# Vector bundles and tight closure (Triest 2023)/Lecture 4/latex

\setcounter{section}{4}
After having understood tight closure and plus closure in the two-dimensional situation we proceed to a special three-dimensional situation, namely families of two-dimensional rings parametrized by a one-dimensional base scheme.
\zwischenueberschrift{Affineness under deformations}
We consider a base scheme $B$ and a morphism \mathdisp {Z \longrightarrow B} { } together with an open subscheme \mavergleichskette {\vergleichskette { W } { \subseteq }{ Z } { }{ } { }{ } { }{ } }{.} For every base point \mavergleichskette {\vergleichskette { b } { \in }{ B } { }{ } { }{ } { }{ } } we get the open subset \mavergleichskettedisp {\vergleichskette { 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 \zusatzklammer {the cohomological dimension of a scheme $X$ is the maximal number $i$ such that \mavergleichskette {\vergleichskette { H^{i}(X, {\mathcal F}) } { \neq }{ 0 } { }{ } { }{ } { }{ } } for some quasicoherent sheaf ${\mathcal F}$. A noetherian scheme is affine if and only if its cohomological dimension is $0$. Tight closure can be characterized by the cohomological dimension of torsors} {.} may vary.
In the algebraic setting, we have a commutative $K$-algebra $D$, a commutative $D$-algebra $S$ and an ideal \mavergleichskette {\vergleichskette { {\mathfrak a} } { \subseteq }{ S } { }{ } { }{ } { }{ } } \zusatzklammer {so \mavergleichskettek {\vergleichskettek { B } { = }{ \operatorname{Spec} { \left( D \right) } } { }{ } { }{ } { }{ } } \mavergleichskettek {\vergleichskettek { Z } { = }{ \operatorname{Spec} { \left( S \right) } } { }{ } { }{ } { }{ } } and \mavergleichskettek {\vergleichskettek { W } { = }{ D( {\mathfrak a} ) } { }{ } { }{ } { }{ } }} which defines for every prime ideal \mavergleichskette {\vergleichskette { {\mathfrak p}
} { \in }{ \operatorname{Spec} { \left( D \right) } } { }{ } { }{ } { }{ } } the extended ideal \mathl{{\mathfrak a}_{ {\mathfrak p} }} in \mathl{S \otimes_{ D } \kappa( {\mathfrak p} )}{.} Then in this situation, \mavergleichskette {\vergleichskette { D ( {\mathfrak a}_{\mathfrak p} ) } { \subseteq }{ \operatorname{Spec} { \left( S \otimes_D \kappa({\mathfrak p}) \right) } } { }{ } { }{ } { }{ } } is the fiber over ${\mathfrak p}$.
This question is already interesting when \mavergleichskette {\vergleichskette { B } { = }{ \operatorname{Spec} { \left( D \right) } } { }{ } { }{ } { }{ } } is an affine one-dimensional integral scheme, in particular in the following two situations. \aufzaehlungzwei { \mavergleichskette {\vergleichskette { B } { = }{ \operatorname{Spec} { \left( \Z \right) } } { }{ } { }{ } { }{ } }{.} Then we speak of an \stichwort {arithmetic deformation} and want to know how affineness varies with the characteristic and what the relation is to characteristic zero. } { \mavergleichskette {\vergleichskette { B } { = }{ {\mathbb A}^{1}_{K} } { = }{ \operatorname{Spec} { \left( K[t] \right) } } { }{ } { }{ } }{,} where $K$ is a field. Then we speak of a \stichwort {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 \zusatzklammer {or a torsor over a punctured two-dimensional 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 \zusatzklammer {see Theorem 2.11 } {,} a weird behavior of the affineness property of torsors is only possible if we have a weird behavior of strong semistability.
\zwischenueberschrift{Arithmetic deformations}
We start with the arithmetic situation, the following example is due to Brenner and Katzman \cite{brennerkatzmanarithmetic}.
