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

Suppose that $I$ admits a divided power structure. On the one hand, $\gamma_p(x_1x_2+x_3x_4+x_5x_6)$ has to be equal to zero because the element $x_1x_2+x_3x_4+x_5x_6$ is zero in our ring, but let's e …

A reference for 1. is https://stacks.math.columbia.edu/tag/07S7 We can then answer question 2. like this: The quotient morphism $E\to E/C$ is faithfully flat (this is a part of the conclusion of the l …

It is not right exact. Assume that $k$ is algebraically closed. If the map $Res_{X/k}B\to Res_{X/k}C$ was surjective as a map of sheaves for the fppf topology, then in particular, the map on sections …

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 …

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

$\newcommand{\Spec}{\mathrm{Spec}}\newcommand{\oL}{\overline{L}}\newcommand{\bG}{\mathbb{G}}\newcommand{\bZ}{\mathbb{Z}}\newcommand{\cL}{\mathcal{L}}$Not in general. The sequence of $p$-divisible grou …

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 …

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

$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical poi …

$\newcommand{\cO}{\mathcal{O}}$It seems that your formula for the etale case indeed gives the answer in general, if it is paraphrased in terms of rings of functions. Consider the coaction map $\Delta: …

Let $X$ be a projective variety over a field $k$ equipped with a very ample line bundle $\mathcal{O}_X(1)$. Suppose that $E, F$ are locally free sheaves of finite rank on $X$ and $c\in \mathrm{Ext}^i( …

$\newcommand{\cO}{\mathcal{O}}\newcommand{\bZ}{\mathbb{Z}}$Let's first work out the case $\mathcal{E}=\mathcal{O}_X$. We want a space $E\to X$ such that $Hom_X(S, E)=\cO_S(S)=Hom(S,\mathbb{A}^1)$. Her …

For a finite flat cover $\pi:Y\to X$ the pushforward $E:=\pi_*\mathcal{O}_Y$ comes with a morphism $F^*E\to E$ induced by the Frobenius on $Y$. If $\pi$ is etale this morphism is an isomorphism: over …

$\newcommand{\cO}{\mathcal{O}}$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 …