We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (𝒜,𝒯){(\mathcal{A},\mathcal{T})} provides for covers, that is when the class 𝒜{\mathcal{A}} is a covering class. We use Hrbek’s bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R . Moreover, we use results of Bazzoni–Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if 𝒢{\mathcal{G}} is the Gabriel topology associated to the 1-tilting cotorsion pair (𝒜,𝒯){(\mathcal{A},\mathcal{T})}, and R𝒢{R_{\mathcal{G}}} is the ring of quotients with respect to 𝒢{\mathcal{G}}, we show that if 𝒜{\mathcal{A}} is covering, then 𝒢{\mathcal{G}} is a perfect localisation (in Stenström’s sense [B. Stenström, Rings of Quotients, Springer, New York, 1975]) and the localisation R𝒢{R_{\mathcal{G}}} has projective dimension at most one as an R -module. Moreover, we show that 𝒜{\mathcal{A}} is covering if and only if both the localisation R𝒢{R_{\mathcal{G}}} and the quotient rings R/J{R/J} are perfect rings for every J∈𝒢{J\in\mathcal{G}}. Rings satisfying the latter two conditions are called 𝒢{\mathcal{G}}-almost perfect.

中文翻译：

---

交换环上的覆盖类和1-倾斜扭曲对

我们感兴趣的是描述一个1倾角扭曲对（𝒜，𝒯）{（\ mathcal {A}，\ mathcal {T}）}提供覆盖的换向环，即当when {\ math { A}}是一个覆盖类。我们在交换环R上的1-倾斜扭曲对与R上忠实有限生成的Gabriel拓扑之间使用Hrbek的双射对应。此外，我们使用Bazzoni–Positselski的结果，尤其是Matlis等价的推广以及它们对由平坦内射环型引起的1-倾斜扭曲对的覆盖类别的表征。明确地，如果𝒢{\ mathcal {G}}是与1-倾斜扭曲对（𝒜，𝒯）{（\ mathcal {A}，\ mathcal {T}）}和R𝒢{R _ {\ mathcal {G}}}是关于𝒢{\ mathcal {G}}的商环，我们表明，如果覆盖𝒜{\ mathcal {A}}，那么𝒢{\ mathcal {G}}就是一个完美的本地化（按照Stenström的观点[B.Stenström，《指环》，Springer，纽约，1975年））和本地化R𝒢{R _ {\ mathcal {G}}}作为R模块的投影维数最多为1。此外，我们证明𝒜{\ mathcal {A}}覆盖且仅当本地化R𝒢{R _ {\ mathcal {G}}}和商环R / J {R / J}对于每个J∈𝒢{J \ in \ mathcal {G}}。满足后两个条件的环称为𝒢{\ mathcal {G}}-几乎完美。我们证明𝒜{\ mathcal {A}}覆盖且仅当局部化R𝒢{R _ {\ mathcal {G}}}和商环R / J {R / J}对每个J∈是完美环时𝒢{J \ in \ mathcal {G}}。满足后两个条件的环称为𝒢{\ mathcal {G}}-几乎完美。我们证明𝒜{\ mathcal {A}}覆盖且仅当局部化R𝒢{R _ {\ mathcal {G}}}和商环R / J {R / J}对每个J∈是完美环时𝒢{J \ in \ mathcal {G}}。满足后两个条件的环称为𝒢{\ mathcal {G}}-几乎完美。