Timeline for Are extensions of linear algebraic groups (over a field) themselves linear algebraic?
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15 events
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| Oct 14, 2012 at 6:29 | comment | added | grp | Dear Michael: I agree with your final comment, but I think that the notion of "reduced group scheme" (of finite type, say) over a field is uninteresting/useless except over a perfect field, for which it implies smoothness (and so is preserved under base change). The point is that if $G$ is a possibly non-reduced group scheme (of finite type) over an imperfect field then $G_{\rm{red}}$ is typically not a subgroup scheme. Results like the one you cite from SGA3 seem to be examples of largely useless generality. | |
| Oct 13, 2012 at 19:29 | comment | added | Michael Thaddeus | But about the imperfect field, once more, the point is that Olivier Benoist doesn't have to say "geometrically reduced." "Reduced" will suffice. | |
| Oct 13, 2012 at 19:27 | comment | added | Michael Thaddeus | Indeed, that's probably what Vistoli was referring to as the "basic result." | |
| Oct 13, 2012 at 18:46 | comment | added | grp | Dear Michael: The point of my Frobenius example was just that the notion of "reduced group" is very delicate when the ground field isn't perfect (and as I guessed, OB meant to say geometric reducedness). Anyway, bringing in stacks for group quotients over a field seems like overkill: by SGA3, for any lft group $G$ over an artin local ring and flat closed subgroup scheme $H$, there is an fppf scheme map $G \rightarrow X$ identifying $X$ with the quotient sheaf $G/H$ (so a group when $H$ is functorially normal) and having all properties one could desire, such as compatibility with base change. | |
| Oct 13, 2012 at 15:53 | vote | accept | Michael Thaddeus | ||
| Oct 13, 2012 at 15:50 | comment | added | Michael Thaddeus | I think the point about imperfect fields is a little bit garbled. "Reduced" should be fine. The reduced but not smooth group $G$ referred to by grp is the subgroup of $G_a \times G_a$ given by $x^p-ty^p=0$ for $t \in F \backslash F^p$. But then the relative Frobenius is a homomorphism $G \to G^{(p)}$, and $G^{(p)}$ is not reduced. On the other hand, SGA3 VIA 6.2 says that if $f: C \to A$ is a quasi-compact and dominant morphism of group schemes over a field with $A$ reduced, then $f$ is faithfully flat. | |
| Oct 13, 2012 at 15:45 | comment | added | Michael Thaddeus | So the upshot of grp's first comment is that it depends how we define a quotient of group schemes? If (following Angelo) we define $C/B$ as the stack-theoretic quotient over, say, the fppf site, then the existence of local fppf sections is built into the definition, but the key theorem is that this is represented by a scheme. Alternatively, one could try to define $C/B$ as a categorical quotient or something else, but one would have to show that this is equivalent to Angelo's definition. If I'm not mistaken, Serre (in Algebraic Groups & Class Fields VII 1.1) artfully evades the question. | |
| Oct 12, 2012 at 18:15 | comment | added | Olivier Benoist | @grp : You're right, I meant geometrically reduced, and 'surjective' might be replaced by 'an epimorphism'. | |
| Oct 12, 2012 at 17:36 | comment | added | grp | @Olivier: Since Michael seems to be working over a general field, one should say "smooth" rather than "reduced". As you know, over any imperfect field there are reduced linear algebraic groups that are not smooth, and their relative Frobenius morphism is a finite surjective homomorphism which is not flat. Also, I see your intent by saying "flatness should somehow be part of the definition of being surjective", but this seems a bit risky since surjective has its own useful (ordinary) meaning for scheme maps. However,"fppf" requires fewer letters than "surjective", so using French solves it. :) | |
| Oct 12, 2012 at 16:26 | history | edited | Angelo | CC BY-SA 3.0 |
added 664 characters in body; added 146 characters in body
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| Oct 12, 2012 at 16:14 | comment | added | Olivier Benoist | The map $C\to A$ is itself fppf (if your algebraic groups are reduced, flatness follows from generic flatness and homogeneity ; if they are not necessarily reduced, flatness should be somehow part of the definition of being surjective). Then you can base change $C\to A$ by the map $C\to A$ itself : here you have a tautological section. | |
| Oct 12, 2012 at 16:08 | comment | added | grp | @Michael: I assumed (and probably Angelo did too) that you know the equivalence of several equivalent definitions of "group extension" (without which it is hard to work with this concept in a nice way). What definition are you using (especially if you aren't assuming smoothness of the groups)? | |
| Oct 12, 2012 at 15:19 | comment | added | Michael Thaddeus | PS: sorry, of course I meant local sections | |
| Oct 12, 2012 at 15:18 | comment | added | Michael Thaddeus | Very good, but that's basically the same argument I briefly indicated, only with the fppf or fpqc topology replacing the Zariski topology. My question for you, and for commenter grp above, is why do sections exist in the fppf or fpqc topologies? That is, why is a group extension a torsor for such topologies? I'll see if I can find this in Oort's book. | |
| Oct 12, 2012 at 15:07 | history | answered | Angelo | CC BY-SA 3.0 |