Contraadjusted modules, contramodules, and reduced cotorsion modules
a Department of Mathematics, Faculty of Natural Sciences, University of Haifa, Mount Carmel, Haifa 31905, Israel
b Laboratory of Algebraic Geometry, National Research University Higher School of Economics, Moscow 119048, Russia
c 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.
Key words and phrases:
cotorsion modules, contraadjusted modules, contramodules, abelian categories, adic completions, flat covers, cotorsion envelopes, abelian groups, Noetherian commutative rings of Krull dimension 1.
|Israel Science Foundation
|Czech Science Foundation
|The author was supported by the ISF grant #446/15 in Israel and by the Grant Agency of the Czech Republic under the grant P201/12/G028 in Prague.
MSC: 13C12, 13C60, 13D07, 13D99, 13J10
Received: May 24, 2016; in revised form July 22, 2017
Leonid Positselski, “Contraadjusted modules, contramodules, and reduced cotorsion modules”, Mosc. Math. J., 17:3 (2017), 385–455
Citation in format AMSBIB
\paper Contraadjusted modules, contramodules, and reduced cotorsion modules
\jour Mosc. Math.~J.
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