Isotropic reductive groups over discrete Hodge algebras

TL;DR

This paper proves that for isotropic reductive groups over regular domains containing an infinite field, certain cohomology maps over discrete Hodge algebras are bijective, extending known results for general linear groups.

Contribution

It establishes new bijectivity results for cohomology and K-theory maps for isotropic reductive groups over discrete Hodge algebras, generalizing prior work on GL_n.

Findings

H^1_Nis(A,G) -> H^1_Nis(R,G) is a bijection for isotropic rank >=1.

H^1_et(A,G) -> H^1_et(R,G) has trivial kernel in characteristic 0.

K_1^G(A) = K_1^G(R) for perfect fields, isotropic rank >=2, and square-free A.

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.