Manfred Kolster

An idelic approach to the wild kernel

Let p be a prime and F an algebraic number field containing a primitive 2p-th root of 1. The main object of this paper is to describe the p-torsion in the wild kernel of the cyclotomic Zp-extension F~ of F in terms of a Tare-twisted idrle-class-group in analogy with the description of the p-torsion in the tame kernel via twisted ideal class groups due to Coates [3]. It turns out that this description depends upon the validity of a conjecture of Gross, and in fact we obtain an equivalent formulation of this conjecture in terms of Iwasawa-theory, which also yields a new interpretation of the Tate-kernel, i.e. the kernel of the surjective map

K2 of rings of algebraic integers

Let F be an algebraic number field with ring of integers Do. For odd primes p and totally real F, Coates [3] has related the p-primary part of the tame kernel K2(oF) to the fixed elements of a Tate-twisted ideal-classgroup, which occurs naturally in Iwasawa-theory. In [9], a similar result was obtained for the prime 2. These connections turned out to be crucial in deducing the Birch-Tate-Conjecture from the Main Conjecture in Iwasawa-theory (cf. [14,9]), which seems to have been settled now completely (cf. [ 14, 12, Appendix, for odd p and abelian number fields]). In the first part of this paper we generalize Coates’ result to arbitrary number fields. The approach is similar to the one used in [7] and [9], combined with results of Mercurjev-Suslin and Levine, which give the relation between torsion in K,(F) and Iwasawa-theory (cf. [ 111). Let E, = F(c,), E, = Un E, and G, = Gal(E,/F). Let U, be the group of units in E, and A, the p-primary part of the class-group. G, acts on WQ U, ( W= l& pP) and on the Tate-twist %a2 A, (Z =b pP) and the p-primary part of KZ(oF) is related to the fixed efements of these groups in the following way: If G, is pro-cyclic (hence either p is odd or p = 2 and F contains i or c2k C2k for some k > 3), there is a natural surjective homomorphism K2(oF)(p) + (ZO,, Am)‘= and the kernel is isomorphic to ( WQ U,)Gm/max.div.subgroup. This maximal divisible subgroup is isomorphic to QP/Z, @ K3(F),d z (QPp/Zp)r2(F), where K3(F)nd = K,(F)/im(Ky(F)) is the indecomposable K,. In other words there is an exact sequence

On higher class groups of orders

The purpose of this paper is to study the torsion in odd-dimensional higher class groups of orders in semi-simple algebras over number fields. We show that for a prime number q these higher class groups can have q-torsion only if the order is not maximal at some prime ideal above q, and we determine part of the structure of this torsion. As an application to integral group rings we show that in dimensions 4n+1 the class groups of the symmetric group Sr have at most 2-torsion and that in dimensions 4n − 1 the possible odd torsion can only occur for primes q such that q−1 2 divides r. In dimensions 4n+1 the same result also holds for Dihedral groups, provided we assume the validity of the local Quillen-Lichtenbaum Conjecture. In a final section we relate the structure of the higher odd-dimensional class groups of a group ring of a finite group G to homomorphisms on the representation ring of G with values in twisted roots of unity and for G abelian also to homogeneous functions on G. Introduction One of the high points in the study of the K-theory of group rings and orders was R. Oliver’s investigation of SK1(Z[G]) for finite groups G, in a series of deep papers, summarized in [Ol]. He defined the higher class group Cl1(Z[G]) = ker ( SK1(Z[G]) −→ ⊕

Improvement of K2-stability under transitive actions of elementary groups

For any ring A with finite stable rank sr(A) the canonical map Kz(n I,A)+K2(A) is known to be surjective, if n r sr(A) + 2 and bijective, if n 2 sr(A) + 3 (cf. [3,4,7,9, 111). Moreover, examples are known where these bounds are sharp (cf. [4]). Thus we have to impose additional assumptions on A to get an improvement of the stability behaviour of Kz. We show that the bounds improve by 1, provided certain actions of elementary groups are transitive. As a consequence we get a result, proved by van der Kallen [5] with a different technique, that the map K2(2,A)*K2(A) is surjective and the map K2(3,A)~Kz(A) is bijective, if A is a Dedekind ring of arithmetic type as in Bass-Milnor-Serre [2] and with infinitely many units. I am indebted to W. van der Kallen, whose very helpfully remarks lead to the final version of this paper.

On torsion in K>2 of fields

Abstract Recent results of Mercurjev-Suslin and Levine relating the torsion and the cotorsion of K 2 and K 3 of a field F to Galois-cohomology are used to express the p -torsion in K 2 ( F )( p ≠ char F ) in terms of the cyclotomic Z p -extension of F .