On quasi-reductive group schemes

Abstract

This paper was motivated by a question of Vilonen, and the main results have been used by Mirković and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over Z ) \mathbb {Z}) of a reductive group. We define a quasi-reductive group over a discrete valuation ring R R to be an affine flat group scheme over R R such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the identity component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over R R , the generic fiber is a reductive group, the special fiber is connected, and the group scheme is smooth over R R in most cases, for example when the residue characteristic is not 2, or when the generic fiber and reduced special fiber are of the same type as reductive groups. We also obtain results about group schemes over a Dedekind scheme or a Noetherian scheme. We show that in residue characteristic 2 there are non-smooth quasi-reductive group schemes with generic fiber SO 2 n + 1 \operatorname {SO}_{2n+1} and they can be classified when R R is strictly Henselian.

A topological property of quasireductive group schemes

“…Combining Propositions 3.4 and 4.3 of [Prasad and Yu 2006], we see that there is a finite extension field K of K (contained in our chosen algebraic closurē K ) with the following property. Let R be the integral closure of R in K , and set…”

“…Gopal Prasad has obtained a stronger conclusion, combining Corollary 2 with the arguments based on [Prasad and Yu 2006]; at his urging, this is included below (Theorem 11).…”

“…We thank Conrad for explaining this argument to us; the reader might compare this with [Prasad and Yu 2006…”

“…This topological result was motivated by the well known Serre-Tate criterion [1968] for good reduction of abelian varieties, which relies ultimately on the theory of Néron models. In a sense, [Prasad and Yu 2006] also relies on some aspects of this theory. Theorem 1.…”

Minimally ramified deformations when

“…Then is a reductive -subgroup scheme of and is a separated étale -group scheme of finite presentation [Con14, Proposition 3.1.3 and Theorem 5.3.5]. Furthermore, by a result of Raynaud is affine as it is a flat, separated, and of finite type with affine generic fiber over the discrete valuation ring [PY06, Proposition 3.1].…”