A dual approach to Černikov modules

Abstract

Let G be a group and A a right G-module. If the additive group A+ of A is a Černikov group, that is, a direct sum of finitely many cyclic and quasi-cyclic groups, we shall call A a Černikov module over G or over the integral group ring . Suppose that A+ is, furthermore, a divisible p-group, where p is a prime. Since the endomorphism ring of a quasi-cyclic p-group is isomorphic to the ring of p-adic integers, we find that is a free -module of finite rank. We can make A* into a right G-module in the usual way, and since A* is actually just the Pontrjagin dual of A, Pontrjagin duality shows that A → A* gives rise to a contravariant equivalence between the categories of divisible Černikov p-torsion modules over and G-modules which are -free of finite rank. Since the latter category is to some extent familiar, at least when G is finite – for its objects determine representations of G over the field of p-adic numbers, a field of characteristic zero – we may hope to exploit this correspondence systematically to study divisible Černikov p-modules. This is our main theme.