# Flat Commutative Ring Epimorphisms of Almost Krull Dimension Zero

@article{Positselski2021FlatCR,
title={Flat Commutative Ring Epimorphisms of Almost Krull Dimension Zero},
author={Leonid Positselski},
journal={Journal of Algebra and Its Applications},
year={2021}
}
• L. Positselski
• Published 7 September 2020
• Mathematics
• Journal of Algebra and Its Applications
We consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the $R$-module $U$ does not exceed $1$. We also describe the Geigle-Lenzing perpendicular subcategory $U^{\perp_{0,1}}$ in $R\mathsf{-Mod}$. Assuming additionally that the ring $U$ and all the rings $R/I$ are perfect, we show that all flat $R$-modules… Expand
1 Citations
A characterisation of enveloping 1-tilting classes over commutative rings
• Mathematics
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#### References

SHOWING 1-10 OF 38 REFERENCES
Enveloping Classes over Commutative Rings.
• Mathematics
• 2019
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 generatedExpand
FLAT RING EPIMORPHISMS OF COUNTABLE TYPE
Abstract Let R→U be an associative ring epimorphism such that U is a flat left R-module. Assume that the related Gabriel topology $\mathbb{G}$ of right ideals in R has a countable base. Then we showExpand
Tilting Modules Arising from Ring Epimorphisms
• Mathematics
• 2008
We show that a tilting module T over a ring R admits an exact sequence 0 → R → T0 → T1 → 0 such that $T_0,T_1\in\text{Add}(T)$ and HomR(T1,T0) = 0 if and only if T has the form S ⊕ S/R for someExpand
Flat morphisms of finite presentation are very flat
• Mathematics
• Annali di Matematica Pura ed Applicata (1923 -)
• 2019
Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring R , R -modules built from the rings of functions on principalExpand
Triangulated Matlis equivalence
This paper is a sequel to arXiv:1503.05523 and arXiv:1605.03934. We extend the classical Harrison-Matlis module category equivalences to a triangulated equivalence between the derived categories ofExpand
A characterisation of enveloping 1-tilting classes over commutative rings
• Mathematics
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 finitelyExpand
Covering classes and 1-tilting cotorsion pairs over commutative rings
• Mathematics
• 2020
Abstract 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}}Expand
Matlis category equivalences for a ring epimorphism
• Mathematics
• 2020
Abstract Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R ⟶ U . Assuming that the ring epimorphism is homological ofExpand
S-almost perfect commutative rings
• Mathematics
• Journal of Algebra
• 2019
Abstract Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only ifExpand
Divisible modules and localization
• Mathematics
• 2005
A Matlis domain is a commutative domain R whose ring of quotients Q has projective dimension one. The properties of these domains have been extensively studied in the literature. In [18], KaplanskyExpand