Contraadjusted modules, contramodules, and reduced cotorsion modules

TL;DR

This paper explores the elementary aspects of contramodules and reduced cotorsion modules, revealing their structure and duality properties in abelian categories, with implications for modules over commutative rings.

Contribution

It provides an elementary introduction to contramodules and reduced cotorsion modules, detailing their structure, duality, and relations over various rings, extending prior technical results.

Findings

Reduced cotorsion abelian groups form an abelian category.

Such groups are isomorphic to products of p-contramodule groups.

p-contramodules are p-adically complete, but the converse is not always true.

Abstract

This paper is devoted to the more elementary aspects of the contramodule story, and can be viewed as an extended introduction to the more technically complicated arXiv:1503.05523. Reduced cotorsion abelian groups form an abelian category, which is in some sense covariantly dual to the category of torsion abelian groups. An abelian group is reduced cotorsion if and only if it is isomorphic to a product of p-contramodule abelian groups over prime numbers p. Any p-contraadjusted abelian group is p-adically complete, and any p-adically separated and complete group is a p-contramodule, but the converse assertions are not true. In some form, these results hold for modules over arbitrary commutative rings, while other formulations are applicable to modules over one-dimensional Noetherian rings.