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Applications of Borel-definable homological algebra to locally compact groups

Martino Lupini

TL;DR

This work establishes a robust homological framework for locally compact Polish abelian groups by proving that the Hom functor has a total right derived functor and a cohomological right derived functor, enabling Ext to be computed in this setting. It leverages the left heart description via groups with a Polish cover and Borel-definable morphisms to obtain explicit, definable models for Ext and cocycles, and to analyze injective and projective objects across LH and its thick subcategories. The paper provides comprehensive classifications of injectives and projectives in these categories (often showing that only trivial injectives occur in many LH categories and giving precise forms in others) and demonstrates countability/vanishing phenomena for Ext in numerous cases. The results significantly advance definable homological algebra for topological groups, connecting Ext theory with cocycle descriptions, dualities, and the structure of Polish covers, with potential applications to definable invariants in topological group theory. The development harmonizes quasi-abelian and enriched category perspectives with descriptive-set-theoretic tools to yield a cohesive, definable approach to derived functors in this non-algebraic setting.

Abstract

We show that the $\mathrm{Hom}$ functor from the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups to the category $\mathbf{PAb}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck's construction of its cohomological right derived functor. We also apply the description of the left heart of subcategories of $\mathbf{PAb}$ in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological $p$-groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological $p$-groups of finite ranks, and type $\mathbb{A}$ groups.

Applications of Borel-definable homological algebra to locally compact groups

TL;DR

This work establishes a robust homological framework for locally compact Polish abelian groups by proving that the Hom functor has a total right derived functor and a cohomological right derived functor, enabling Ext to be computed in this setting. It leverages the left heart description via groups with a Polish cover and Borel-definable morphisms to obtain explicit, definable models for Ext and cocycles, and to analyze injective and projective objects across LH and its thick subcategories. The paper provides comprehensive classifications of injectives and projectives in these categories (often showing that only trivial injectives occur in many LH categories and giving precise forms in others) and demonstrates countability/vanishing phenomena for Ext in numerous cases. The results significantly advance definable homological algebra for topological groups, connecting Ext theory with cocycle descriptions, dualities, and the structure of Polish covers, with potential applications to definable invariants in topological group theory. The development harmonizes quasi-abelian and enriched category perspectives with descriptive-set-theoretic tools to yield a cohesive, definable approach to derived functors in this non-algebraic setting.

Abstract

We show that the functor from the category of locally compact Polish abelian groups to the category of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck's construction of its cohomological right derived functor. We also apply the description of the left heart of subcategories of in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of , as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological -groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological -groups of finite ranks, and type groups.

Paper Structure

This paper contains 26 sections, 74 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Exact and quasi-abelian categories and derived functors
  3. Exact categories
  4. Triangulated category
  5. Derived categories
  6. Quasi-abelian categories
  7. Derived functors
  8. Bifunctors
  9. Injectives and projectives
  10. Enriched (quasi-)abelian categories
  11. Groups with a Polish cover and definable functors
  12. The left heart of abelian Polish groups
  13. Borel-definable sets and groups
  14. Quasi-abelian standard Borel categories
  15. Classes of Polish abelian groups
  16. ...and 11 more sections

Key Result

Lemma 2.7

Suppose that $\mathcal{A}$ is an exact category and $\mathcal{C}$ is a cogenerating fully exact subcategory of $\mathcal{A}$ closed under quotients. Then for any bounded complex $A$ in $\mathcal{A}$ there exists a bounded complex $D$ in $\mathcal{D}$ and a quasi-isomorphism $\eta :A\rightarrow C$ wi

Theorems & Definitions (146)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Theorem 3.1
  • ...and 136 more