Benutzer:Bocardodarapti/Talk in Zürich Oktober 2008

Deformations of vector bundles and the localization problem in tight closure theory

Let ${}R$ be a noetherian domain of positive characteristic, let

$F\colon R\longrightarrow R,\,f\longmapsto f^{p},$

be the Frobenius homomorphism and

$F^{e}\colon R\longrightarrow R,\,f\longmapsto f^{q}$

(mit ${}q=p^{e}$) its ${}e$th iteration. Let ${}I$ be an ideal and set

${}I^{[q]}={\text{ extended ideal of }}I{\text{ under }}F^{e}\,.$

Then define the tight closure of ${}I$ to be the ideal

${}I^{*}={\left\{f\in R\mid {\text{ there exists }}z\neq 0{\text{ such that }}zf^{q}\in I^{[q]}{\text{ for all }}q=p^{e}\right\}}\,.$

The localization problem

Let ${}S\subseteq R$ be a multiplicative system and ${}I$ an ideal in ${}R$. Then the localization problem of tight closure is the question whether the identity

${}(I^{*})_{S}=(IR_{S})^{*}\,$

holds.

Here the inclusion ${}\subseteq$ is always true and ${}\supseteq$ is the problem. The problem means explicitly:

if ${}f\in (IR_{S})^{*}$, can we find an ${}h\in S$ such that ${}hf\in I^{*}$ holds in ${}R$?

Why does localization often holds?

Because an element belongs to the tight closure because of a certain reason, and this reason often localizes/globalizes.

A typical reason for belonging to the tight closure is through plus closure.

For an ideal ${}I\subseteq R$ in a domain ${}R$ define its plus closure by

${}I^{+}={\left\{f\in R\mid {\text{there exists a finite domain extension }}R\subseteq T{\text{ such that }}f\in IT\right\}}\,.$

Equivalent: Let ${}R^{+}$ be the absolute integral closure of ${}R$. This is the integral closure of ${}R$ in an algebraic closure of the quotient field ${}Q(R)$ (first considered by Artin). Then

$f\in I^{+}{\text{ if and only if }}f\in IR^{+}.$

The plus closure commutes with localization.

We also have the inclusion ${}I^{+}\subseteq I^{*}$. Here the question arises:

Question: Is ${}I^{+}=I^{*}$?

This question is known as the tantalizing question in tight closure theory.

One should think of the left hand side as a geometric condition and of the right hand side as a cohomological conditon (later).

In dimension two tight closure always localizes. However, a positive answer to the tantalizing question in dimenson two in general would imply the localization property in dimension three.

It is difficult in general to get hold of tight closure and plus closure, they are difficult to compute. We will describe now a specific situation where one has a kind of geometric interpretation for and a detailed understanding of tight closure.

Forcing algebras

Forcing algebras, introduced by Hochster, give a new interpretation of tight closure.

Definition

Let ${}R$ be a commutative ring and let ${}f_{1},\ldots ,f_{n}$ and ${}f$ be elements in ${}R$. Then the ${}R$-algebra

$R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}-f\right)}$

is called the forcing algebra of these elements (or these data).

If we fix a point ${}P\in \operatorname {Spec} \,R$, then the fiber over ${}P$ of the spectrum of the forcing algebra, ${}\operatorname {Spec} \,A$, is just

$\operatorname {Spec} \,\kappa (P)[T_{1},\ldots ,T_{n}]/(f_{1}(P)T_{1}+\ldots +f_{n}(P)T_{n}+f(P)).$

This is the solution set to an inhomogeneous linear equation over ${}\kappa (P)$, so it is an affine space (in the sense of linear algebra) (may be empty). If the ideal ${}(f_{1},\ldots ,f_{n})$ is primary to ${}{\mathfrak {m}}$, and ${}f\in {\mathfrak {m}}$, then the fiber over ${}{\mathfrak {m}}$ has dimension ${}n$, whereas the dimensions of the other fibers are ${}n-1$.

The relation between tight closure and forcing algebras is given in the following theorem.

