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Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$.

These statements are valid in the fpqc topology: that is, they are true when considering algebraic groups as group objects in the category of sheaves on the big (affine) fpqc site of $\mathrm{Spec}\;\mathbb{C}$. My question is:

Can I lift this to the étale topology?

Even if this cannot be done in general, is there a know set of conditions on $G$ (maybe on $N$ too) that guarantees that the map $G \to G/N$ is also a quotient in the étale topology?

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What does "any normal subgroup $N$" mean? Not every subgroup sheaf of the functor represented by $G$ is represented by an affine algebraic group (e.g., can make $N$ representable not by an affine group scheme). You meant to ask: for a closed normal subgroup scheme $N$ of $G$, is there an fppf (hence fpqc!) homomorphism of affine algebraic groups $G\rightarrow Q$ with kernel $N$, and if so then is $G\rightarrow Q$ also etale? The 1st is affirmative over any field (use generic flatness and geometric translations), the 2nd is an exercise in fppf descent theory since $N$ is smooth (char. 0!). – user29283 Jun 11 '13 at 23:55
The phrase "According to SGA3" could use some more precision. – user29720 Jun 12 '13 at 5:48
@xuhan: I did mean closed. Edited. Thanks. – Alberto García-Raboso Jun 12 '13 at 14:49
@kreck: the citation that I've seen is to SGA3-I VI-B 9.2, but it is a bit difficult to parse... – Alberto García-Raboso Jun 12 '13 at 14:51
@Alberto: Referring to that part of SGA3 for this is puzzling, since Theorem 3.2 in VI$_{\rm{A}}$ is easier to parse. But the usual textbooks construct $G \rightarrow Q$ with kernel $N$ in down-to-earth ways, so "all" you need to identify $Q$ with a sheaf quotient $G/N$ is that a surjective homomorphism $f$ between smooth affine groups is flat (and even smooth in char. 0). As in my previous comment: use translations to get flatness from generic flatness, and then fppf descent to get smoothness of $f$ from that of $\ker f$. And as Angelo notes, smooth surjections have etale-local sections.QED – user29720 Jun 13 '13 at 3:20
up vote 10 down vote accepted

The map $G\to G/N$ is always a quotient in the étale topology. Since you are in characteristic $0$, the group scheme $N$ is smooth. Since $G \to G/N$ is an $N$-torsor, because the action of $N$ on $G$ is free, and an fpqc torsor under a smooth group is also and étale torsor, because a smooth map always has local sections in the étale topology.

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