Spectra of the Γ-invariant of Uniform Modules

For a ring R, denote by SpecΓ(κ,R) the κ-spectrum of the Γ-invariant of strongly uniform right R-modules. Recent realization techniques of Goodearl and Wehrung show that SpecΓ(א1 , R) is full for a suitable von Neumann regular algebra R, but the techniques do not extend to cardinals κ > א1. By a direct construction, we prove that for any field F and any regular uncountable cardinal κ there is an F -algebra R such that SpecΓ(κ,R) is full. We also derive some consequences for the Γ-invariant of strongly dense lattices of two-sided ideals, and for the complexity of Ziegler spectra of infinite dimensional algebras. The Γ-invariant method introduced by Eklof in [E1] and [E2] provides an efficient tool for classification of algebraic objects which are defined by existence of infinite filtrations of particular forms. The method has been used to develop a structure theory of almost free groups [EM], uniserial modules [Sa], and bilinear spaces [A], [BFS]. More recently, Γ-invariants were defined also in the dual setting, for objects possessing dual filtrations. This resulted in a classification of dense lattices [ET], and of strongly uniform modules [T1], [T2]. For a regular uncountable cardinal κ, denote by B(κ) the Boolean algebra consisting of all subsets of κ modulo the filter of subsets containing a closed unbounded set. The Γ-invariant of objects of dimension κ takes values in B(κ). The value measures a caveat for an object of dimension κ to have a certain algebraic property. For example, for almost free groups, the property is “to be a free group” [E3]. For bilinear spaces, the property is “to decompose orthogonally into subspaces of dimension < κ” [BFS]. For dense lattices, it is “to be relatively complemented” [ET], etc. For each Γ-invariant, two natural problems arise: (1) Given a regular uncountable cardinal κ and i ∈ B(κ), is there an object of dimension κ whose Γ-invariant value equals i ? The set of all i ∈ B(κ) for which the answer to (1) is positive is called the κspectrum of the Γ-invariant, and denoted by SpecΓ(κ). The κ-spectrum is said to be full provided that SpecΓ(κ) = B(κ), [BFS]. (2) For i ∈ SpecΓ(κ), describe all the objects of dimension κ whose Γ-invariant value equals i. First author publication number 693. Research of the second author supported by a Fulbright Scholarship at UCI. His thanks are due to Professor Paul Eklof for many stimulating discussions on the subject, and for his constant help. Thanks are also due to Rutgers University for supporting the second author’s trip to Rutgers. Typeset by AMS-TEX 1