Reduction map in the higher K-theory of the rings of integers in number fields

Abstract

This article studies the reduction maps in the higher K-theory of the ring of integers in a number field arising from the canonical reduction maps at nonzero prime ideals. It proves an explicit density estimate for the subset of primes where the images of a fixed collection of elements vanish. Our result applies to a collection of elements possibly having different degrees and suggests that the linearly independent elements of global K-theory exhibit mutually independent reduction patterns. We also relate the reduction map in K-theory to the reduction map in stable cohomology of general linear groups. This connection allows us to examine the pullback of Quillen's e-classes in the cohomology of the stable general linear group over a finite field. During the proof of the main result, we construct the smallest Galois extension which trivializes a Galois cohomology class of degree one, and show that the linear independence of classes results in disjointness of corresponding field extensions.