Theorem

Let ${}R$ be a normal excellent local domain with maximal ideal ${}{\mathfrak {m}}$ over a field of positive characteristic. Let ${}f_{1},\ldots ,f_{n}$ generate an ${}{\mathfrak {m}}$-primary ideal ${}I$ and let ${}f$ be another element in ${}R$. Then

${}f\in I^{*}$ if and only if

${}H_{\mathfrak {m}}^{\dim(R)}(A)\neq 0\,,$

where ${}A=R[T_{1},\ldots ,T_{n}]/{\left(f_{1}T_{1}+\cdots +f_{n}T_{n}+f\right)}$

denotes the forcing algebra of these elements.

If the dimension ${}d$ is at least two, then

$H_{\mathfrak {m}}^{d}(R)\longrightarrow H_{\mathfrak {m}}^{d}(A)\cong H_{{\mathfrak {m}}A}^{d}(A)\cong H^{d-1}(D({{\mathfrak {m}}A}),{\mathcal {O}}_{A}).$

This means that we have to look at the cohomological properties of the complement of the exceptional fiber over the closed point. If the dimension is two, then we have to look whether the first cohomology of the structure sheaf vanishes. This is true if and only if the open subset ${}D({\mathfrak {m}}A)$ is an affine scheme (the spectrum of a ring).

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.

Geometric interpretation in dimension two

Let ${}R$ be a two-dimensional standard-graded normal domain over an algebraically closed field ${}K$. Let ${}C=\operatorname {Proj} {\left(R\right)}$ be the corresponding smooth projective curve and let

${}I={\left(f_{1},\ldots ,f_{n}\right)}\,$

be an ${}R_{+}$-primary homogeneous ideal with generators of degrees ${}d_{1},\ldots ,d_{n}$. Then we get on ${}C$ the short exact sequence

$0\longrightarrow \operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)\longrightarrow \bigoplus _{i=1}^{n}{\mathcal {O}}_{C}(m-d_{i}){\stackrel {f_{1},\ldots ,f_{n}}{\longrightarrow }}{\mathcal {O}}_{C}(m)\longrightarrow 0.$

Here ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m)$ is a vector bundle, called the syzygy bundle, of rank ${}n-1$ and of degree

$((n-1)m-\sum _{i=1}^{n}d_{i})\operatorname {deg} \,(C).$

An element

${}f\in R_{m}=\Gamma (C,{\mathcal {O}}_{C}(m))\,$

defines a cohomology class

${}c=\delta (f)\in H^{1}(C,\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))\,.$

With this notation we have

$f\in I^{*}{\text{ if and only if there exists }}z\neq 0$

(homogeneous of some degree ${}k$) such that ${}zF^{e*}(c)=0$ for all ${}e$. This cohomology class lives in

${}H^{1}(C,F^{e*}(\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}(m))\otimes {\mathcal {O}}(k))=H^{1}(C,\operatorname {Syz} {\left(f_{1}^{q},\ldots ,f_{n}^{q}\right)}(qm+k))\,.$

For the plus closure we have a similar correspondence:

${}f\in I^{+}$ if and only if there exists a curve ${}\varphi \colon C'\rightarrow C$ such that the pull-back ${}\varphi ^{*}(c)$ of the cohomology class vanishes.

For a vector bundle ${}{\mathcal {S}}$ and a cohomology class ${}c\in H^{1}(C,{\mathcal {S}})$ one can construct a geometric object: Because of ${}H^{1}(C,{\mathcal {S}})\cong \operatorname {Ext} ^{1}({\mathcal {O}}_{C},{\mathcal {S}})$, the class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {{\mathcal {S}}'}\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0$

and hence projective bundles

${\mathbb {P} }({\mathcal {S}}^{*})\subset {\mathbb {P} }({{\mathcal {S}}'}^{*})$

and the corresponding torsor

$T={\mathbb {P} }({{\mathcal {S}}'}^{*})-{\mathbb {P} }({\mathcal {S}}^{*}).$

This geometric object is the torsor or principal fiber bundle or affine-linear bundle given by ${}c$. The sheaf ${}{\mathcal {S}}$ acts on it by translations.