\inputbeispiel {
Consider \mathl{\Z[x,y,z]/ { \left( x^7+y^7+z^7 \right) }} and take the ideal \mavergleichskette {\vergleichskette { I } { = }{ { \left( x^4,y^4, z^4 \right) } } { }{ } { }{ } { }{ } } and the element \mavergleichskette {\vergleichskette { f } { = }{ x^3y^3 } { }{ } { }{ } { }{ } }{.} Consider reductions \mathl{\Z \rightarrow \Z/(p)}{.} Then \mathdisp {f \in I^* \text{ holds in } \Z/(p) [x,y,z]/(x^7+y^7+z^7) \text{ for } p \equiv 3 \! \! \! \mod 7} { } and \mathdisp {f \not\in I^* \text{ holds in } \Z/(p) [x,y,z]/(x^7+y^7+z^7) \text{ for } p \equiv 2 \! \! \! \mod 7} { . }
In particular, the bundle \mathl{\operatorname{Syz} { \left(x^4,y^4,z^4 \right) }} is semistable in the generic fiber, but not strongly semistable for any reduction \mavergleichskette {\vergleichskette { 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 \zusatzklammer {or local cohomology} of quasiaffine schemes, this example has the following properties: the open subset given by the ideal \mavergleichskettedisphandlinks {\vergleichskettedisphandlinks { (x,y,z) } { \subseteq} { \Z/(p) [x,y,z,s_1,s_2,s_3]/ { \left( x^7+y^7+z^7, s_1x^4+s_2y^4+s_3z^4+ x^3y^3 \right) }
} { } { } { } { } { } { } } has cohomological dimension $1$ if \mavergleichskette {\vergleichskette { p } { = }{ 3 \mod 7 } { }{ } { }{ } { }{ } } and has cohomological dimension $0$ \zusatzklammer {equivalently, \mathlk{D(x,y,z)} is an affine scheme} if \mavergleichskette {\vergleichskette { p } { = }{ 2 \mod 7 } { }{ } { }{ } { }{ } }{.}
\zwischenueberschrift{Geometric deformations - A counterexample to the localization problem}
Let \mavergleichskette {\vergleichskette { S } { \subseteq }{ R } { }{ } { }{ } { }{ } } be a multiplicative system and $I$ an ideal in $R$. Then the \betonung{localization problem} of tight closure is the question whether the identity \mavergleichskettedisp {\vergleichskette { (I^*)_S } { =} { (IR_S)^* } { } { } { } { } { } { } } holds.
Here the inclusion $\subseteq$ is always true and $\supseteq$ is the problem. The problem means explicitly:
If \mavergleichskette {\vergleichskette { f } { \in }{ (IR_S)^* } { }{ } { }{ } { }{ } }{,} can we find an \mavergleichskette {\vergleichskette { h } { \in }{ S } { }{ } { }{ } { }{ } } such that \mavergleichskette {\vergleichskette { hf } { \in }{ I^* } { }{ } { }{ } { }{ } } holds in $R$?
\faktsituation {Let \mavergleichskette {\vergleichskette { \Z/(p) } { \subset }{ D } { }{ } { }{ } { }{ } } be a one-dimensional domain, \mavergleichskette {\vergleichskette { D } { \subseteq }{ R } { }{ } { }{ } { }{ } } of finite type and $I$ an ideal in $R$. Suppose that localization holds and that \mathdisp {f \in I^* \text{ holds in } R \otimes_DQ(D) =R_{D^*} = R_{Q(D)}} { } \zusatzklammer {\mavergleichskettek {\vergleichskettek { S } { = }{ D^* } { = }{ D \setminus \{0\} } { }{ } { }{ } } is the multiplicative system} {.}} \faktfolgerung {Then \mavergleichskette {\vergleichskette { f } { \in }{ I^* } { }{ } { }{ } { }{ } } holds in \mathl{R \otimes_D \kappa( {\mathfrak p} )} for almost all ${\mathfrak p}$ in Spec $D$.} \faktzusatz } {
By localization, there exists \mavergleichskette {\vergleichskette { h } { \in }{ D } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { h } { \neq }{ 0 } { }{ } { }{ } { }{ } }{,} such that \mavergleichskette {\vergleichskette { hf } { \in }{ I^* \text{ in } R } { }{ } { }{ } { }{ } }{.} By persistence of tight closure \zusatzklammer {under a ring homomorphism} {,} we get \mathdisp {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 \mathl{R_{\kappa( {\mathfrak p} )}} and hence \mathdisp {f \in I^* \text{ in } R_{\kappa({\mathfrak p})}} { } for almost all ${\mathfrak p}$.
