Tilting Preenvelopes and Cotilting Precovers in General Abelian Categories

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 Ext1-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 \({\mathscr{B}}:=\text {Sub}(\mathcal {T})\) 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., \({\mathscr{B}}\)). In a final section, we apply the results when \(\mathcal {A}=\text {mod-}R\) 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.