Pruefer modules in filtration categories of semibricks

Abstract

Let $R$ be a ring with unity and $\mathcal{X}$ a semibrick in the module category $\mathrm{Mod}\,R$, that is, a class of pairwise orthogonal finitely presented modules whose endomorphism rings are division rings. We study the full subcategory $\mathrm{Filt}(\mathcal{X})$ consisting of all modules admitting a filtration with factors in $\mathcal{X}$. We show that $\mathrm{Filt}(\mathcal{X})$ is a wide subcategory of $\mathrm{Mod}\,R$. For the Ext-orthogonal class \[ \mathcal{X}^{\perp} = \{M \in \mathrm{Mod}\,R \mid \operatorname{Ext}^1_R(X,M)=0 \text{ for all } X \in \mathcal{X}\} \] we construct, for every module $Y$, an $\mathcal{X}^{\perp}$-envelope $Y_{\mathcal{X}}(\infty)$ as a direct limit of iterated universal short exact sequences. Assume that every $X \in \mathcal{X}$ has projective dimension at most one and that $\operatorname{Hom}_R(X,R)=0$ for all $X \in \mathcal{X}$. Then the envelope $R_{\mathcal{X}}(\infty)$ of the regular module is isomorphic to the universal localization $R_{\mathcal{X}}$ of $R$ at $\mathcal{X}$ in the sense of Schofield. The $\mathcal{X}^{\perp}$-envelopes of modules in $\mathcal{X}$ are called Pr\"ufer modules since they share many properties with classical Pr\"ufer groups and with Pr\"ufer modules over tame hereditary algebras. We prove that every injective object in $\mathrm{Filt}(\mathcal{X})$ is a direct sum of such Pr\"ufer modules.