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Contraadjusted modules, contramodules, and reduced cotorsion modules

Leonid Positselski

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.

Contraadjusted modules, contramodules, and reduced cotorsion modules

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.

Paper Structure

This paper contains 13 sections, 87 theorems, 181 equations.

Table of Contents

  1. Orthogonality Classes as Abelian Categories
  2. $s$-Contraadjustedness, $s$-Contramoduleness, and $s$-Completeness
  3. $s$-Power Infinite Summation Operations
  4. $[s,t]$-Power Infinite Summation Operations
  5. $[s_1,\dotsc,s_m]$-Contraadjustedness, $I$-Contramoduleness, and $I$-Adic Completeness
  6. $s$-Torsion Modules and the Functor $\Gamma_s$, $s$-Contramodules and the Functor $\Delta_s$
  7. $I$-Torsion Modules and the Functor $\Gamma_I$, $I$-Contramodules and the Functor $\Delta_I$
  8. Covers, Envelopes, and Cotorsion Theories
  9. Contramodules and Relatively Cotorsion Modules
  10. Flat, Projective, and Free Contramodules
  11. Free Contramodules and Flat Cotorsion Modules
  12. Cotorsion and Contraadjusted Abelian Groups
  13. Noetherian Rings of Krull Dimension $1$

Key Result

Theorem 1.1

(a) Let $U$ be a projective left $R$-module. Then the full subcategory $\mathsf C\subset R{\operatorname{\mathsf{--mod}}}$ formed by all the left $R$-modules $C$ for which $\mathop{\mathrm{Hom}}\nolimits_R(U,C)=0$ is closed under subobjects, quotient objects, extensions, and infinite products in $R{

Theorems & Definitions (198)

  • Theorem 1.1
  • proof
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 188 more