Homotopy invariance of non-stable K_1-functors

Details Homotopy invariance of non-stable K_1-functors Homotopy invariance of non-stable K_1-functors

2011-11-20

arxiv.org/abs/1111.4664v3

Mathematics

Algebraic Geometry

24 pages; restrictions on the root system type removed, the title changed

Scientific paper

Let G be an isotropic simply connected simple algebraic group over a field k, having isotropic rank >=2. Let K_1^G be the non-stable K_1-functor associated to G (also called the Whitehead group of G in the field case). We show that K_1^G(k)=K_1^G(k[X_1,...,X_n]) for any n>= 1. This implies that K_1^G is A^1-homotopy invariant on the category of regular k-algebras, if k is perfect. If k is infinite perfect, one also deduces that K_1^G(R)-> K_1^G(K) is injective for any regular local k-algebra R with the fraction field K.

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