Enveloping Classes over Commutative Rings
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2019
- DOI:
- 10.48550/arXiv.1901.07921
- arXiv:
- arXiv:1901.07921
- Bibcode:
- 2019arXiv190107921B
- Keywords:
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- Mathematics - Commutative Algebra
- E-Print:
- Updated version includes improvement of main theorem due to recent results, generalisation of techniques which were moved to Section 4 from Section 5, updates and additions to references and a change of abstract