Vektorbündel/Abschlussoperationen/Problemliste/Textabschnitt/en

Problem

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

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

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

Problem

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

Problem

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

$T\longrightarrow Y.$

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

$\varphi \colon Y'\longrightarrow Y$

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

Problem

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

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

Problem (Miyaoka)

Let

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

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

Problem

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