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Moscow Mathematical Journal

Volume 17, Issue 3, July–September 2017  pp. 385–455.

Contraadjusted Modules, Contramodules, and Reduced Cotorsion Modules

Authors:  Leonid Positselski (1)
Author institution:(1) Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa 31905, Israel; and
Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow 119048; and
Sector of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127051, Russia


This paper is devoted to the more elementary aspects of the contramodule story, and can be viewed as an extended introduction to our more technically complicated paper “Dedualizing complexes and MGM duality”. 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.

2010 Math. Subj. Class. 13C12, 13C60, 13D07, 13D99, 13J10.

Keywords: Cotorsion modules, contraadjusted modules, contramodules, abelian categories, adic completions, flat covers, cotorsion envelopes, abelian groups, Noetherian commutative rings of Krull dimension 1.

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