Relative 𝑆-invariants

Abstract

Abstract. Warfield has defined the concept of a T*-module over a discrete valuation ring and has proved a classification theorem for these modules. In this paper, the invariant 5 defined by the author is extended. This allows a generalization of the classification theorem of Warfield.1. Preliminaries. R will be a discrete valuation ring throughout, and /» will represent a generator of its maximal ideal. The word "module" will mean R -module. Standard terminology of abelian groups will be used (see Fuchs [1], especially §79). The exception to this is that we will define the height of 0 to be oo', and assume a < co < oo' for all ordinals a. The indicator of an element x will be denoted H(x). An indicator is called proper if it does not contain oo'. If A is a submodule of M and a is an ordinal or co, the ath relative Ulm invariant will be denoted f(a, M, A).In [5], (announced in [4]), Warfield defined a T-module as a module that can be defined in terms of generators and relations in such a way that the only relations are of the form/»* = 0 orpx = y. A summand of a F-module is called a T*-module. A module M has torsion free rank one if, for any two elements x and y of infinite order in M, there are nonzero elements r and s in R such that rx = sy. Warfield has shown that if M is a F-module, it is either a torsion module, or a direct sum of modules of torsion free rank one.A subset X of a module M is a decomposition basis if [X] is the free module on X, M/[X] is torsion, and h(^r¡x¡) = min(rj(/-,x,)}, for r¡ E R, x¡ G X. (Here /i(x) denotes the height of x.) It has been shown by Warfield that a 7^-module M has a decomposition basis X such that [X] is nice in M.In §2, we define the concept of stability, which is stronger than niceness. (ii) every coset m + A of infinite order contains an element x such that /» 'x

The fully invariant subgroups of local Warfield groups

“…Let x ∈ H have infinite order. Then G/ x is a simply presented p-group, and it is straightforward to verify (or see [10,Fact F]…”