Tilting preenvelopes and cotilting precovers in general Abelian categories

TL;DR

This paper generalizes classical tilting and cotorsion theories to arbitrary Abelian categories, characterizing special preenveloping subcategories and their relation to cotorsion pairs, with applications to module categories over rings.

Contribution

It introduces a broad framework for tilting and cotilting subcategories in Abelian categories, extending classical module theory results and providing new characterizations and applications.

Findings

Characterization of (semi-)special preenveloping subcategories as cotorsion pairs.

Identification of subcategories generated by Ext-universal objects.

Application of results to module categories over coherent rings.

Abstract

We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories $\mathcal{T}$ of $\mathcal{A}$ closed under extensions and direct summands are precisely those for which $(_{}^{\perp_1}\mathcal{T},\mathcal{T})$ is a right complete cotorsion pair, where $_{}^{\perp_1}\mathcal{T}:=\text{Ker} (\text{Ext}_{\mathcal{A}}^1(-,\mathcal{T}))$. Particular cases appear when $\mathcal{T}=V^{\perp_1}:=\text{Ker}(\text{Ext}_{\mathcal{A}}^1(V,-))$, for an $\text{Ext}^1$-universal object $V$ such that $\text{Ext}_{\mathcal{A}}^1(V,-)$ vanishes on all (existing) coproducts of copies of $V$. For many choices of $\mathcal{A}$, we show that these latter examples exhaust all the possibilities. We then show that, when $\mathcal{A}$ has an epi-generator, the (semi-)special preenveloping torsion classes $\mathcal{T}$ given by (quasi-)tilting objects are exactly those for which any object $T\in\mathcal{T}$ is the epimorphic image of some object in $_{}^{\perp_1}\mathcal{T}$ (and the subcategory $\mathcal{B}:=\text{Sub}(\mathcal{T})$ of subobjects of objects in $\mathcal{T}$ is reflective) and they are, in turn, the right constituents of complete cotorsion pairs in $\mathcal{A}$ (resp., $\mathcal{B}$). In a final section, we apply the results when $\mathcal{A}=\mathrm{mod}\text{-}R$ is the category of finitely presented modules over a right coherent ring $R$, something that gives new results and raises new questions even at the level of classical tilting theory in categories of modules.