Math 249b. Structure of Solvable Groups over Fields

Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T nRu(G) for any maximal torus T of G. Over more general k, an analogous such semi-direct product structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and 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. The commutative k-group G 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 the group 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 semi-direct 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 Grothendieck’s theorem that over a general field k, if S is a maximal k-torus in a smooth affine k-group H then Sk is maximal in Hk. Thus, by the conjugacy of maximal tori in Gk, G = T n U 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. Our exposition in §1–§4 is a refinement of Appendix B of [CGP]. The general solvable case is addressed in §5, where we include applications to general smooth connected affine k-groups. Throughout the discussion below, k is an arbitrary field with characteristic p > 0.