Linear group homology properties of the inclusion of a ring of integers into a number field

Abstract

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences

$$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$

for all positive integers n, where θ _# is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Théorème 1 of [S2]); in particular, θ _# is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K _n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and étale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K _n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K _n F is a large torsion group but all its divisible elements belong to the image of θ _# because ⊕ _m K _n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

The second author wishes to thank the Swiss National Science Foundation for financial support while this research was being carried out.