We study the behaviour of modules M that fit into a short exact sequence 0 → M → C → M → 0, where C belongs to a class of modules C, the so-called C-periodic modules. We find a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. In the second part we will combine techniques of hereditary cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if M is any module and C is cotorsion, then M will be also cotorsion. This will lead to some meaningful consequences in the category Ch(R) of unbounded chain complexes and in Gorenstein homological algebra. For example we show that every acyclic complex of cotorsion modules has cotorsion cycles, and more generally, every map F → C where C is a complex of cotorsion modules and F is an acyclic complex of flat cycles, is null-homotopic. In other words, every complex of cotorsion modules is dg-cotorsion.
We explore the interlacing between model category structures attained to classes of modules of finite Xdimension, for certain classes of modules X . As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.Let Λ be a finite dimensional algebra over a field k (or more generally, let Λ be an Artin ring). The big finitistic dimension of Λ, Findim(Λ), is defined as the supremum of the projective dimensions of all modules having finite projective dimension. And the little finitistic dimension of Λ, findim(Λ), is defined in a similar way by restricting to the subclass of all finitely generated modules of finite projective dimension. In 1960, Bass stated the so-called Finitistic Dimension Conjectures: (I) Findim(Λ) = findim(Λ), and (II) findim(Λ) is finite. The first conjecture was proved to be false by Huisgen-Zimmermann in [19], but the second one still remains open. It has been proved to be true, for instance, for finite-dimensional monomial algebras [16], for Artin algebras with vanishing cube radical [20], or Artin algebras with representation dimension bounded by 3 [22].In [21] and [5], Huisgen-Zimmermann, Smalø, Auslander and Reiten proved that the finitistic dimension conjectures hold for Artin algebras in which the class P < ∞ of all finitely generated modules of finite projective dimension is contravariantly finite (equivalently, it is a precovering class in the sense of [9], [15]). In general, P < ∞ does not need to be contravariantly finite, even for Artin algebras satisfying the finitistic dimension conjectures. But, as Angeleri-Hügel and Trlifaj have noticed in [3], it cogenerates a cotorsion pair (F, C) in which the class F is precovering in R -Mod. By means of this idea, the authors are able to extend Auslander-Reiten's approach to arbitrary artinian rings and obtain a general criterium for an artinian ring to satisfy the finitistic dimension conjectures in terms of Tilting Theory (see [3]). This type of arguments has also been recently extended to more general homologies induced by arbitrary hereditary cotorsion pairs (see [1]).On the other hand, Hovey has recently shown in [18] that there exists a quite strong relation between the construction of hereditary cotorsion pairs in module categories and the existence of model structures in the sense of Quillen in the associated categories of unbounded chain complexes. Moreover, under very general hypotheses, the cohomology functors defined from these model structures coincide with the absolute cohomology functors defined from the injective model structure (in the sense of [18, Example 3.2])]. Recall that a model category is a category with three distinguished classes of morphisms (fibrations, cofibrations and weak equivalences) satisfying a certain number of axioms. We refer to [17] for a complete definition and main properties of model categories. One of the main advantages of these model categories is that they allow the construction of the ...
We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.
Let R R be any ring. We prove that all direct products of flat right R R -modules have finite flat dimension if and only if each finitely generated left ideal of R R has finite projective dimension relative to the class of all F \mathcal F -Mittag-Leffler left R R -modules, where F \mathcal F is the class of all flat right R R -modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if X \mathcal X is any class of left R R -modules closed under filtrations that contains all projective modules, then R R has finite left global projective dimension relative to X \mathcal X if and only if each left ideal of R R has finite projective dimension relative to X \mathcal X . This result contains, as particular cases, the well-known results concerning the classical left global, weak and Gorenstein global dimensions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.