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Structure theorems for the heart of LCA

Oliver Braunling, Fei Ren

TL;DR

The paper constructs and analyzes the abelian envelope (left and right hearts) of the category of locally compact abelian groups, revealing that the added cokernel objects can be realized as Hausdorff topological abelian groups up to lattice isogenies. It introduces a canonical exact functor $\Theta$ from precompactly generated/precompact abelian groups to two-term complexes, establishing an exact equivalence between $\mathsf{PGA}/\mathsf{Ab}_{\mathrm{fg}}$ (and $\mathsf{PCA}/\mathsf{Ab}_{\mathrm{fin}}$) and the torsion subcategory of the left/right hearts, i.e., the ghost category. The work provides rewriting rules for roofs and objects to prove essential surjectivity, connects with condensed-abelian-group frameworks, and extends duality to Pontryagin duality, identifying precise correspondences for type $d$-c objects and illuminating the structure of non-LCA cokernels. These results demystify the abelian envelopes in this setting and yield concrete, lattice-isogeny–aware models for abstract cokernels, with broader implications for related topological-modules contexts and duality theories.

Abstract

Cohomology theories with values in LCA (locally compact abelian) groups suffer from the problem that the latter do not form an abelian category. However, the category LCA has a canonical abelian category envelope, the heart of a suitable t-structure. It adds formal cokernel objects. We show the surprising result that these abstract cokernels can also be interpreted as Hausdorff topological abelian groups, at least up to lattice isogenies. These need not be locally compact.

Structure theorems for the heart of LCA

TL;DR

The paper constructs and analyzes the abelian envelope (left and right hearts) of the category of locally compact abelian groups, revealing that the added cokernel objects can be realized as Hausdorff topological abelian groups up to lattice isogenies. It introduces a canonical exact functor from precompactly generated/precompact abelian groups to two-term complexes, establishing an exact equivalence between (and ) and the torsion subcategory of the left/right hearts, i.e., the ghost category. The work provides rewriting rules for roofs and objects to prove essential surjectivity, connects with condensed-abelian-group frameworks, and extends duality to Pontryagin duality, identifying precise correspondences for type -c objects and illuminating the structure of non-LCA cokernels. These results demystify the abelian envelopes in this setting and yield concrete, lattice-isogeny–aware models for abstract cokernels, with broader implications for related topological-modules contexts and duality theories.

Abstract

Cohomology theories with values in LCA (locally compact abelian) groups suffer from the problem that the latter do not form an abelian category. However, the category LCA has a canonical abelian category envelope, the heart of a suitable t-structure. It adds formal cokernel objects. We show the surprising result that these abstract cokernels can also be interpreted as Hausdorff topological abelian groups, at least up to lattice isogenies. These need not be locally compact.

Paper Structure

This paper contains 22 sections, 43 theorems, 152 equations.

Table of Contents

  1. Overview
  2. Variations of the correspondence
  3. Abelian envelopes of other categories of topological modules
  4. Quasi-abelian categories
  5. Definitions
  6. Quotients
  7. Cotilting torsion pairs
  8. Categories of topological groups
  9. Definitions
  10. Basic properties
  11. Examples
  12. Group-theoretical compact generation
  13. Exactness of the Completion Functor
  14. Completion
  15. Consequences of the exactness of completion
  16. ...and 7 more sections

Key Result

Theorem 1

Every object $X\in\mathcal{LH}(\mathsf{LCA})$ is a unique extension with $B\in\mathsf{LCA}$ a classical LCA group and a non-classical part $A$. All objects which show up as non-classical parts form a full subcategory $\mathsf{Ghost}$ in the heart which is itself quasi-abelian and there is an exact equivalence of categories where $\mathsf{PGA}[S_{\operatorname*{Lattice}}^{-1}]$ is the category $\ma

Theorems & Definitions (124)

  • Theorem 1: Main Correspondence
  • Theorem 2: Correspondence, Vector-free Variant
  • Theorem 3: Correspondence, Right Heart Versions
  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5: hr
  • Definition 3.1: MR1996800, MR1327209
  • ...and 114 more