Baer module hulls of certain modules over a Dedekind domain

The notion of a Baer ring, introduced by Kaplansky, has been extended to that of a Baer module using the endomorphism ring of a module in recent years. There do exist some results in the literature on Baer ring hulls of given rings. In contrast, the study of a Baer module hull of a given module remains wide open. In this paper we initiate this study. For a given module [Formula: see text], the Baer module hull, [Formula: see text], is the smallest Baer overmodule contained in a fixed injective hull [Formula: see text] of [Formula: see text]. For a certain class of modules [Formula: see text] over a commutative Noetherian domain, we characterize all essential overmodules of [Formula: see text] which are Baer. As a consequence, it is shown that Baer module hulls exist for such modules over a Dedekind domain. A precise description of such hulls is obtained. It is proved that a finitely generated module [Formula: see text] over a Dedekind domain has a Baer module hull if and only if the torsion submodule [Formula: see text] of [Formula: see text] is semisimple. Further, in this case, the Baer module hull of [Formula: see text] is explicitly described. As applications, various properties and examples of Baer hulls are exhibited. It is shown that if [Formula: see text] are two modules with Baer hulls, [Formula: see text] may not have a Baer hull. On the other hand, the Baer module hull of the [Formula: see text]-module [Formula: see text] ([Formula: see text] a prime integer) is precisely given by [Formula: see text]. It is shown that infinitely generated modules over a Dedekind domain may not have Baer module hulls.

Infinite Dimensional Representations of Canonical Algebras

The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to themore general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Prüfer modules, and how other modules are determined by these modules.

Complexity and varieties for infinitely generated modules, II

It has now been almost twenty years since Alperin introduced the idea of the complexity of a finitely generated kG-module, when G is a finite group and k is a field of characteristic p > 0. In proving one of the first major results in the area [1], Alperin and Evens demonstrated the connection of the study of complexity for modules to the group cohomology. That connection eventually led to the categorization of modules according to their associated varieties in the maximal ideal spectrum of the cohomology ring H*(G, k). In all of the work that has followed, two principles have proved to be extremely important. The first is that the associated variety of a module is directly related to the structure of the module through the rank variety which is defined by the matrix representation of the module. The second major result is the tensor product theorem which says that the variety associated to a tensor product M ⊗kN is the intersection of the varieties associated to the modules M and N. In this paper we generalize these results to infinitely generated kG-modules.

Complexity and varieties for infinitely generated modules

In the past fifteen years the theory of complexity and varieties of modules has become a standard tool in the modular representation theory of finite groups. Moreover the techniques have been used in the study of integral representations [8] and have been extended to the representation theories of objects such as groups of finite virtual cohomological dimension [1], infinitesimal subgroups of algebraic groups and restricted Lie algebras [14, 16]. In all cases some sort of finiteness condition on the module category has been required to make the theory work. Usually this comes in the form of stipulating that all modules under consideration be finitely generated. While the restrictions have been efficient for most applications to date, there are very good reasons for wanting to develop a theory that will accommodate infinitely generated modules. One reason might be the possibility of extending the techniques of representations to other classes of infinite groups. Another reason is that some recent work has revealed a few of the defects of the finiteness requirement. One such problem can be summarized as follows.