In order to get a counterexample to the localization property we will look now at geometric deformations: \mavergleichskettedisp {\vergleichskette { D } { =} { {\mathbb F}_p[t] } { \subset} { {\mathbb F}_p[t][x,y,z]/(g) } { =} { S } { } { } }{,} where $t$ has degree $0$ and \mathl{x,y,z} have degree $1$ and $g$ is homogeneous. Then \zusatzklammer {for every homomorphism ${\mathbb F}_p[t] \rightarrow K$ to a field} \mathdisp {S \otimes_{ {\mathbb F}_p [t]} K} { }
is a two-dimensional standard-graded ring over $K$. For the residue class fields of points of \mavergleichskette {\vergleichskette { {\mathbb A}^{1}_{ {\mathbb F}_p } } { = }{ \operatorname{Spec} { \left( {\mathbb F}_p [t] \right) } } { }{ } { }{ } { }{ } } we have basically two possibilities. \auflistungzwei{ \mavergleichskette {\vergleichskette { K } { = }{ {\mathbb F}_p (t) } { }{ } { }{ } { }{ } }{,} the function field. This is the \betonung{generic} or \betonung{transcendental} case. }{ \mavergleichskette {\vergleichskette { K } { = }{ {\mathbb F}_q } { }{ } { }{ } { }{ } }{,} the \betonung{special} or \betonung{algebraic} or \betonung{finite} case.}
How does \mavergleichskette {\vergleichskette { 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 \cite{monskypoints4quartics}.
\inputbeispiel {
Let \mavergleichskettedisp {\vergleichskette { g } { =} { z^4 +z ^2xy +z { \left( x^3+y^3 \right) } + { \left( t+t^2 \right) } x^2y^2 } { } { } { } { } { } { } }{.} Consider \mavergleichskettedisp {\vergleichskette { S } { =} { {\mathbb F}_2[t,x,y,z]/(g) } { } { } { } { } { } { } }{.} Then Monsky proved the following results on the \betonung{Hilbert-Kunz multiplicity} of the maximal ideal \mathl{(x,y,z)} in \mathl{S \otimes_{ {\mathbb F}_2[t]} L}{,} $L$ a field: \mavergleichskettedisphandlinks {\vergleichskettedisphandlinks { 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} } { } { } { } { } { } { } }
We consider $S$ as an ${\mathbb F}_2 [t]$-algebra, the corresponding morphism $\operatorname{Spec} { \left( S \right) } \rightarrow {\mathbb A}^{1}_{ {\mathbb F}_2 }$ and the corresponding smooth projective relative curve $C = \operatorname{Proj} { \left( S \right) } \rightarrow {\mathbb A}^{1}_{ {\mathbb F}_2 }$. The fibers are \mathl{\operatorname{Spec} { \left( S_{\kappa({\mathfrak p})} \right) }} and $C_{\kappa({\mathfrak p})}$ respectively.
By the geometric interpretation of Hilbert-Kunz theory, the computations mentioned in Example 4.3
mean that the restricted cotangent bundle \mavergleichskettedisp {\vergleichskette { \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 \mavergleichskette {\vergleichskette { d } { = }{ \deg(\alpha) } { }{ } { }{ } { }{ } }{,} \mathl{t \mapsto \alpha}{,} where \mavergleichskette {\vergleichskette { L } { = }{ \mathbb F_2(\alpha) } { }{ } { }{ } { }{ } }{,} the $d$-th Frobenius pull-back destabilizes \zusatzklammer {meaning that it is not semistable anymore} {.}
The maximal ideal \mathl{(x,y,z)} can not be used directly, as it is tightly closed. However, we look at the second Frobenius pull-back which is \zusatzklammer {characteristic two} just \mavergleichskettedisp {\vergleichskette { 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 \mavergleichskette {\vergleichskette { f } { = }{ y^3z^3 } { }{ } { }{ } { }{ } }{.} This is our example \zusatzklammer {\mathlk{x^3y^3} does not work} {.} First, by strong semistability in the transcendental case, we have \mathdisp {f \in I^* \text{ in } S \otimes {\mathbb F}_2(t)} { } by the degree formula. If localization would hold, then by Proposition 4.2 , $f$ would also belong to the tight closure of $I$ for almost all algebraic instances \mavergleichskette {\vergleichskette { {\mathbb F}_q } { = }{ {\mathbb F}_2(\alpha) } { }{ } { }{ } { }{ } }{,} \mathl{t \mapsto \alpha}{.} Contrary to that we show that for all algebraic instances, the element $f$ never belongs to the tight closure of $I$.