In the situation of a forcing algebra for homogeneous elements, this torsor ${}T$ can also be obtained as ${}\operatorname {Proj} \,A$, where ${}A$ is the (not necessarily positively) graded forcing algebra. In particular, it follows that the containment ${}f\in I^{*}$ is equivalent to the property that ${}T$ is not an affine variety (and ${}f\in I^{+}$ if and only if ${}T$ contains a projective curve). For these properties, positivity (ampleness) properties of the syzygy bundle are crucial. We need the concept of semistability.

Definition (semistable and strongly semistable)

Let ${}{\mathcal {S}}$ be a vector bundle on a smooth projective curve ${}C$. It is called semistable, if ${}\mu ({\mathcal {T}})={\frac {\deg({\mathcal {T}})}{\operatorname {rk} ({\mathcal {T}})}}\leq {\frac {\deg({\mathcal {S}})}{\operatorname {rk} ({\mathcal {S}})}}=\mu ({\mathcal {S}})$ for all subbundles ${}{\mathcal {T}}$.

Suppose that the base field has positive characteristic ${}p>0$. Then ${}{\mathcal {S}}$ is called strongly semistable, if all (absolute) Frobenius pull-backs ${}F^{e*}({\mathcal {S}})$ are semistable.

For strongly semistable syzygy bundles we get the following degree formula; the rational number occuring in it is called the degree bound for tight closure.

Theorem (Brenner)

Suppose that ${}\operatorname {Syz} {\left(f_{1},\ldots ,f_{n}\right)}$ is strongly semistable. Then

$R_{m}\subseteq I^{*}{\text{ for }}m\geq {\frac {\sum d_{i}}{n-1}}{\text{ and (for almost all prime numbers) }}R_{m}\cap I^{*}\subseteq I{\text{ for }}m<{\frac {\sum d_{i}}{n-1}}.$

In general, there exists an exact criterion depending on ${}c=\delta (f)$ and the strong Harder-Narasimhan filtration of ${}\operatorname {Syz}$.

If ${}K$ is finite, then the same criterion holds for plus closure. Therefore we get:

Theorem (Brenner)

Let ${}R$ be a standard-graded, two-dimensional normal domain over (the algebraic closure of) a finite field. Let ${}I$ be an ${}R_{+}$-primary graded ideal.

Then

${}I^{*}=I^{+}\,.$

The first chance for a counterexample to the localization property is in dimension three. We will look at a special case of the localization problem. Since we understand now the two-dimensional situation quite well, it is natural to look at a family of two-dimesional rings and corresponding families of projective curves.

Deformations of two-dimensional tight closure problems

Let ${}D\subseteq R$ flat, ${}f$ and ${}I$ in ${}R$. For every ${}D\rightarrow K$, ${}K$ a field, we can consider ${}f$ and ${}I$ in ${}R\otimes _{D}K$. In particular for ${}K=\kappa ({\mathfrak {p}})$, ${}{\mathfrak {p}}\in \operatorname {Spec} \,D$.

How does the property

$f\in I^{*}{\text{ in }}R\otimes _{D}\kappa ({\mathfrak {p}})\,$

vary with ${}{\mathfrak {p}}$?

There are two cases:

• ${}D$ contains ${}\mathbb {Z} /(p)$. This is a geometric or equicharacteristic deformation. Example ${}D=\mathbb {Z} /(p)[t]$.

• ${}D$ contains ${}\mathbb {Z} \subseteq D$. This is an arithmetic or mixed characteristic deformation. Example ${}D=\mathbb {Z}$.

We deal first with the arithmetic situation.

Example (Brenner-Katzman)

Consider ${}\mathbb {Z} [x,y,z]/{\left(x^{7}+y^{7}+z^{7}\right)}$ and take the ideal ${}I={\left(x^{4},y^{4},z^{4}\right)}$ and the element ${}f=x^{3}y^{3}$. Consider reductions ${}\mathbb {Z} \rightarrow \mathbb {Z} /(p)$. Then

$f\in I^{*}{\text{ holds in }}\mathbb {Z} /(p)[x,y,z]/(x^{7}+y^{7}+z^{7}){\text{ for }}p\equiv 3\!\!\!\mod 7$

and

$f\not \in I^{*}{\text{ holds in }}\mathbb {Z} /(p)[x,y,z]/(x^{7}+y^{7}+z^{7}){\text{ for }}p\equiv 2\!\!\!\mod 7.$

In particular, the bundle ${}\operatorname {Syz} {\left(x^{4},y^{4},z^{4}\right)}$ is semistable in the generic fiber, but not strongly semistable for any reduction ${}p\equiv 2\!\!\!\mod 7$. The corresponding torsor is an affine scheme for infinitely many prime reductions and not an affine scheme for infinitely many prime reductions.

