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chevalley commutator formula

Journal of K-Theory (1)

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Homotopy invariance of non-stable K1-functors
A. Stavrova
Journal:

Journal of K-Theory / Volume 13 / Issue 2 / April 2014

Published online by Cambridge University Press:

10 October 2013, pp. 199-248

Print publication:

April 2014
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Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.