Structure theorems for the heart of LCA

TL;DR

This paper explores the structure of the heart of a t-structure in the category of locally compact abelian groups, revealing that certain cokernels can be interpreted as Hausdorff topological abelian groups, expanding understanding of cohomology theories.

Contribution

It demonstrates that the abstract cokernels in the abelian envelope of LCA can be realized as Hausdorff topological abelian groups, even if they are not locally compact.

Findings

Abstract cokernels correspond to Hausdorff topological abelian groups.

These groups are determined up to lattice isogenies.

The results provide new insights into cohomology theories with LCA coefficients.

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.