In particular, in this example, the syzygy bundle is semistable on the generic fiber in characteristic zero, but not strongly semistable for infinitely many prime numbers. A question of Miyaoka asks whether in such a situation it is always strongly semistable for at least infinitely many prime reductions (it is not for almost all prime reductions).

We now show that the geometric deformations are a special case of the localization problem.

Proposition

Let

${}\mathbb {Z} /(p)\subset D$ be a one-dimensional domain and ${}D\subseteq R$ of finite type, and ${}I$ an ideal in ${}R$. Suppose that localization holds and that

$f\in I^{*}{\text{ holds in }}R\otimes _{D}Q(D)=R_{D^{*}}=R_{Q(D)}$

(${}S=D^{*}=D\setminus \{0\}$ is the multiplicative system). Then ${}f\in I^{*}$ holds in ${}R\otimes _{D}\kappa ({\mathfrak {p}})$ for almost all ${}{\mathfrak {p}}$ in Spec ${}D$.

Proof

By localization, there exists

${}h\in D$, ${}h\neq 0$, such that ${}hf\in I^{*}{\text{ in }}R$.

By persistence of tight closure (under a ring homomorphism) we get

$hf\in I^{*}{\text{ in }}R_{\kappa ({\mathfrak {p}})}.$

The element ${}h$ does not belong to ${}{\mathfrak {p}}$ for almost all ${}{\mathfrak {p}}$, so ${}h$ is a unit in ${}R_{\kappa ({\mathfrak {p}})}$ and hence

$f\in I^{*}{\text{ in }}R_{\kappa ({\mathfrak {p}})}$

for almost all ${}{\mathfrak {p}}$.

$\Box$

We will look at the easiest possibility of such a deformation:

$D={\mathbb {F} }_{p}[t]\subset {\mathbb {F} }_{p}[t][x,y,z]/(g)$

where ${}t$ has degree ${}0$ and ${}x,y,z$ have degree one and ${}g$ is homogeneous. Then (for ${}{\mathbb {F} }_{p}[t]\rightarrow K$)

$R\otimes _{{\mathbb {F} }_{p}[t]}K$

is a two-dimensional standard-graded ring over ${}K$. For residue class fields of points of ${}{\mathbb {A} }_{{\mathbb {F} }_{p}}^{1}=\operatorname {Spec} \,{\mathbb {F} }_{p}[t]$ we have basically two possibilities.

• ${}K={\mathbb {F} }_{p}(t)$, the function field. This is the generic or transcendental case.

• ${}K={\mathbb {F} }_{q}$, the special or algebraic or finite case.

Note that in the second case tight closure is plus closure. To analyze the behavior of tight closure in such a family we can use what we know in the two-dimensional standard-graded situation.

A counterexample to the localization problem

In order to establish an example where tight closure does not behave uniformly under a geometric deformation we first need a situation where strong semistability does not behave uniformly. Such an example was given by Paul Monsky in 1997.

Example (Monsky)

Let

${}g=z^{4}+z^{2}xy+z(x^{3}+y^{3})+(t+t^{2})x^{2}y^{2}\,.$

Consider

${}S={\mathbb {F} }_{2}[t,x,y,z]/(g)\,.$

Then Monsky proved the following results on the Hilbert-Kunz multiplicity of the maximal ideal ${}(x,y,z)$ in ${}S\otimes _{{\mathbb {F} }_{2}[t]}L$, ${}L$ a field:

${}e_{HK}(S\otimes _{\mathbb {F} _{2}[t]}L)={\begin{cases}3{\text{ for }}L={\mathbb {F} }_{2}(t)\\3+{\frac {1}{4^{d}}}{\text{ for }}L={\mathbb {F} }_{q}={\mathbb {F} }_{2}(\alpha ),\,(t\mapsto \alpha ,\,d=\deg(\alpha ))\,.\end{cases}}\,$

