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 $\text{Ext}^1$-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.