# Vektorbündel/Abschlussoperationen/Problemliste/Textabschnitt/en

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## Problem

Let ${\displaystyle {}K}$ denote a finite field and let ${\displaystyle {}R}$ be a domain of finite type over ${\displaystyle {}K}$. Is

${\displaystyle I^{*}=I^{+}}$

for every ideal ${\displaystyle {}I\subseteq R}$?

## Problem

Let ${\displaystyle {}Y}$ be an abelian variety in positive characteristic ${\displaystyle {}p}$ and let ${\displaystyle {}c\in H^{1}(Y,{\mathcal {S}})}$ be a cohomology class of a vector bundle such that the cohomological dimension of the corresponding torsor equals the dimension of ${\displaystyle {}Y}$. Is it true that ${\displaystyle {}c}$ can be annihilated by an iteration of the ${\displaystyle {}[p]}$-multiplication?

## Problem

Let ${\displaystyle {}Y}$ denote a smooth projective variety of dimension ${\displaystyle {}d}$ over a finite field ${\displaystyle {}K}$. Let ${\displaystyle {}{\mathcal {S}}}$ be a vector bundle on ${\displaystyle {}Y}$ and ${\displaystyle {}c\in H^{1}(Y,{\mathcal {S}})}$ be a cohomolgy class with corresponding torsor

${\displaystyle T\longrightarrow Y.}$

Does ${\displaystyle {}T}$ has cohomological dimension ${\displaystyle {}d}$ if and only if there exists a finite morphism

${\displaystyle \varphi \colon Y'\longrightarrow Y}$

(where ${\displaystyle {}Y'}$ denotes another (smooth?) projective variety of dimension ${\displaystyle {}d}$) such that ${\displaystyle {}\varphi ^{*}(c)=0}$?

## Problem

Give, for every prime number ${\displaystyle {}p\geq 3}$, an example of an ${\displaystyle {}{\mathbb {F} }_{p}[t]}$-algebra ${\displaystyle {}S}$, an ideal ${\displaystyle {}I\subseteq S}$ and an element ${\displaystyle {}f\in S}$ such that ${\displaystyle {}f\not \in I^{*}}$ in ${\displaystyle {}S_{{\mathbb {F} }_{p}(t)}}$, but ${\displaystyle {}f\in I^{*}}$ in ${\displaystyle {}S\otimes _{{\mathbb {F} }_{p}[t]}{\mathbb {F} }_{p}[t]/{\mathfrak {m}}}$ for every maximal ideal ${\displaystyle {}{\mathfrak {m}}}$.

Is this possible for the ring ${\displaystyle {}S={\mathbb {F} }_{p}[t][x,y,z]/(z^{4}-xy(x+y)(x+ty))}$?

## Problem (Miyaoka)

Let

${\displaystyle Y\longrightarrow \operatorname {Spec} _{}^{}{\left(D\right)}}$

denote a relative smooth projective curve over an onedimensional affine integral base of mixed characteristic (of finite type over ${\displaystyle {}\mathbb {Z} }$). Suppose that ${\displaystyle {}{\mathcal {S}}}$ is a vector bundle on ${\displaystyle {}Y}$ such that the generic bundle ${\displaystyle {}{\mathcal {S}}_{\eta }}$ over the generic curve ${\displaystyle {}Y_{\eta }}$ (in characteristic ${\displaystyle {}0}$) is semistable. Is it true that for infinitely many points ${\displaystyle {}{\mathfrak {m}}\in \operatorname {Spec} _{}^{}{\left(D\right)}}$ the bundle ${\displaystyle {}{\mathcal {S}}_{\mathfrak {m}}}$ over the curve ${\displaystyle {}Y_{\mathfrak {m}}}$ (in positive characteristic) is strongly semistable?

## Problem

Let ${\displaystyle {}D}$ denote an onedimensional domain of mixed characteristic (of finite type over ${\displaystyle {}\mathbb {Z} }$) and let ${\displaystyle {}D\subseteq S}$ be a domain of finite type of relative dimension ${\displaystyle {}2}$. Let ${\displaystyle {}I\subseteq S}$ be an ideal and ${\displaystyle {}f\in S}$. Suppose that ${\displaystyle {}f}$ belongs to the solid closure of ${\displaystyle {}I}$ inside the generic ring ${\displaystyle {}S_{Q(D)}}$ (in characteristic ${\displaystyle {}0}$). Is it true that ${\displaystyle {}f\in I^{*}}$ holds for infinitely many points ${\displaystyle {}{\mathfrak {m}}\in \operatorname {Spec} _{}^{}{\left(D\right)}}$ (in positive characteristic)?