By the geometric interpretation of Hilbert-Kunz theory this means that the restricted cotangent bundle

$\operatorname {Syz} \,(x,y,z)=(\Omega _{{\mathbb {P} }^{2}})_{C}$

is strongly semistable in the transcendental case, but not strongly semistable in the algebraic case. In fact, for ${}d=\deg(\alpha )$, ${}t\mapsto \alpha$, where ${}K=\mathbb {F} _{2}(\alpha )$, the ${}d$-th Frobenius pull-back destabilizes.

The maximal ideal ${}(x,y,z)$ can not be used directly. However, we look at the second Frobenius pull-back which is (characteristic two) just

$I=(x^{4},y^{4},z^{4}).$

By the degree formula we have to look for an element of degree ${}6$. Let's take

$f=y^{3}z^{3}.$

This is our example ($x^{3}y^{3}$ does not work). First, by strong semistability in the transcendental case we have

$f\in I^{*}{\text{ in }}R\otimes {\mathbb {F} }_{2}(t)$

by the degree formula. If localization would hold, then ${}f$ would also belong to the tight closure of ${}I$ for almost all algebraic instances ${}{\mathbb {F} }_{q}={\mathbb {F} }_{2}(\alpha )$, ${}t\mapsto \alpha$. Contrary to that we show that for all algebraic instances the element ${}f$ belongs never to the tight closure of ${}I$.

Lemma (Monsky)

Let

${}{\mathbb {F} }_{q}={\mathbb {F} }_{p}(\alpha )$, ${}t\mapsto \alpha$, ${}\deg(\alpha )=d$. Set ${}Q=2^{d-1}$. Then

${}xyf^{Q}\notin I^{[Q]}\,.$

Proof

This is an elementary but tedious computation.

$\Box$

Theorem (Brenner-Monsky)

Tight closure does not commute with localization.

Proof

One knows in our situation that ${}xy$ is a so-called test element. Hence the previous Lemma shows that ${}f\notin I^{*}$.

$\Box$

Corollary

Tight closure is not plus closure in graded dimension two for fields with transcendental elements.

Proof

Consider

${}R={\mathbb {F} }_{2}(t)[x,y,z]/(g)\,.$

In this ring ${}y^{3}z^{3}\in I^{*}$,

but it can not belong to the plus closure. Else there would be a curve morphism ${}Y\to C_{{\mathbb {F} }_{2}(t)}$ which annihilates the cohomology class ${}c$ and this would extend to a morphism of relative curves almost everywhere.

$\Box$

Corollary

There is an example of a smooth projective

(relatively over the affine line) variety ${}Z$ and an effective divisor ${}D\subset Z$ and a morphism

$Z\longrightarrow {\mathbb {A} }_{{\mathbb {F} }_{2}}^{1}$

such that ${}(Z\setminus D)_{\eta }$ is not an affine variety over the generic point ${}\eta$, but for every algebraic point ${}x$ the fiber ${}(Z\setminus D)_{x}$ is an affine variety.

Proof

Take

${}C\rightarrow {\mathbb {A} }_{{\mathbb {F} }_{2}}^{1}$ to be the Monsky quartic and consider the syzygy bundle

${}{\mathcal {S}}=\operatorname {Syz} {\left(x^{4},y^{4},z^{4}\right)}(6)\,$

together with the cohomology class ${}c$ determined by ${}f=y^{3}z^{3}$. This class defines an extension

$0\longrightarrow {\mathcal {S}}\longrightarrow {\mathcal {S}}'\longrightarrow {\mathcal {O}}_{C}\longrightarrow 0$

and hence ${}{\mathbb {P} }({\mathcal {S}}^{*})\subset {\mathbb {P} }({{\mathcal {S}}'}^{*})$.

Then ${}{\mathbb {P} }({{\mathcal {S}}'}^{*})\setminus {\mathbb {P} }({\mathcal {S}}^{*})$ is an example with the stated properties by the previous results.

$\Box$