论文标题
覆盖课和$ 1 $ - 在交换戒指上使用可交角对
Covering classes and $1$-tilting cotorsion pairs over commutative rings
论文作者
论文摘要
我们有兴趣表征$ 1 $的cotorsion对$（\ Mathcal {a}，\ Mathcal {t}）$提供封面的换向戒指，也就是$ \ Mathcal {a a} $是一个封面。我们在$ 1 $ r $上使用$ 1 $的$ 1 $ listing cotorsion对与忠实生成的Gabriel Topologies在$ r $上使用的Cotorsion对。此外，我们使用Bazzoni-Positselki的结果，特别是Matlis等效性的概括及其对覆盖类的表征，价格为$ 1 $ - 替代由平坦的注射式环形表达产生的cotorsion对。明确的是，如果$ \ nathcal {g} $是与$ 1 $ - 替代cotorsion对$（\ Mathcal {a}，\ Mathcal {t}）$相关的Gabriel拓扑，和$ r_ \ r_ \ nathcal {g} $是与$ \ nath $ \ nathcal of s. $ \ MATHCAL {a} $覆盖，然后$ \ Mathcal {g} $是一个完美的本地化（从Stenström的意义上讲），而本地化$ r_ \ Mathcal {g} $最多具有投影维度。此外，我们表明$ \ Mathcal {a} $在且仅当notization $ r_ \ r_ \ mathcal {g} $和商戒指$ r/j $都是每个$ j \ in \ mathcal {g} $的完美戒指。满足后两个条件的戒指称为$ \ MATHCAL {G} $ - 几乎是完美的。
We are interested in characterising the commutative rings for which a $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$ provides for covers, that is when the class $\mathcal{A}$ 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 $\mathcal{G}$ is the Gabriel topology associated to the $1$-tilting cotorsion pair $(\mathcal{A}, \mathcal{T})$, and $R_\mathcal{G}$ is the ring of quotients with respect to $\mathcal{G}$, we show that if $\mathcal{A}$ is covering then $\mathcal{G}$ is a perfect localisation (in Stenström's sense) and the localisation $R_\mathcal{G}$ has projective dimension at most one. Moreover, we show that $\mathcal{A}$ is covering if and only if both the localisation $R_\mathcal{G}$ and the quotient rings $R/J$ are perfect rings for every $J \in \mathcal{G}$. Rings satisfying the latter two conditions are called $\mathcal{G}$-almost perfect.
