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- W2911906095 abstract "Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $mathcal{G}$ associated to the $1$-tilting class $mathcal{T}$ over a commutative ring as illustrated by Hrbek. We prove that a $1$-tilting class $mathcal{T}$ is enveloping if and only if $ mathcal{G}$ is a perfect Gabriel topology (that is, it arises from a perfect localisation) and $R/J$ is a perfect ring for each $J in mathcal{G}$, or equivalently $mathcal{G}$ is a perfect Gabriel topology and the discrete quotient rings of the topological ring $mathfrak R=$End$(R_ mathcal{G}/R)$ are perfect rings where $R_mathcal{G}$ denotes the ring of quotients with respect to $mathcal{G}$. Moreover, if the above equivalent conditions hold it follows that pdim$R_mathcal{G} leq 1$ and $mathcal{T}$ arises from a flat ring epimorphism." @default.
- W2911906095 created "2019-02-21" @default.
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- W2911906095 date "2019-01-23" @default.
- W2911906095 modified "2023-09-27" @default.
- W2911906095 title "Enveloping Classes over Commutative Rings" @default.
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