Search Results

$\def\C{\mathbb{C}}\def\OO{\mathcal{O}}\def\cL{\mathcal{E}}\def\cF{\mathcal{F}}$Let $X_0$ be a variety over $\C$ and $X=X_0\times_{\C}R$. Denote by $i:X_0\to X$ the obvious closed immersion. Let $\cL …

It seems to me that the following arguement doses not depend on characteristic. Consider a general fiber of $\varphi:\mathbb{A}^N\dashrightarrow X$. Take its closure in $\mathbb{P}^N$. It is a subvari …

$\newcommand{\bQ}{\mathbb{Q}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\fp}{\mathfrak{p}}\newcommand{\bF}{\mathbb{F}}\newcommand{\bP}{\mathbb{P}}$Here is a variation on the theme of Will Sawin's answer …

$\newcommand\Vect{\mathit{Vect}}\newcommand\Hom{\mathit{Hom}}$At least the birational tyie of a smooth projective variety can be recovered from the monoidal category of vector bundles on it. (the prev …

If the base field $k$ is algebraically closed then a section exists because of a group-theoretic property of the tame fundamental group (considering this question over an algebraically closed field $k …

Suppose that $\mathrm{char}\, k=p>0$. It is easy to give an example of a stack $V$ with an endomorphism that violates this property. Take $V=B\alpha_p$, the scaling action of $\mathbb{G}_m$ on $\alpha …

Any variety which admits a cell decomposition gives an example. Cell decomposition of $X$ is a stratification $X=\bigcup\limits_n X^n$ such that $X^n\setminus X^{n-1}=\coprod\limits_{i=1}^{k_n}\mathbb …

It seems possible to adapt Kim's example to the p-adic setting, although there is a significant discrepancy in how Picard groups of affine curves behave. We want to construct a non-trivial line bundle …

I'm not sure if this is the kind of evidence you are looking for, but since you mention the Fontaine-Mazur conjecture, let me remark that the relative version of the Fontaine-Mazur conjecture implies …

$\newcommand{\bG}{\mathbb{G}}$Let $X$ be any smooth scheme over an algebraically closed field $k$ of characteristic $p$. From the short exact sequence $0\to\mu_p\to \bG_m\to\bG_m\to 0$ of sheaves on t …

In the Russian edition it is $$(\xi_0,\xi_1,\xi_2,\dots)\leftrightarrow f(\xi_0)+pf(\xi_1)+p^2f(\xi_2)+\dots$$ where $f$ is the Teichmuller map.

$\newcommand{\cO}{\mathcal{O}}$Consider exact sequence of trivial vector bundles $$0\to\cO\xrightarrow{\left(\begin{matrix}x \\ y\end{matrix}\right)}\cO\oplus\cO\xrightarrow{\left(\begin{matrix}y & -x …

$\newcommand{\Spec}{\mathrm{Spec}\,}\newcommand{\cO}{{\cal{O}}}\newcommand{\cE}{{\cal{E}}}\DeclareMathOperator{\Sect}{Sect}$Here is an example where $\pi:E'\to X$ cannot be given a structure of a vect …

Let $K$ be a finite extension of $\mathbb{Q}_p$ with the ring of integers $\mathcal{O}_K$ and the residue field $k$. By a theorem of Berthelot and Ogus(https://link.springer.com/article/10.1007%2FBF0 …

$\newcommand{tot}{\mathbb{C}^n\setminus 0}\newcommand{tan}{\mathcal{T}_{\mathbb{P}^{n-1}}}$ Since morphism $\pi$ is affine, for any quasicoherent sheaf $\mathcal{F}$ on $\mathbb{C}^n\setminus 0$ its h …