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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?

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  • $\begingroup$ Dear Pooya, It might help to clarify exactly what you mean by "maximal torus" in the relative context. Regards, $\endgroup$
    – Emerton
    Commented Sep 4, 2012 at 3:08
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    $\begingroup$ Are there smooth commutative unipotent groups that are not vector spaces? $\endgroup$
    – Will Sawin
    Commented Sep 4, 2012 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
    Commented Sep 4, 2012 at 3:38
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    $\begingroup$ 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
    Commented Sep 4, 2012 at 6:19
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    $\begingroup$ @S. Carnahan: True, but Witt groups are nonetheless "split" (in the sense that they have a composition series over $k$ with successive quotients $k$-isomorphic to $\mathbf{G}_a$), so a more striking kind of example answering Will's question in the negative is one which doesn't even contain $\mathbf{G}_a$ as a $k$-subgroup. Of course there are no such examples when $k$ is perfect, but for imperfect $k$ Rosenlicht gave the classic example $y^p = x - a x^p$ for $a \in k - k^p$ with any $p = {\rm{char}}(k) > 0$. Tits studied this phenomenon in detail (in any dimension) in his Yale lecture notes. $\endgroup$
    – grp
    Commented Sep 5, 2012 at 18:14

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