TITLE: Some basic questions on quotient of group schemes
QUESTION [7 upvotes]: Let $S$ be a fixed base scheme and $G, H$ be group schemes over $S$. Since I am mainly interested in commutative group schemes over fields, we may assume that $G,H$ are commutative and $S$ is a field if this helps.
(1) Let $f:G\to H$ be a morphism of group schemes. To define the cokernel of this map, we need to choose which topology to work with. Some people use the fppf topology (as in van der Geer & Moonen's book) and other people use the fpqc topology (as in Cornell-Silverman). My question is: what is the difference of those two topologies in terms of group schemes? Is fppf quotient and fpqc quotient of group schemes different? Which topology do people prefer when they are working with group schemes?
(2) Let $H$ be a (normal) closed subgroup scheme of $G$. I think there are at least three plausible definitions of the quotient $G/H$:

Categorical quotient: Since $H$ naturally acts on $G$, we can think categorical quotient $G/H$ of the action $H\times G\to G$.

Fppf/fpqc quotient: $G/H$ represents the quotient of $H\to G$ in the category of fppf/fpqc sheaves.

Naive quotient: A group scheme $G/H$ with a surjective (wrt fppf/fpqc topology) map $p:G\to G/H$ such that kernel of $p$ is the inclusion $H\to G$


Are they equivalent in some good situations? In van der Geer & Moonen's book, it is proved that a fppf quotient is also a categorical quotient. But I cannot find proof nor prove other directions.
context of the question (2): Let $f:A\to B$ be an isogeny of abelian varieties with kernel $\ker f$. Then we have the dual exact sequence $0\to \widehat{B}\to \widehat{A}\to \widehat{\ker f}\to 0$. In Milne's book on abelian variety, to prove the dual exact sequence, consider $0\to \ker f\to A\to B\to 0$ as an exact sequence in the category of commutative group schemes over a field and use a long exact sequence with $\text{Hom}(-, \mathbb{G}_m)$. To use the long exact sequence, we need to prove $B$ is $A/\ker f$ as a fppf/fpqc quotient (In fact I don't know which topology to work with. This is why I ask the question (1)...). However, I only know that $B$ is the `naive quotient (3)' $A/\ker f$.
(3) Is the category of commutative group schemes over a field an abelian category? This statement is in  Milne's book on abelian variety, but I cannot find proof. The main point is existence of cokernel, i.e. representability of fppf/fpqc quotient. However, I only know the following theorem in Cornell & Silverman,
Theorem. Let $G$ be a finite type $S$-group scheme and let $H$ be a closed subgroup scheme of $G$. If $H$ is proper and flat over $S$ and if $G$ is quasi-projective over $S$, then the quotient sheaf $G/H$ is representable.
and this is too weak to prove our statement.
Also one more quick question: do you know any good reference dealing with sufficiently general group schemes? I know Shatz's paper in Cornell-Silverman, Tate's paper in Cornell-Silvermann-Stevens, and Stix's lecture note, but they focus on finite flat group schemes. Also, I know some other articles & books which mainly focus on affine algebraic groups. Are there some more general references?
Thank you for reading my stupid questions.

REPLY [2 votes]: So I think for this sort of question (quotients of flat finitely presented group schemes) the best is to use the theory of algebraic stacks and spaces. I am not an expert, so if someone could double check this that would be great.
Let $G$ be an fppf group scheme over a scheme $S$, and $H$ an fppf subgroup scheme of $G$.
Let $\mathcal{X}=[G/H]$ be the stack quotient. Since $G \times H \to G \times G$ is an fppf groupoid, it is algebraic and $G \to \mathcal{X}$ is an fppf presentation of $\mathcal{X}$. Since the inertia is $H$, it is fppf, so $\mathcal{X}$ is a gerbe over the fppf sheaf quotient $G/H$ (which is an algebraic space), and so $\mathcal{X} \to G/H$ is smooth. So $G \to G/H$ is fppf, where $G/H$ is the quotient in algebraic spaces (or in fppf sheafs).
Now if $G/H$ is a nice space, for instance qs (this is always the case in practice, for instance it is if $H \to G$ is qc), then it contains an open subscheme. If the base $S$ is a field, then since $G$ acts transitively in $G/H$ by acting on this subscheme we get that $G/H$ is a scheme (this is the same trick as for proving that a group algebraic space over a field is a group scheme. In fact we also have that an abelian algebraic space over a base $S$ is always an abelian scheme but this is harder to prove).
Remark: if $H \to G$ is proper, then $[G/H]$ is separated.