一般阿贝尔范畴中的倾斜预覆盖和余倾斜预覆盖
Tilting preenvelopes and cotilting precovers in general Abelian categories
摘要：
我们考虑一个任意阿贝尔范畴$\mathcal{A}$和一个子范畴$\mathcal{T}$封闭下的扩展和直接和项，并刻画这些$\mathcal{T}$是（半）特殊的preenveloping在$\mathcal{A}$ ;作为一个副产品，我们推广到这一设置几个经典的结果类别的模块。例如，我们得到$\mathcal{A}$的特殊预包络子范畴$\mathcal{T}$在扩张和直和项下是闭的，并且$(_{}^{\perp_1}\mathcal{T},\mathcal{T})$是右完全余挠对，其中$_{}^{\perp_1}\mathcal{T}:=\text{Ker} (\text{Ext}_{\mathcal{A}}^1(-,\mathcal{T}))$ .当$\mathcal{T}=V^{\perp_1}:=\text{Ker}(\text{Ext}_{\mathcal{A}}^1(V,-))$，对于$\text{Ext}^1$ -通用对象$V$，使得$\text{Ext}_{\mathcal{A}}^1(V,-)$在$V$的拷贝的所有（现有）副产物上消失时，出现特殊情况。对于$\mathcal{A}$的许多选择，我们表明，这些后面的例子用尽所有的可能性。然后，我们证明了当$\mathcal{A}$有epi-generator时，由（准）倾斜对象给出的（半）特殊预包络挠类$\mathcal{T}$恰好是这样的挠类：对于这些挠类，任何对象$T\in\mathcal{T}$都是$_{}^{\perp_1}\mathcal{T}$中某个对象的epimorphic像（并且$\mathcal{T}$中对象的子范畴$\mathcal{B}:=\text{Sub}(\mathcal{T})$是反射的），并且它们又是$\mathcal{A}$中完全余挠对的右成分（分别为，$\mathcal{B}$）。在最后一节中，我们应用了当$\mathcal{A}=\mathrm{mod}\text{-}R$是右相干环$R$上有限表现模的范畴时的结果，即使在模范畴中的经典倾斜理论的水平上，也给出了新的结果并提出了新的问题。
英文摘要：
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.
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