A logarithmic approach to classgroups of integral group rings

Abstract

Let 1 be a fixed rational prime number. We denote the ring of rational l-adic integers by Z, , and the rational Z-adic field by Q, . For any ring R (always with a 1) we denote the group of units of R by Ii*. For a finite group r, FrGhlich has defined a group Det(Br~*) (cf. p. 383 of [ZJ). We give a precise definition of this group below, though essentially Det(&I’*) can be thought of as the reduced norms of units of the local group ring &r. This group plays a vital role in the description of various classgroups of modules over ZI’. The aim of this paper is to give a new and beautiful description of Det(&P) in terms of an “integral logarithm” obtained by use of Adams operations. In particular this description will enable us to answer certain questions concerning the Galois cohomology of such groups. This is very important because such cohomological questions play a crucial part in trying to describe the Galois module structure of rings of integers (cf. (13)). Before we can proceed to state the main theorems and give the new logarithmic description, we must introduce the concept of a Galois order over Zr , and then define Det(&r*). Let K/Q be a finite, unramified extension, let OK be the ring of integers of K and let A = Gal(K,Q). If A is a finite group with A ~4 A, then we define the Galois order over Zr