Kurs:Vector bundles, forcing algebras and local cohomology (Tehran 2012)/Lecture 8

Affineness under deformations

We consider a base scheme ${}B$ and a morphism

$Z\longrightarrow B$

together with an open subscheme ${}W\subseteq Z$. For every base point ${}b\in B$ we get the open subset

${}W_{b}\subseteq Z_{b}\,$

inside the fiber ${}Z_{b}$. It is a natural question to ask how properties of ${}W_{b}$ vary with ${}b$. In particular we may ask how the cohomological dimension of ${}W_{b}$ varies and how the affineness may vary.

In the algebraic setting we have a ${}D$-algebra ${}S$ and an ideal ${}{\mathfrak {a}}\subseteq S$ (so ${}B=\operatorname {Spec} {\left(D\right)}$, ${}Z=\operatorname {Spec} {\left(S\right)}$ and ${}W=D({\mathfrak {a}})$) which defines for every prime ideal ${}{\mathfrak {p}}\in \operatorname {Spec} {\left(D\right)}$ the extended ideal ${}{\mathfrak {a}}_{\mathfrak {p}}$ in ${}S\otimes _{D}\kappa ({\mathfrak {p}})$.

This question is already interesting when ${}B=\operatorname {Spec} {\left(D\right)}$ is an affine one-dimensional integral scheme, in particular in the following two situations.

1. ${}B=\operatorname {Spec} {\left(\mathbb {Z} \right)}$. Then we speak of an arithmetic deformation and want to know how affineness varies with the characteristic and what the relation is to characteristic zero.
2. ${}B={\mathbb {A} }_{K}^{1}=\operatorname {Spec} {\left(K[t]\right)}$, where ${}K$ is a field. Then we speak of a geometric deformation and want to know how affineness varies with the parameter ${}t$, in particular how the behavior over the special points where the residue class field is algebraic over ${}K$ is related to the behavior over the generic point.

It is fairly easy to show that if the open subset in the generic fiber is affine, then also the open subsets are affine for almost all special points.

We deal with this question where ${}W$ is a torsor over a family of smooth projective curves (or a torsor over a punctured twodimensional spectrum). The arithmetic as well as the geometric variant of this question are directly related to questions in tight closure theory. Because of the above mentioned degree criteria in the strongly semistable case, a weird behavior of the affineness property of torsors is only possible if we have a weird behavior of strong semistability.

Arithmetic deformations

We start with the arithmetic situation, the following example is due to Brenner and Katzman.

Example

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 terms of affineness (or local cohomology) this example has the following properties: the ideal

${}(x,y,z)\subseteq \mathbb {Z} /(p)[x,y,z,s_{1},s_{2},s_{3}]/{\left(x^{7}+y^{7}+z^{7},s_{1}x^{4}+s_{2}y^{4}+s_{3}z^{4}+x^{3}y^{3}\right)}\,$

has cohomological dimension ${}1$ if ${}p=3\mod 7$ and has cohomological dimension ${}0$ (equivalently, $D(x,y,z)$ is an affine scheme) if ${}p=2\mod 7$.

Geometric deformations - A counterexample to 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 ${}R$?

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 ${}{\mathfrak {p}}$.

$\Box$

In order to get a counterexample to the localization property we will look now at geometric deformations:

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

where ${}t$ has degree ${}0$ and ${}x,y,z$ have degree ${}1$ and ${}g$ is homogeneous. Then (for every field ${}{\mathbb {F} }_{p}[t]\subseteq K$)

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

is a two-dimensional standard-graded ring over ${}K$. For residue class fields of points of ${}{\mathbb {A} }_{{\mathbb {F} }_{p}}^{1}=\operatorname {Spec} {\left({\mathbb {F} }_{p}[t]\right)}$ 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.

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, in terms of Hilbert-Kunz theory, by Paul Monsky in 1998.

Example

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

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

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 ${}L=\mathbb {F} _{2}(\alpha )$, the ${}d$-th Frobenius pull-back destabilizes (meaning that it is not semistable anymore).

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

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

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 ${}I$.

Lemma

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

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 ${}f\notin I^{*}$.

$\Box$

In terms of affineness (or local cohomology) this example has the following properties: the ideal

${}(x,y,z)\subseteq {\mathbb {F} }_{2}(t)[x,y,z,s_{1},s_{2},s_{3}]/{\left(g,s_{1}x^{4}+s_{2}y^{4}+s_{3}z^{4}+y^{3}z^{3}\right)}\,$

has cohomological dimension ${}1$ if ${}t$ is transcendental and has cohomological dimension ${}0$ (equivalently, $D(x,y,z)$ is an affine scheme) if ${}t$ is algebraic.

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.

$\Box$

It is an open question whether such an example can exist in characteristic zero.

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