The Structure of Solvable Groups over General Fields

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T n Ru(G) for any maximal torus T of G [Bo, 10.6(4)]. Over more general k, an analogous such structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and an element a ∈ k − kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s := k ′ ⊗k ks = ks(a) is a separable closure of k′, and k′ s p ⊂ ks. The affine Weil restriction G = Rk′/k(Gm) is an open subscheme of Rk′/k(A 1 k′) = A p k, so it is a smooth connected affine k-group of dimension p > 1. Loosely speaking, G is “k′× viewed as a k-group”. More precisely, for k-algebras R we have G(R) = (k⊗k R)× functorially in R. This commutative k-group contains an evident 1-dimensional torus T ' Gm corresponding to the subgroup R× ⊂ (k′ ⊗k R)×, and G/T is unipotent because (G/T )(ks) = (k ′ s) /(ks) × is p-torsion. In particular, T is the unique maximal torus of G. Since G(ks) = k ′ s × has no nontrivial p-torsion, G contains no nontrivial unipotent smooth connected k-subgroup. Thus, G is a commutative counterexample over k to the analogue of the semidirect product structure for connected solvable smooth affine groups over k. The appearance of imperfect fields in the preceding counterexample is essential. To explain this, recall that over a general field k, if S is a maximal k-torus in G then Sk is maximal in Gk by a theorem of Grothendieck (see [C, A.1.2]). Thus, by the conjugacy of maximal tori in Gk, G = TnU for a k-torus T and a unipotent smooth connected normal k-subgroup U ⊂ G if and only if the subgroup Ru(Gk) ⊂ Gk is defined over k (i.e., descends to a k-subgroup of G). In such cases, the semi-direct product structure holds for G over k using any maximal k-torus T of G (and U is unique: it must be a k-descent of Ru(Gk)). If k is perfect then by Galois descent we may always descend Ru(Gk) to a k-subgroup of G. The main challenge is the case of imperfect k. In these notes, we explain Tits’ structure theory for unipotent smooth connected groups over general fields of positive characteristic (especially imperfect fields), and we use it to establish a general structure theorem for solvable smooth connected groups that replaces (and generalizes) the semidirect product structure over perfect k. The bulk of the work is in the unipotent case, for which our exposition is a mild revision of [CGP, App. B] (which develops some fundamental results of Tits, building on earlier work of Rosenlicht, concerning the structure of smooth connected unipotent groups and torus actions on such groups over an arbitrary ground field of positive characteristic). The results on unipotent groups were presented by Tits in a course at Yale University in 1967, and lecture notes [Ti1] for that course were circulated but never published. Much of the course was concerned with general results on linear algebraic groups that are available now in many standard references (such as [Bo], [Hum], and [Spr]). The original account (with proofs) of Tits’ structure theory of unipotent groups is his unpublished Yale lecture notes, and a summary of the results is given in [Oes, Ch. V]. In some parts we have simply reproduced arguments from Tits’ lecture notes. Throughout the discussion below, k is an arbitrary field with characteristic p > 0.