Kaplansky Classes and Cotorsion Theories of Complexes

Asadollahi, Javad; Hafezi, Rasool

doi:10.1080/00927872.2012.751601

Cited by 3 publications

(3 citation statements)

References 23 publications

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“…• the class of complexes with components from F ( Šť ovíček [35] and Asadollahi-Hafezi [1]); • the class of objects having an F -resolution of length n (for any fixed n ∈ N; Slavik-Trlifaj [33]).…”

Section: Av = Bmentioning

confidence: 99%

“…The notion of deconstructibility of a class of modules was implicit in the solution of the Flat Cover Conjecture around the year 2000 (see [7] and [3], and [36] for some historical remarks). A class F of modules is deconstructible (in the sense of Göbel-Trlifaj [20]) if there is a set S such that S ⊂ F and every member of F is a "transfinite extension" of members of S. 1 Deconstructibility is now one of the main tools used to show that a class is precovering, which is an important property in relative homological algebra (e.g. to derive relative Ext groups; see the introductions to [8] and [22] for a discussion).…”

Section: Introductionmentioning

confidence: 99%

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Maximum deconstructibility in module categories

Cox¹

2020

Preprint

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If F is a subclass of the objects in an abelian category C, an essential requirement for certain aspects of relative homological algebra is that F be a precovering class (a.k.a. right-approximating class) in C. An often-used fact is that deconstructible classes (in the sense of Saorín-Št'ovíček [31]) are always precovering classes. We provide a new characterization of deconstructibility for classes of modules and of complexes of modules, in terms of traces of partially elementary submodels of the universe of sets; this provides a kind of "top-down" way of verifying deconstructibility. We use this characterization to isolate a certain necessary condition for a class to be deconstructible, and prove that if there are enough large cardinals in the universe, this necessary condition is also sufficient; so large cardinals imply, in a certain sense, the maximum possible amount of deconstructibility. In particular, large cardinals imply that for every ring R and every class X of R-modules, the class of X-Gorenstein Projective modules is deconstructible (and hence precovering). In the case X = {projectives}, this yields an alternate proof of a recent theorem of Šaroch, though from a stronger large cardinal hypothesis than his.

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Section: Av = Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Maximum deconstructibility in module categories

Cox¹

2020

Preprint

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“…Note that U M and X X are two κ-Kaplansky classes of chain complexes by [AH,Theorem 3.4]. Thus there exists a chain subcomplex U 1 of U such that S ⊆ U 1 , Card(U 1 ) ≤ κ, and U 1 and U/U 1 are contained in U M .…”

Section: Preliminariesmentioning

confidence: 99%