On the cogeneration of cotorsion pairs

Eklof, Paul C.; Shelah, Saharon; Trlifaj, Jan

doi:10.1016/j.jalgebra.2003.09.018

Cited by 8 publications

(6 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, the latter fact is not provable in ZFC, because it is also consistent that ⊥ {Z} = P 0 , [18]. Further consistency results on the nondeconstructibility of the classes of the form ⊥ C were proved in [9], but it is still an open problem whether there exists (in ZFC) a non-deconstructible class of modules of the form ⊥ C.…”

Section: Introductionmentioning

confidence: 99%

Approximations and locally free modules

Slavik

Trlifaj

2013

Bulletin of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

For any set of modules 풮, we prove the existence of precovers (right approximations) for all classes of modules of bounded 풞‐resolution dimension, where 풞 is the class of all 풮‐filtered modules. In contrast, we use infinite‐dimensional tilting theory to show that the class of all locally free modules induced by a non‐∑‐pure‐split tilting module is not precovering. Consequently, the class of all locally Baer modules is not precovering for any countable hereditary artin algebra of infinite representation type.

show abstract

Section: Introductionmentioning

confidence: 99%

Approximations and locally free modules

Slavik

Trlifaj

2013

Bulletin of the London Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…We conjecture that the deconstructibility of ⊥ (D ⊥ Q ) is equivalent to the fact that F = lim − → D Q = ⊥ (D ⊥ Q ). It is interesting to note that if R is a countable ring such that ⊥ (D ⊥ Q ) = F for a class of left R-modules Q, then by Corollary 7.5 it follows that the class ⊥ (D ⊥ Q ) is not deconstructible; this would yield a first known example of the class of all roots of Ext that is not deconstructible in ZFC (examples of such classes in extensions of ZFC have however been constructed in [11]).…”

Section: Non-deconstructibility Of Flat Mittag-leffler Modules and Co...mentioning

confidence: 99%

Almost free modules and Mittag--Leffler conditions

Herbera,

Trlifaj

2009

Preprint

Self Cite

View full text Add to dashboard Cite

Drinfeld recently suggested to replace projective modules by the flat Mittag-Leffler ones in the definition of an infinite dimensional vector bundle on a scheme X, [8]. Two questions arise: (1) What is the structure of the class D of all flat Mittag-Leffler modules over a general ring? (2) Can flat Mittag-Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi-coherent sheaves on X?We answer (1) by showing that a module M is flat Mittag-Leffler, if and only if M is ℵ 1 -projective in the sense of Eklof and Mekler [10]. We use this to characterize the rings such that D is closed under products, and relate the classes of all Mittag-Leffler, strict Mittag-Leffler, and separable modules. Then we prove that the class D is not deconstructible for any non-right perfect ring. So unlike the classes of all projective and flat modules, the class D does not admit the homotopy theory tools developed recently by Hovey [25]. This gives a negative answer to (2).

show abstract

“…However, the deconstructibility of some of the classes of modules depends on the extension of ZFC that we work in. We finish this section by a sample result of this kind from [22] and [24] (see also [29, §10]):…”

Section: 5])mentioning

confidence: 99%

“…Then ⊥ C is κ-deconstructible for κ = card(R)+ℵ 0 . (ii) [22] Assume that R has countably many maximal ideals, and N is a non-cotorsion module. Then the deconstructibility of ⊥ N is independent of ZFC.…”

Section: Theorem 26 Let R Be a Dedekind Domain (= A Hereditary Commmentioning

confidence: 99%

Filtrations for the Roots of Ext

Trlifaj

2007

Milan j. math.

Self Cite

View full text Add to dashboard Cite

We survey the set-theoretic methods of module theory that make it possible to equip roots of the contravariant Ext functor with filtrations built from the small roots. The power of these methods is illustrated by several applications: a solution to the Kaplansky problem on Baer modules and some of the related problems for relative Baer modules, the structure of tilting modules and classes, the structure of Matlis localizations of commutative rings, and in particular cases, proofs of the finitistic dimension conjectures, and of the telescope conjecture for module categories.The bifunctor Ext 1 R plays a crucial role in understanding properties of the category Mod-R of all right R-modules over an associative unital ring R.Though Ext 1 R (A, B) = 0 whenever A is a projective module or B is an injective module, in general it is quite difficult to characterize the zeros of Ext, that is, the pairs (A, B) such that Ext 1 R (A, B) = 0. Given a class of modules C, we will call a module A a root of Ext for C provided that Ext 1 R (A, C) = 0 for all C ∈ C. For example, if R is a commutative domain and C is the class of all torsion modules, then the roots of Ext for C are called Baer modules. Kaplansky asked for the structure of these modules in 1962 [35], but only recently a complete answer was given in [2]: the Baer modules coincide

show abstract

On the cogeneration of cotorsion pairs

Cited by 8 publications

References 14 publications

Approximations and locally free modules

Approximations and locally free modules

Almost free modules and Mittag--Leffler conditions

Filtrations for the Roots of Ext

Contact Info

Product

Resources

About