Enveloping Classes over Commutative Rings

Bazzoni, Silvana; Gros, Giovanna Le

doi:10.48550/arxiv.1901.07921

Cited by 2 publications

(4 citation statements)

References 7 publications

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“…Let G denote the Gabriel filter of ideals in R related to u. We show that pd R U ≤ 1 whenever the quotient ring R/I is semilocal of Krull dimension zero for every ideal I ∈ G. The argument is based on the notion of I-contramodule R-modules for an ideal I in a commutative ring R. This result of ours has already found its uses in the work of Bazzoni and Le Gros on envelopes and covers in the tilting cotorsion pairs related to 1-tilting modules over commutative rings; see [4,Remark 8.6 and Theorem 8.7] and [5,Remark 8.2 and Theorem 8.17].…”

Section: Introductionmentioning

confidence: 77%

Flat commutative ring epimorphisms of almost Krull dimension zero

Positselski

2020

Preprint

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We consider flat epimorphisms of commutative rings R −→ U such that, for every ideal I ⊂ 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 ⊥0,1 in R-Mod. Assuming additionally that the ring U and all the rings R/I are perfect, we show that all flat R-modules are U -strongly flat. Thus we obtain a generalization of some results of the paper [6], where the case of the localization U = S −1 R of the ring R at a multiplicative subset S ⊂ R was considered.

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Section: Introductionmentioning

confidence: 77%

Flat commutative ring epimorphisms of almost Krull dimension zero

Positselski

2020

Preprint

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“…More properties of Gabriel topologies. We refer to [BLG19] for more properties of right Gabriel topologies. Many of them hold in the noncommutative case.…”

Section: Perfect Localisationsmentioning

confidence: 99%

“…In particular, in the case of R = Z Theorem 8.7 in [BLG19] implies that every 1-tilting class T is enveloping as Z is semihereditary and for any proper ideal aZ of Z, Z/aZ is artinian.…”

Section: When R Is a G-almost Perfect Ringmentioning

confidence: 99%

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Covering classes and $1$-tilting cotorsion pairs over commutative rings

Bazzoni¹,

Gros²

2020

Preprint

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We are interested in characterising the commutative rings for which a 1-tilting cotorsion pair (A, T ) provides for covers, that is when the class 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 is the Gabriel topology associated to the 1-tilting cotorsion pair (A, T ), and RG is the ring of quotients with respect to G, we show that if A is covering then G is a perfect localisation (in Stenström's sense [Ste75]) and the localisation RG has projective dimension at most one. Moreover, we show that A is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈ G. Rings satisfying the latter two conditions are called G-almost perfect.

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Enveloping Classes over Commutative Rings

Cited by 2 publications

References 7 publications

Flat commutative ring epimorphisms of almost Krull dimension zero

Flat commutative ring epimorphisms of almost Krull dimension zero

Covering classes and $1$-tilting cotorsion pairs over commutative rings

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