\faktsituation {Let \mavergleichskette {\vergleichskette { {\mathbb F}_q } { = }{ {\mathbb F}_p(\alpha) } { }{ } { }{ } { }{ } }{,} \mathl{t \mapsto \alpha}{,} \mavergleichskette {\vergleichskette { \deg(\alpha) } { = }{ d } { }{ } { }{ } { }{ } }{.} Set \mavergleichskette {\vergleichskette { Q } { = }{ 2^{d-1} } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then \mavergleichskettedisp {\vergleichskette { xy f^Q
} { \notin} { I^{[Q]} } { } { } { } { } { } { } }{.}} \faktzusatz } {
This is an elementary but tedious computation.
\faktsituation \faktfolgerung {Tight closure does not commute with localization.} \faktzusatz } {
One knows in our situation that $xy$ is a so-called test element. Hence Lemma 4.4
shows that \mavergleichskette {\vergleichskette { f } { \notin }{ I^* } { }{ } { }{ } { }{ } }{.}
In terms of affineness of quasiaffine schemes \zusatzklammer {or local cohomology} {,} this example has the following properties: the open subset given by the ideal \mavergleichskettedisphandlinks {\vergleichskettedisphandlinks { (x,y,z) } { \subseteq} { {\mathbb F}_2(t) [x,y,z,s_1,s_2,s_3]/ { \left( g, s_1x^4+s_2y^4+s_3z^4+ y^3z^3 \right) }
} { } { } { } { } { } { } } has cohomological dimension $1$ if $t$ is transcendental and has cohomological dimension $0$ \zusatzklammer {equivalently, \mathlk{D(x,y,z)} is an affine scheme} if $t$ is algebraic.
\faktsituation \faktfolgerung {Tight closure is not plus closure in graded dimension two for fields with transcendental elements.} \faktzusatz } {
Consider \mavergleichskettedisp {\vergleichskette { R } { =} { {\mathbb F}_2(t)[x,y,z]/(g) } { } { } { } { } { } { } }{.} In this ring \mavergleichskette {\vergleichskette { y^3z^3 } { \in }{ I^* } { }{ } { }{ } { }{ } }{,} but it can not belong to the plus closure. Else there would be a curve morphism \mathdisp {Y \longrightarrow C_{ {\mathbb F}_2(t)}} { }
which annihilates the cohomology class $c$ and this would extend to a morphism of relative curves over ${\mathbb A}^{1}_{{\mathbb F}_2}$ almost everywhere.
\faktsituation \faktfolgerung {There is an example of a smooth variety $Z$ and an effective divisor \mavergleichskette {\vergleichskette { D } { \subset }{ Z } { }{ } { }{ } { }{ } } and a projective morphism \mathdisp {Z \longrightarrow {\mathbb A}^{1}_{{\mathbb F}_2}} { } such that \mathl{(Z \setminus D)_\eta} is not an affine variety over the generic point $\eta$, but for every algebraic point $x$ the fiber \mathl{(Z \setminus D)_x} is an affine variety.} \faktzusatz } {
Take $C \rightarrow {\mathbb A}^{1}_{{\mathbb F}_2}$ to be the Monsky quartic and consider the syzygy bundle \mavergleichskettedisp {\vergleichskette { {\mathcal S} } { =} { \operatorname{Syz} { \left(x^4,y^4,z^4 \right) } (6) } { } { } { } { } { } { } } together with the cohomology class $c$ determined by \mavergleichskette {\vergleichskette { f } { = }{ y^3z^3 } { }{ } { }{ } { }{ } }{.} This class defines an extension \mathdisp {0 \longrightarrow {\mathcal S} \longrightarrow {\mathcal S}' \longrightarrow {\mathcal O}_C \longrightarrow 0} { } and hence \mavergleichskette {\vergleichskette { {\mathbb P}({\mathcal S}^{\vee}) } { \subset }{ {\mathbb P}({ {\mathcal S}' }^{\vee}) } { }{ } { }{ } { }{ } }{.} Then \mathl{{\mathbb P}({ {\mathcal S}' }^{\vee}) \setminus {\mathbb P}({\mathcal S}^{\vee})} is an example with the stated properties by the previous results.
It is an open question whether such an example exist in characteristic zero.