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, AuslanderBuchweitz 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 AuslanderBuchweitz approximation theory from a homotopytheoretical 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 ncoherent rings using closure properties and constructing hereditary cotorsion pairs which involve the classes of modules of type FPn, the FPninjective modules and the FPnflat 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. 
Relative homological algebra:
Currently most of my research interests are focused on relative versions of cotorsion theories. The idea is to find left and right approximations for objects in an certain subcategory of an abelian category, not necessarily for the whole ambient category. These relativizations have connections with Frobenius pairs and AuslanderBuchweitz contexts. Some applications in mind include: more particular descriptions of the orthogonal classes forming a cotorsion pair, the study of finitistic dimensions and Serre quotients, etc.

Gorenstein homological algebra:
Among the objects in categories that I study the most are the relative Gorenstein objects. So far I have focused on Gorenstein objects relative to FPninjective and FPnflat objects (that is, injective and flat objects relative to objects of finite type), but I am currently interested in more relative versions of Gorensteiness.
Publications
SUBMITTED WORKS AND PREPRINTS:

Cut cotorsion pairs.
With Mindy Huerta and Octavio Mendoza.
arxiv:2006.11189.
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 AuslanderBuchweitz approximation theory, by considering relative analogs for Frobenius pairs and AuslanderBuchweitz contexts. Several applications are given in the settings of relative Gorenstein homological algebra, chain complexes and quasicoherent sheaves, but also 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 cotstructures.

Locally type FPn and ncoherent categories.
With Daniel Bravo and James Gillespie.
arxiv:1908.10987.
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 Extorthogonal to the class of objects of type FPn, called FPninjective objects, which will be the right half of a complete cotorsion pair.
As a generalization of the category of modules over an ncoherent ring, we present the concept of ncoherent 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 FPninjective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by FPninjective objects. Moreover, we see how ncoherent categories provide a suitable framework for a nice theory of Gorenstein homological algebra with respect to the class of FPninjective modules. We define Gorenstein FPninjective objects and construct two different model category structures (one abelian and the other one exact) in which these Gorenstein objects are the fibrant objects.
PUBLISHED, ACCEPTED OR IN PRESS:

nCotorsion pairs.
With Mindy Huerta and Octavio Mendoza.
Journal of Pure and Applied Algebra. Volume 225, Issue 5.
arXiv:1902.10863.
Motivated by some properties satisfied by Gorenstein projective and Gorenstein injective modules over an IwanagaGorenstein ring, we present the concept of left and right ncotorsion pairs in an abelian category C. Two classes A and B of objects of C form a left ncotorsion 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 Bresolution 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 ncotorsion 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 mcluster tilting subcategories. 
Balanced pairs, cotorsion triplets and quiver representations
With Sergio Estrada and Haiyan Zhu.
Proceedings of the Edinburgh Mathematical Society. Volume 63, Issue 1, pp. 6790.
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. 
Relative FPinjective and FPflat complexes and their model structures.
With Tiwei Zhao.
Communications in Algebra. Volume 47, Issue 4, pp. 17081730.
arXiv:1703.10703.
In this paper, we introduce the notions of FPninjective and FPnflat complexes in terms of complexes of type FPn. We show that some characterizations analogous to that of injective, FPinjective and flat complexes exist for FPninjective and FPnflat complexes. We also introduce and study FPninjective and FPnflat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of preenvelopes and covers in this setting is discussed, and we prove that any complex has an FPnflat cover and an FPnflat preenvelope, and in the case n ≥ 2 that any complex has an FPninjective cover and an FPninjective preenvelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded FPninjective and FPnflat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.

Frobenius pairs in abelian categories:
correspondences with cotorsion pairs, exact model categories, and AuslanderBuchweitz contexts.
With Víctor Becerril, Octavio Mendoza and Valente Santiago.
Journal of Homotopy Related Structures. Volume 14, Issue 1, 2019, pp. 150.
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 Xresolution 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.

Finiteness conditions and cotorsion pairs.
With Daniel Bravo.
Journal of Pure and Applied Algebra. Volume 221, Issue 6, June 2017, pp. 12491267.
arXiv:1510.08966.
We study the interplay between the notions of ncoherent rings and finitely npresented modules, and also study the relative homological algebra associated to them. We show that the ncoherency of a ring is equivalent to the thickness of the class of finitely npresented 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 ncoherent rings.

Homological dimensions and Abelian model structures on chain complexes.
Rocky Mountain Journal of Mathematics. Volume. 46, Issue 3, 2016, pp. 9511010.
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.
BOOK CHAPTERS:

Model structures and relative Gorenstein flat modules and chain complexes.
With Sergio Estrada and Alina Iacob.
Contemporary Mathematics. Volume 751.
Categorical, Homological and Combinatorial Methods in Algebra. pp. 135175.
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 Rmodules, 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 Bflat modules, where B is a class of right Rmodules. 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 Bmodules 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 Rmodules, study other model structures on complexes constructed from relative Gorenstein flat modules, and compare these models via computing their homotopy categories.
RESEARCH MONOGRAPHS:

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 selfcontained as possible, this monographs presents new results in relative homological algebra and model category theory. We also reprove 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:

Relationship between Abelian model structures and Gorenstein homological dimensions.
PhD thesis. (xxviii + 336 pp.).
Université du Québec à Montréal. 
Pares de cotorsión y las conjeturas de la dimensión finitista.
Master thesis. (vi + 106 pp.).
Universidad Simón Bolívar. 
Cohomología de los espacios proyectivos complejos.
Bachelor thesis. (v + 52 pp.).
Expository lecture notes
 The nprojective cotorsion pair.
 Tensor products and internal Homs for chain complexes.
 Baer sums.
 The RiemannRoch Theorem.
 The maximal torus of a compact Lie group.
 The Freyd's Adjoint Functor Theorem.
 The de Rham's Theorem.
 The RadonNikodym Theorem.
 On the Exactness Axiom in Čech Cohomology.
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.