Isotropic reductive groups over discrete Hodge algebras

Abstract

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra A=R[x_1,...,x_n]/I over R, the map H^1_Nis(A,G) -> H^1_Nis(R,G) induced by evaluation at x_1=...=x_n=0, is a bijection. If k has characteristic 0, then, moreover, the map H^1_et(A,G) -> H^1_et(R,G) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is >=2, and A is square-free, then K_1^G(A)=K_1^G(R), where K_1^G(R)=G(R)/E(R) is the corresponding non-stable K_1-functor, also called the Whitehead group of G. The corresponging statements for G=GL_n were previously proved by Ton Vorst.