The structure of ${\lim }^1$-groups

If $(A_n{)}_n$ is a decreasing filtration of a module $A$ and $\widehat{A}={\lim }_{n}A\mathrm{/}{A}_n$, then ${\lim }_n^1{A}_n$ is identified with the cokernel of the canonical map $A⟶\widehat{A}$. 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 $(A_n{)}_n⟶(X_n{)}_n$, 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.