$\mathcal P_1$-covers over commutative rings

Bazzoni, Silvana; Gros, Giovanna Le

doi:10.4171/rsmup/54

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2022

2024

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Cited by 3 publications

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“…In [3], Enochs Conjecture has been verified for the class P 1 of modules of projective dimension at most 1 over a commutative semihereditary ring.…”

Section: Introductionmentioning

confidence: 99%

Enochs Conjecture for cotorsion pairs and more

Bazzoni¹,

Šaroch²

2023

Preprint

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Enochs Conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. In this paper, we prove the conjecture for the left hand-class of any cotorsion pair generated by (a class of) ℵn-presented modules for a fixed n < ω.We also show that it is consistent with ZFC that Enochs Conjecture holds for all classes of the form Filt(S) where S is a set of modules.

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“…In [3], Enochs Conjecture has been verified for the class P 1 of modules of projective dimension at most 1 over a commutative semihereditary ring.…”

Section: Introductionmentioning

confidence: 99%

Enochs Conjecture for cotorsion pairs and more

Bazzoni¹,

Šaroch²

2023

Preprint

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Enochs’ conjecture for small precovering classes of modules

Šaroch¹

2022

Isr. J. Math.

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Enochs’ conjecture for cotorsion pairs and more

Bazzoni,

Šaroch

2024

Forum Mathematicum

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Enochs’ conjecture asserts that each covering class of modules (over any ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full generality. In this paper, we prove the conjecture for the classes Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} consists of ℵ n {\aleph_{n}} -presented modules for some fixed n < ω {n<\omega} . In particular, this applies to the left-hand class of any cotorsion pair generated by a class of ℵ n {\aleph_{n}} -presented modules. Moreover, we also show that it is consistent with ZFC that Enochs’ conjecture holds for all classes of the form Filt ⁡ ( 𝒮 ) {\operatorname{Filt}(\mathcal{S})} , where 𝒮 {\mathcal{S}} is a set of modules. This leaves us with no explicit example of a covering class where we cannot prove that Enochs’ conjecture holds (possibly under some additional set-theoretic assumption).

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$\mathcal P_1$-covers over commutative rings

Cited by 3 publications

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Enochs Conjecture for cotorsion pairs and more

Enochs Conjecture for cotorsion pairs and more

Enochs’ conjecture for small precovering classes of modules

Enochs’ conjecture for cotorsion pairs and more

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