Homotopy invariance of non-stable K₁-functors

2011/11/20 by Stavrova, Anastasia · 2 citations
#19B99 #20G07 #20G15 #20G35 #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.1111.4664

Abstract

Let G be reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank >=2. Let K₁^G be the non-stable K₁-functor associated to G (also called the Whitehead group of G in the field case). We show that K₁^G(k)=K₁^G(k[X₁,...,X_n]) for any n>= 1. This implies that K₁^G is A¹-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K₁^G(R)-> K₁^G(K) is injective for any regular local k-algebra R with the fraction field K.

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• Non-Stable K₁-Functors of Discrete Valuation Rings Containing a Field
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