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 \(\ge n\), if every normal semisimple reductive R-subgroup of G contains \(({{\mathrm{{{\mathbf {G}}}_m}}}_{,R})^n\). We prove that if G has isotropic rank \(\ge 1\) and R is a regular domain containing an infinite field k, then for any discrete Hodge algebra \(A=R[x_1,\ldots ,x_n]/I\) over R, the map \(H^1_{\mathrm {Nis}}(A,G)\rightarrow H^1_{\mathrm {Nis}}(R,G)\) induced by evaluation at \(x_1=\cdots =x_n=0\), is a bijection. If k has characteristic 0, then, moreover, the map \(H^1_{\acute{\mathrm{e}}\mathrm {t}}(A,G)\rightarrow H^1_{\acute{\mathrm{e}}\mathrm {t}}(R,G)\) has trivial kernel. We also prove that if k is perfect, G is defined over k, the isotropic rank of G is \(\ge 2\), 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 corresponding statements for \(G={{\mathrm{GL}}}_n\) were previously proved by Ton Vorst.