomials
\mavergleichskette
{\vergleichskette
{ f
}
{ \notin }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
but inside the continuous closure of $I$. For
\mathl{{\mathbb C}[X]}{} it is easy to show that the continuous closure is
\zusatzklammer {like the integral closure} {} {}
just the ideal itself. We also remark that when we would only allow holomorphic functions
\mathl{g_1 , \ldots , g_n}{} then we could not get something larger. However, with continuous functions
\maabbdisp {g_1, g_2} { {\mathbb C}^2 } { {\mathbb C}
} {}
we can for example write
\mavergleichskettedisp
{\vergleichskette
{ X^2Y^2
}
{ =} { g_1 X^3 + g _2 Y^3
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Continuous closure is always inside the integral closure and hence also inside the radical. The element $XY$ does not belong to the continuous closure of
\mavergleichskette
{\vergleichskette
{ I
}
{ = }{ (X^2,Y^2)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
though it belongs to the integral closure of $I$. In terms of forcing algebras, an element $f$ belongs to the continuous closure if and only if the complex forcing mapping
\maabbdisp {\varphi_{\mathbb C}} { \operatorname{Spec} { \left( B \right) }_{\mathbb C} } { \operatorname{Spec} { \left( R \right) }_{\mathbb C}
} {}
\zusatzklammer {between the corresponding complex spaces} {} {}
admits a continuous section.