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It seems likely that you are assuming the curves are smooth and geometrically connected (e.g., you don't have in mind generalized Jacobians for singular curves), but you have omitted hypotheses on the …

The answer to the first question is affirmative if we assume the generic fiber is normal. First, some general setup (so it is well-posed to speak of "compatibility with any base change" below). Let …

Let $X$ be a smooth projective irreducible curve over an algebraically closed field $k$, and let $\omega$ be a rational 1-form on $X$, so $\sum_x {\rm{res}}_x(\omega) = 0$ (sum over $x \in X(k)$). For …

The assertion (X) is false in any characteristic $p > 0$ for any $Y$ with genus at least 2. (It is true and easy for $Y$ of genus 1, and true and easy and uninteresting for $Y$ of genus 0.) To see …

To avoid notational confusion between translation in the derived category (of abelian fppf sheaves on the category of lfp $S$-schemes) and torsion in abelian schemes, I'll denote the $n$-torsion in $A …

No. Let $R'$ be any non-excellent dvr whatsoever, hence of equicharacteristic $p > 0$, and let $t \in R'$ be a uniformizer. Let $R = \mathbf{F}_p[t]_{(t)}$. The local inclusion $R \hookrightarrow R'$ …

(This was meant to be a comment elaborating on Dan Petersen's answer, but got too long.) The error is that you cooked up an isomorphism between $F_x \circ f^{\ast}$ and $F_x$ and by means of that simp …

Since the fppf cohomology group ${\rm{H}}^1(F, \alpha_p) = F/F^p$ is visibly uncountable (where $p = {\rm{char}}(F) > 0$), perhaps you meant to assume $G$ is smooth (and then fppf cohomology coincides …

In view of Laurent's answer, one may ask more generally for a criterion on a connected reductive $K$-group $G$ to ensure that for all compact open subgroups $U$ of $G(\mathbf{A}_K^S)$ (for a finite se …

[I originally gave what I thought to be a counterexample in the affine case, but I realized it violates universal schematic dominance, so below I give a non-qc counterexample that was originally a com …

Do you really mean to consider an abelian scheme over the entire ring of integers, and not just a localization thereof? Either way, every finite flat group scheme $G$ over a domain $R$ with fraction …

This question is most "reasonable" when $V$ is projective, since at least then the automorphism functor is represented by a (locally finite type) $K$-scheme $G_V := {\rm{Aut}}_{V/K}$ and so one can tr …

This is addressing the final question at the end of the edited version of the posted question, which finally gets to the heart of the matter (and without which the earlier parts of the question don't …

It should be that $G(K)^0$ is the group of $g \in G(K)$ such that $|\chi(g)| = 1$ for all $K$-rational characters $\chi:G \rightarrow {\rm{GL}}_1$. Once this is shown, it follows that $G(K)/G(K)^0$ …

Presumably you know that $f$ is separable over $k(t)$ or else you wouldn't pose the question. Scale $x$ by $k(t)^{\times}$ so that $f$ becomes $x$-monic in $k[x,t]$ with $df/dx \ne 0$. Now the irreduc …