Torsors of isotropic reductive groups over Laurent polynomials

TL;DR

This paper proves a conjecture relating isotropic ranks of reductive groups over Laurent polynomial rings and their fields of fractions, establishing key properties of torsors and K-theory maps.

Contribution

It establishes the equivalence of isotropic rank conditions over Laurent polynomial rings and their fields of fractions, confirming a conjecture and analyzing K-theory maps.

Findings

Isotropic rank >=1 over R iff over the field of fractions.

The natural map H^1_{et}(R,G) to H^1_{et}(k(x_1,...,x_n),G) has trivial kernel.

Injectivity and isomorphism of K_1-functors for groups of isotropic rank >=2.

Abstract

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions k(x_1,...,x_n) of R, and if this is the case, then the natural map H^1_{et}(R,G)\to H^1_{\et}(k(x_1,...,x_n),G) has trivial kernel, and G is loop reductive, i.e. contains a maximal R-torus. In particular, we settle in positive the conjecture of V. Chernousov, P. Gille, and A. Pianzola that H^1_{Zar}(R,G)=* for such groups G. We also deduce that if G is a reductive group over R of isotropic rank >=2, then the natural map of non-stable K_1-functors K_1^G(R)\to K_1^G( k((x_1))...((x_n)) ) is injective, and an isomorphism if G is moreover semisimple.