On torsion free and cotorsion discrete modules (Q527037)

Let \(G\) be a profinite group, i.e. a topological group that is the inverse limit of a system of discrete finite groups, and let \(\text{DMod}(G)\) be the category of discrete \(G\)-modules, i.e. the \(G\)-modules \(X\) satisfying that for every \(x \in X\) there is an open normal subgroup \(N\) of \(G\) such that \(x\) is invariant with respect to \(N\). It is well-known that \(\text{DMod}(G)\) is a Grothendieck category, so it has enough injectives, and it is also locally noetherian, so it has injective covers. Let \(\mathcal{F}\) be the full subcategory formed by the \(G\)-modules that are torsion free as abelian groups, and let \(\mathcal{F}^{\perp}\) be the right perpendicular category of \(\mathcal{F}\) given by \[ \Big\{ C \in \text{DMod}(G) : \text{Ext}^{1}_{\text{DMod}(G)}(F,C) = 0, \text{ for all \(F \in \mathcal{F}\)} \Big\}. \] The objects in \(\mathcal{F}^{\perp}\) are called cotorsion discrete modules. The authors show that the pair \((\mathcal{F},\mathcal{F}^{\perp})\) is a complete cotorsion pair, i.e. a cotorsion pair with enough projectives and injectives (see Prop. 3.1). Moreover, using an equivalent characterization of cotorsion discrete modules in terms of the behaviour of the submodules of \(N\)-invariants, for all open normal subgroups \(N\) of \(G\) (see Prop. 3.6), together with well-known results (some of them proved by the authors), they also describe the class of all (simultaneously) cotorsion and torsion free discrete \(G\)-modules (see Thm. 3.8), and the underlying abelian group structure of any finitely generated cotorsion discrete \(G\)-module (see Thm. 3.9).