2
$\begingroup$

The structure of a commutative affine algebraic group $G$ over a field $k$ is understood (SGA 3): $G$ has a maximal subgroup of multiplicative type $M$ and the quotient $G/M$ is unipotent.

I am looking for any generalization of this structure theory for commutative group schemes over an integral base scheme, $X$. I'd appreciate any general comments/references but for sake of an explicit question:

By a unipotent group scheme I mean a group scheme over $X$ that has a normal series with $\mathbb G_a$-factors. Then the question is: let $G$ be a smooth commutative group scheme of finite type $G$ over an integral $\mathbb C$-scheme $X$ admitting a maximal torus $T$. Does there exist a Zariski open $U \subset X$ over which $G/T$ is unipotent in this sense?

$\endgroup$
  • $\begingroup$ Dear Pooya, It might help to clarify exactly what you mean by "maximal torus" in the relative context. Regards, $\endgroup$ – Emerton Sep 4 '12 at 3:08
  • $\begingroup$ Are there smooth commutative unipotent groups that are not vector spaces? $\endgroup$ – Will Sawin Sep 4 '12 at 3:18
  • $\begingroup$ Dear Emerton: yes! A maximal torus is a subgroup scheme $T$ of $G$ which is a torus (i.e. fpqc locally isomorphic to $\mathbb G_m^r$) and point-wise maximal: for any point $x \in X$, $T_{\bar x}$ is the maximal torus of $G_{\bar x}$ where $\bar x$ is the spectrum of the algebraic closure of $\kappa (x)$ [SGA 3-IIX, Definition 1.3]. Thanks, $\endgroup$ – Pooya Sep 4 '12 at 3:38
  • $\begingroup$ Dear Will: I think they are vector spaces (i.e. Zariski locally trivial) by my definition [Kambayashi-Miyanishi, on flat fibrations by the affine line]. $\endgroup$ – Pooya Sep 4 '12 at 3:48
  • 2
    $\begingroup$ The quotient $G/T$ exists as a smooth affine group, and by design it is unipotent on geometric fibers over the integral base in char. 0. The generic fiber is unipotent over the char. 0 function field of the base. Your definition of unipotence is wrong in char. $> 0$ (where there are many smooth connected commutative unipotent groups containing no $\mathbf{G}_a$), but in char. 0 it is equivalent to the "right" definition (the filtration condition on geometric fibers). Such a composition series on the actual (not just geometric) generic fiber spreads out over some dense open of the base. $\endgroup$ – grp Sep 4 '12 at 6:19

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.