Covers, envelopes, and cotorsion theories in locally presentable abelian categories and contramodule categories

TL;DR

This paper extends the theory of cotorsion theories, covers, and envelopes from module and Grothendieck categories to locally presentable abelian categories, with applications to contramodule categories over topological rings.

Contribution

It generalizes the existence and completeness results of cotorsion theories to broader categories and applies these to contramodule categories, providing new examples and counterexamples.

Findings

Generalized cotorsion theory completeness in locally presentable abelian categories

Established existence of covers and envelopes in these categories

Provided applications to contramodule categories over topological rings

Abstract

We prove general results about completeness of cotorsion theories and existence of covers and envelopes in locally presentable abelian categories, extending the well-established theory for module categories and Grothendieck categories. These results are then applied to the categories of contramodules over topological rings, which provide examples and counterexamples.