Homological algebra of pro-Lie Polish abelian groups
Matteo Casarosa, Alessandro Codenotti, Martino Lupini
TL;DR
We develop homological algebra for pro-Lie Polish abelian groups, proving that $\mathbf{proLiePAb}$ is a thick subcategory of Polish abelian groups and providing a full injective/projective classification with homological dimension $1$. The work extends Hoffmann–Spitzweck type decompositions to the pro-Lie setting, introduces a layered type decomposition (including $\mathbb{Z}$, $\mathbb{S}^{1}$, and $\mathbb{A}$ types), and shows that Ext is enriched via left hearts and Polish covers. It also analyzes several thick subcategories (e.g., pro-$p$ and topological torsion) and provides explicit descriptions of injectives and projectives therein, proving the existence of enough projectives (but not injectives) in $\mathbf{proLiePAb}$, while establishing parallel results with $\mathrm{PAb}_{\mathrm{nA}}$ and non-Archimedean cases. The Ext framework is connected to derived functors $\mathrm{RHom}$ and left-heart theory, enabling a unified approach to extensions, cohomology, and computations across locally compact, pro-Lie, and non-Archimedean Polish abelian groups with potential implications for definable topology and enriched homological invariants.
Abstract
In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$.