Description of my research field

My work focuses mainly on category theory and homological algebra, specifically on the relation between cotorsion theories and abelian model structures, the study of properties of such structures, applications, and the process of obtaining them in different contexts, such as relative homological algebra, Auslander-Buchweitz approximation theory, finiteness conditions of modules, etc.



In a more precise way, my work can be described in the following research lines:
  • Construction of abelian model structures from homological dimensions:

    The relation between cotorsion theories and model structures is described by a result known as the Hovey Correspondence, which I used in my Ph.D. thesis two obtain nine new model structures which involve classical and Gorenstein homological dimensions.

    Apart from the previous model structures, another contribution I have made to relative homological algebra has been the study of Auslander-Buchweitz approximation theory from a homotopy-theoretical point of view, by defining and studying the concept of Frobenius pairs.

  • Finiteness conditions:

    I also work on finiteness conditions over rings. In this field, one of my contributions is a characterisation for n-coherent rings using closure properties and constructing hereditary cotorsion pairs which involve the classes of modules of type FPn, the FPn-injective modules and the FPn-flat modules. These concepts are in turn generalisations of finitely presented, injective and flat modules, respectively.

    I have also used these classes of modules in the process of obtaining new model structures on the category of chain complexes over a ring. One important aspect on this matter is looking for conditions under which a ring homomorphism yields Quillen adjunctions between these structures.



Publications


SUBMITTED WORKS AND PREPRINTS:

  • n-Cotorsion pairs.
    With Mindy Huerta and Octavio Mendoza.
    arXiv:1902.10863.

    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.

  • Model structures and relative Gorenstein flat modules and chain complexes.
    With Sergio Estrada and Alina Iacob.
    arXiv:1709.00658.

    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.


PUBLISHED, ACCEPTED OR IN PRESS:

  1. Balanced pairs, cotorsion triplets and quiver representations
    With Sergio Estrada and Haiyan Zhu.
    Proceedings of the Edinburgh Mathematical Society. In press.
    arXiv:1802.09989.

    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.

  2. Relative FP-injective and FP-flat complexes and their model structures.
    With Tiwei Zhao.
    Communications in Algebra. In press.
    arXiv:1703.10703.

    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.

  3. Frobenius pairs in abelian categories: correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts.
    With Víctor Becerril, Octavio Mendoza and Valente Santiago.
    Journal of Homotopy Related Structures. Volume 14, Issue 1, 2019, pp. 1-50.
    arXiv:1602.0732.

    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.

  4. Finiteness conditions and cotorsion pairs.
    With Daniel Bravo.
    Journal of Pure and Applied Algebra. Volume 221, Issue 6, June 2017, pp. 1249-1267.
    arXiv:1510.08966.

    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.

  5. Homological dimensions and Abelian model structures on chain complexes.
    Rocky Mountain Journal of Mathematics. Volume. 46, Issue 3, 2016, pp. 951-1010.
    arXiv:1202.2481.

    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.


RESEARCH MONOGRAPHS / BOOKS:

  1. Introduction to Abelian Model Structures and Gorenstein Homological Dimensions.
    Monographs and Research Notes in Mathematics. CRC Press. Taylor & Francis Group.
    August 17, 2016. (xxviii + 344 pp.)
    .

    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.


THESES:

  1. Relationship between Abelian model structures and Gorenstein homological dimensions.
    PhD thesis. (xxviii + 336 pp.).
    Université du Québec à Montréal.
  2. Pares de cotorsión y las conjeturas de la dimensión finitista.
    Master thesis. (vi + 106 pp.).
    Universidad Simón Bolívar.
  3. Cohomología de los espacios proyectivos complejos.
    Bachelor thesis. (v + 52 pp.).


Expository lecture notes




Collaborators


THESES SUPERVISORS:

  • André Joyal.
    Professeur émérite.
    Département de mathématiques. Université du Québec à Montréal.
    joyal[DOT]andre[AT]uqam[DOT]ca.
    Collaborations: Ph.D. thesis' supervisor.
  • Juan RADA.
    Profesor asociado.
    Instituto de Matemáticas. Universidad de Antioquia.
    pablo[DOT]rada[AT]udea[DOT]edu[DOT]co
    Collaborations: Master thesis' supervisor.
  • Fermín Metodio DALMAGRO.
    Profesor.
    Escuela de Matemática. Universidad Central de Venezuela.
    fermin[DOT]dalmagro[AT]ciens[DOT]ucv[DOT]ve
    Collaborations: Bachelor thesis' supervisor.

COAUTHORS:

  • Víctor BECERRIL.
    Ph. D.
    Instituto de Matemáticas. Universidad Nacional Autónoma de México.
    victor[DOT]becerril[AT]matem[DOT]unam[DOT]mx
    Collaborations: 1 research paper.
  • Daniel BRAVO.
    Académico.
    Instituto de Ciencias Físicas y Matemáticas. Universidad Austral de Chile (Isla Teja Campus).
    daniel[DOT]bravo[AT]uach[DOT]cl
    Collaborations: 1 research paper.
  • Sergio ESTRADA.
    Profesor Titular.
    Departamento de Matemáticas. Universidad de Murcia.
    sestrada[AT]um[DOT]es
    Collaborations: 1 research paper.
  • Octavio MENDOZA.
    Investigador.
    Instituto de Matemáticas. Universidad Nacional Autónoma de México.
    omendoza[AT]matem[DOT]unam[DOT]mx
    Collaborations: 1 research paper.
  • Valente SANTIAGO.
    Profesor de carrera asociado C.
    Departamento de Matemáticas. Universidad Nacional Autónoma de México.
    valente[DOT]santiago[DOT]v[AT]gmail[DOT]com
    Collaborations: 1 research paper.
  • Tiwei ZHAO.
    Professor.
    Qufu Normal University. China.
    tiweizhao[AT]qfnu[DOT]edu[DOT]cn
    Collaborations: 1 research paper.
  • Haiyan ZHU.
    Professor.
    College of Science. Zhejiang University of Technology.
    hyzhu[AT]zjut[DOT]edu[DOT]cn
    Collaborations: 1 research paper.