Benutzer:Holger Brenner/Talk in Bengaluru December 2010

Arithmetic and geometric deformations of tight closure problems

We consider a linear homogeneous equation

f_{1}t_{1}+\cdots +f_{n}t_{n}=0

and also a linear inhomogeneous equation

f_{1}t_{1}+\cdots +f_{n}t_{n}=f_{0},

where ${}f_{1},\ldots ,f_{n},f_{0}$ are elements in a field ${}K$ . The solution set to the homogeneous equation is a vector space ${}V$ over ${}K$ of dimension ${}n-1$ or ${}n$ (if all ${}f_{i}$ are ${}0$ ). For the solution set ${}T$ of the inhomogeneous equation there exists an action

V\times T\longrightarrow T,\,(v,t)\longmapsto v+t,

and if we fix one solution ${}t_{0}\in T$

(supposing that one solution exists), then there exists a bijection

V\longrightarrow T,\,v\longmapsto v+t_{0}.

Suppose now that ${}X$ is a geometric object (a topological space, a manifold, a variety, the spectrum of a ring) and that

f_{1},\ldots ,f_{n},f_{0}\colon X\longrightarrow K

are functions on ${}X$ . Then we get the space

T={\left\{(P,t_{1},\ldots ,t_{n})\mid f_{1}(P)t_{1}+\cdots +f_{n}(P)t_{n}=f_{0}(P)\right\}}\subseteq X\times K^{n}

together with the projection to ${}X$ . For a fixed point ${}P\in X$ , the fiber of ${}T$ over ${}P$ is the solution set to the corresponding inhomogeneous equation. For ${}f_{0}=0$ , we get a solution space

V\longrightarrow X,

where all fibers are vector spaces

(maybe of non-constant dimension) and where again ${}V$ acts on ${}T$ . Locally, there are bijections ${}V\cong T$ . Let

U={\left\{Q\in X\mid f_{i}(Q)\neq 0{\text{ for at least one }}i\right\}}.

Then ${}V{|}_{U}$ is a vector bundle and ${}T{|}_{U}$ is a ${}V{|}_{U}$ -principal fiber bundle.

${}T$ is fiberwise an affine space over the base and locally an affine space over ${}U$ , so locally it is an easy object. We are interested in global properties of ${}T$ and of ${}T{|}_{U}$ .

We describe now the algebraic setting.

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).

This algebra was introduced by Hochster. The forcing algebra forces that ${}f$ belongs to the extended ideal ${}(f_{1},\ldots ,f_{n})B$ . It yields a scheme morphism

\varphi \colon {\rm {Spec}}\,B\longrightarrow X={\rm {Spec}}\,R.

We are interested in the relationship:

How is ${}f$ related to ${}I$ ?

Does ${}f$ belong to certain closure operations of ${}I$ ? ${}\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Longleftrightarrow \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$ Properties of ${}\varphi$ .

Examples

f\in I\Longleftrightarrow \varphi {\text{ has a scheme-section}}.

f\in {\rm {rad}}\,I\Longleftrightarrow \varphi {\text{ is surjective}}.

f\in {\overline {I}}\,\,({\text{integral closure}})\Longleftrightarrow \varphi {\text{ is a universal submersion}}.

Tight closure

We want to deal with tight closure, a closure operation introduced by Hochster and Huneke.

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

Theorem (Hochster)

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).

In this talk we will focus on four problems of tight closure.

Is there a geometric interpretation for tight closure?
Are there situations where one can determine the tight closure of an ideal?
How does tight closure depend on the prime characteristic?
The localization problem of tight closure.

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

?

Geometric interpretation in dimension two

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.

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 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. For this 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 ${}\operatorname {Syz}$ .

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 possibilty of such a deformation:

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

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

S\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.

How does ${}f\in I^{*}$ vary with ${}K$ ? 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 }}S\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 ${}I$ .