Research

Abstract: We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type FPn and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type FPn categories as a generalization of locally finitely generated and locally finitely presented categories. We also define and study the injective objects that are Ext-orthogonal to the class of objects of type FPn, called FPn-injective objects, which will be the right half of a complete cotorsion pair.

As a generalization of the category of modules over an n-coherent ring, we present the concept of n-coherent categories, which also recovers the notions of locally noetherian and locally coherent categories for n = 0, 1. Such categories will provide a setting in which the FPn-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by FPn-injective objects. Moreover, we see how n-coherent categories provide a suitable framework for a nice theory of Gorenstein homological algebra with respect to the class of FPn-injective modules. We define Gorenstein FPn-injective objects and construct two different model category structures (one abelian and the other one exact) in which these Gorenstein objects are the fibrant objects.

Abstract: We present and study the concept of m-periodic Gorenstein objects relative to a pair (A,B) of classes of objects in an abelian category, as a generalization of m-strongly Gorenstein projective modules over associative rings. We prove several properties when (A,B) satisfies certain homological conditions, like for instance when (A,B) is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relative to these pairs are also established.

Abstract: In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of FPn-injective objects to be a torsion class. By doing so, we propose a notion of n-hereditary categories. We also define and study the class of FPn-flat objects in Grothendieck categories with a generating set of small projective objects, and provide several equivalent conditions for this class to be torsion-free. At the end, we present several applications and examples of n-hereditary categories in the contexts modules over a ring, chain complexes of modules and categories of additive functors from an additive category to the category of abelian groups. Concerning the latter setting, we find a characterization of when these functor categories are n-hereditary in terms of the domain additive category.

Abstract: We present the concept of cotorsion pairs cut along subcategories of an abelian category. This provides a generalization of complete cotorsion pairs, and represents a general framework to find approximations restricted to certain subcategories. We also exhibit some connections between cut cotorsion pairs and Auslander–Buchweitz approximation theory, by considering relative analogs for Frobenius pairs and Auslander–Buchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes, and quasi-coherent sheaves, as well as to characterize some important results on the Finitistic Dimension Conjecture, the existence of right adjoints of quotient functors by Serre subcategories, and the description of cotorsion pairs in triangulated categories as co-t-structures.

Abstract: Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an Iwanaga-Gorenstein ring, we present the concept of left and right n-cotorsion pairs in an abelian category C. Two classes A and B of objects of C form a left n-cotorsion pair (A,B) in C if the orthogonality relation Ext^i(A,B) = 0 is satisfied for indexes 1 ≤ i ≤ n, and if every object of C has a resolution by objects in A whose syzygies have B-resolution dimension at most n−1. This concept and its dual generalise the notion of complete cotorsion pairs, and has an appealing relation with left and right approximations, especially with those having the so called unique mapping property.

The main purpose of this paper is to describe several properties of n-cotorsion pairs and to establish a relation with complete cotorsion pairs. We also give some applications in relative homological algebra, that will cover the study of approximations associated to Gorenstein projective, Gorenstein injective and Gorenstein flat modules and chain complexes, as well as m-cluster tilting subcategories.

Abstract: Balanced pairs appear naturally in the realm of Relative Homological Algebra associated to the balance of right derived functors of the 𝖧𝗈𝗆 functor. A natural source to get such pairs is by means of cotorsion triplets.

In this paper we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories having enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also give a short proof of the lack of balance for derived functors of 𝖧𝗈𝗆 computed by using flat resolutions which extends the one showed by Enochs in the commutative case.

Abstract: In this paper, we introduce the notions of FPn-injective and FPn-flat complexes in terms of complexes of type FPn. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for FPn-injective and FPn-flat complexes. We also introduce and study FPn-injective and FPn-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of pre-envelopes and covers in this setting is discussed, and we prove that any complex has an FPn-flat cover and an FPn-flat pre-envelope, and in the case n ≥ 2 that any complex has an FPn-injective cover and an FPn-injective pre-envelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded FPn-injective and FPn-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.

Abstract: We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs (X,ω) in an abelian category C. We show how to construct from (X,ω) a projective exact model structure on Xˆ, the subcategory of objects in C with finite X-resolution dimension, via cotorsion pairs relative to a thick subcategory of C. We also establish correspondences between these model structures, relative cotorsion pairs, Frobenius pairs, and Auslander–Buchweitz contexts. Some applications of this theory are given in the context of Gorenstein homological algebra, and connections with perfect cotorsion pairs, covering subcategories and cotilting modules are also presented and described.

Abstract: We study the interplay between the notions of n-coherent rings and finitely n-presented modules, and also study the relative homological algebra associated to them. We show that the n-coherency of a ring is equivalent to the thickness of the class of finitely n-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to the Ext and Tor functors. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of n-coherent rings.

Abstract: We construct Abelian model structures on the category of chain complexes over a ring R, from the notion of homological dimensions of modules. Given an integer n > 0, we prove that the left modules over a ringoid R with projective dimension at most n form the left half of a complete cotorsion pair. Using this result, we prove that there is a unique Abelian model structure on the category of chain complexes over R, where the exact complexes are the trivial objects and the complexes with projective dimension at most n form the class of trivially cofibrant objects.

In a previous work by D. Bravo et. al., the authors construct an Abelian model structure on chain complexes, where the trivial objects are the exact complexes, and the class of cofibrant objects is given by the complexes whose terms are all projective. We extend this result by finding a new Abelian model structure with the same trivial objects where the cofibrant objects are given by the class of complexes whose terms are modules with projective dimension at most n. We also prove similar results concerning flat dimension.

Abstract: A recent result by J. Šaroch and J. Šťovíček asserts that there is a unique abelian model structure on the category of left R-modules, for any associative ring R with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules.

In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein B-flat modules, where B is a class of right R-modules. Using some of the techniques considered by Šaroch and Šťovíček, plus some other arguments coming from model theory, we determine some conditions for B so that the class of Gorenstein B-modules is closed under extensions. This will allow us to show approximation properties concerning these modules, and also to obtain a relative version of the model structure described before. Moreover, we also present and prove our results in the category of complexes of left R-modules, study other model structures on complexes constructed from relative Gorenstein flat modules, and compare these models via computing their homotopy categories.

Abstract: This monograph provides a starting point to study the relationship between homological and homotopical algebra, a very active branch of mathematics. We show how to obtain new model structures in homological algebra by constructing a pair of compatible complete cotorsion pairs related to a specific homological dimension and then applying the Hovey Correspondence to generate an abelian model structure.

The first part of the monograph introduces the definitions and notations of the universal constructions most often used in category theory. The next part presents a proof of the Eklof and Trlifaj theorem in Grothedieck categories and covers M. Hovey’s work that connects the theories of cotorsion pairs and model categories. The final two parts study the relationship between model structures and classical and Gorenstein homological dimensions and explore special types of Grothendieck categories known as Gorenstein categories.

As self-contained as possible, this monographs presents new results in relative homological algebra and model category theory. We also re-prove some established results using different arguments or from a pedagogical point of view. In addition, we prove folklore results that are difficult to locate in the literature.