Low dimensional homology of linear groups over Hensel local rings, by Kevin P. Knudson

In this paper we prove that if R is an augmented Hensel local k-algebra, then the natural map Hi(GLn(k),Z/p) --> Hi(GLn(R),Z/p) is an isomorphism for i<=3, (p,char k) = 1, k infinite. We use this to derive the following rigidity result. Suppose that X is a smooth affine curve over an algebraically closed field k and let x,y be closed points on X. Then the corresponding specialization homomorphisms

sx,sy:Hi(GLn(k[X]),Z/p) --> Hi(GLn(k),Z/p)
coincide for i<=3. This, in turn, implies the Friedlander-Milnor conjecture in positive characteristic for Hi(GLn) for i<=3.

Kevin P. Knudson <knudson@math.nwu.edu>