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A contramodule generalization of Neeman's flat and projective module theorem

Leonid Positselski

TL;DR

The paper extends Neeman’s flat/projective periodicity and BCE’s cotorsion periodicity from rings to complete, separated topological rings by developing a contramodule framework. It proves that for a complex of flat $\mathfrak R$-contramodules, six Becker-type conditions are equivalent, linking contraacyclicity with acyclicity in the exact category of flat contramodules and with directed colimit constructions. It establishes that the contraderived category of $\mathfrak R$-contramodules is equivalent to derived categories of the exact category of flat contramodules, and to the homotopy category of flat cotorsion contramodules; dual coderived results are obtained for cotorsion contramodules. The work builds complete hereditary cotorsion pairs in complex categories, proves pure acyclicity implies contraacyclicity (and vice versa), and presents five equivalent constructions of the contraderived category, unifying several classical and modern frameworks. These results provide a robust, generalizable bridge between flat, projective, and cotorsion theories in the contramodule setting with potential geometric interpretations as pro-objects and pro-sheaves on ind-schemes.

Abstract

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules, and also to the homotopy category of flat cotorsion left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat $\mathfrak R$-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada.

A contramodule generalization of Neeman's flat and projective module theorem

TL;DR

The paper extends Neeman’s flat/projective periodicity and BCE’s cotorsion periodicity from rings to complete, separated topological rings by developing a contramodule framework. It proves that for a complex of flat -contramodules, six Becker-type conditions are equivalent, linking contraacyclicity with acyclicity in the exact category of flat contramodules and with directed colimit constructions. It establishes that the contraderived category of -contramodules is equivalent to derived categories of the exact category of flat contramodules, and to the homotopy category of flat cotorsion contramodules; dual coderived results are obtained for cotorsion contramodules. The work builds complete hereditary cotorsion pairs in complex categories, proves pure acyclicity implies contraacyclicity (and vice versa), and presents five equivalent constructions of the contraderived category, unifying several classical and modern frameworks. These results provide a robust, generalizable bridge between flat, projective, and cotorsion theories in the contramodule setting with potential geometric interpretations as pro-objects and pro-sheaves on ind-schemes.

Abstract

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left -contramodules is equivalent to the derived category of the exact category of flat left -contramodules, and also to the homotopy category of flat cotorsion left -contramodules. In other words, a complex of flat -contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat -contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat -contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada.
Paper Structure (11 sections, 44 theorems, 27 equations)

This paper contains 11 sections, 44 theorems, 27 equations.

Table of Contents

  1. Accessible Subcategories
  2. Contraderived Categories
  3. Topological Rings
  4. Countably Presentable Contramodules
  5. Acyclic Complexes of Flat Contramodules are Contraacyclic
  6. Contraacyclic Complexes of Flat Contramodules are Acyclic
  7. Cotorsion Pairs
  8. Cotorsion Pair with Complexes of Cotorsion Contramodules
  9. Coderived Category of Flat Contramodules
  10. Contraderived Category of Cotorsion Contramodules
  11. Five Constructions of the Contraderived Category

Key Result

Proposition 1.1

Let $\mathsf K$ be a $\kappa$-accessible category and $\mathsf T$ be a set of (some) $\kappa$-presentable objects in $\mathsf K$. Then the full subcategory $\varinjlim_{(\kappa)}\mathsf T\subset\mathsf K$ is closed under $\kappa$-directed colimits in $\mathsf K$. The category $\mathsf L=\varinjlim_{

Theorems & Definitions (90)

  • Proposition 1.1
  • proof
  • Proposition 1.2
  • proof
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 80 more