The structure of $\lim^1$-groups

TL;DR

This paper demonstrates that any $ ext{lim}^1$-group can be represented as a cokernel of a canonical map from a module to its completion, using a functorial construction based on Quillen's small object argument.

Contribution

It shows that all $ ext{lim}^1$-groups are canonically realizable via inverse sequences of modules and functorial morphisms, extending the understanding of their structure.

Findings

Any $ ext{lim}^1$-group is of the form cokernel of a canonical map.

Constructs inverse sequences of modules functorially inducing isomorphisms on $ ext{lim}^1$.

Uses Quillen's small object argument and cotorsion pair theory in the proof.

Abstract

If $(A_n)_n$ is a decreasing filtration of a module $A$ and $\widehat{A} = \lim_n A/A_n$, then $\lim^1_n A_n$ is identified with the cokernel of the canonical map $A \longrightarrow \widehat{A}$. In this note, we show that any $\lim^1$-group is canonically of that form: For any inverse sequence of modules $(X_n)_n$ there exists an inverse sequence $(A_n)_n$ as above and a morphism $(A_n)_n \longrightarrow (X_n)_n$, depending functorially on $(X_n)_n$, that induces an isomorphism on $\lim^1$. The proof is based on Quillen's small object argument, as formulated by Eklof and Trlifaj in their investigation of the existence of enough injective objects in certain cotorsion pairs, and also uses a construction by Salce that provides enough projective objects therein.