arXiv subjects

Marco A. Pérez

Publications and source records attributed to Marco A. Pérez.

18 recordsLinked to original sources

arXiv2026

Cotorsion pairs and model structures induced by resolution dimensions relative to Frobenius pairs

We explore some methods to obtain cotorsion pairs and model category structures from a Frobenius pair. Specifically, we consider a type of Frobenius pair $(\mathcal{X},ω)$ in an abelian category $\mathfrak{C}$ for which there exists a nonnegative integer $m$ such that every object in the right orthogonal complement $\mathcal{X}^{\perp_1}$ of $\mathcal{X}$ under $\text{Ext}^1_{\mathfrak{C}}(-,\sim)$ has a special $ω^\wedge_m$-precover and a special $[ω^\wedge_m]^{\perp_1}$-preenvelope, where $ω^\wedge_m$ denotes the class of objects in $\mathfrak{C}$ with resolution dimension relative to $ω$ at most $m$. The cotorsion pairs and model structures obtained from these Frobenius pairs will involve the classes $\mathcal{X}$, $ω$, $\mathcal{X}^\wedge_m$, $ω^\wedge_m$, and their orthogonal complements, in the description of its cofibrant, fibrant and trivial objects. As applications of our results, we obtain several cotorsion pairs and model structures related to relative and absolute Gorenstein homological dimensions.

math.CT↗

arXiv2025

Gorensteinness from duality pairs induced via Foxby equivalences

We define and study induced duality pairs under Foxby equivalences. Given a semidualizing $(S,R)$-bimodule ${}_S C_R$, if $(\mathcal{A}_C(R),\mathcal{B}_C(R^{\rm op}))$ and $(\mathcal{A}_C(S^{\rm op}),\mathcal{B}_C(S))$ denote the duality pairs formed by the corresponding classes of Auslander and Bass modules, and if $(\mathcal{M,N})$ is a duality pair over $R$, we study the duality pair formed by the essential images of the restricted Foxby equivalences $(C \otimes_R \sim)|_{\mathcal{A}_C(R) \cap \mathcal{M}}$ and $\mathrm{Hom}_{R^{\rm op}}(C,\sim) |_{\mathcal{B}_C(R^{\rm op}) \cap \mathcal{N}}$, denoted by $\mathcal{M}^C(S)$ and $\mathcal{N}^C(S^{\rm op})$. We investigate which additional properties of the duality pair $(\mathcal{M,N})$ are transferred to $(\mathcal{M}^C(S),\mathcal{N}^C(S^{\rm op}))$. We also study several versions of Gorenstein injective and Gorenstein flat modules relative to the pairs $(\mathcal{A}_C(R) \cap \mathcal{M},\mathcal{B}_C(R^{\rm op}) \cap \mathcal{N})$ and $(\mathcal{M}^C(S),\mathcal{N}^C(S^{\rm op}))$. For instance, we explore the relation between these classes of modules under Foxby equivalences and under Pontryagin duality.

math.RA↗

arXiv2025

Gorenstein categories relative to G-admissible triples

We present the notion of Gorenstein categories relative to G-admissible triples. This is a relativization of the concept of Gorenstein category (an abelian category with enough projective and injective objects, in which the suprema of the sets $\{ {\rm pd}(I) \ \text{:} \ I \text{ is injective} \}$ and $\{ {\rm id}(P) \ \text{:} \ P \text{ is projective} \}$ are finite). Such categories turn out to be a suitable setting on which it is possible to obtain hereditary abelian model structures where the (co)fibrant objects are Gorenstein injective (resp., Gorenstein projective) objects relative to GI-admissible (resp., GP-admissible) pairs. Applications and examples of these structures are given. Moreover, we link relative Gorenstein categories with tilting theory and obtain relations between different relative homological dimensions.

math.CT↗

arXiv2024

Homological and homotopical aspects of Gorenstein flat modules and complexes relative to duality pairs

We study homological and homotopical aspects of Gorenstein flat modules over a ring with respect to a duality pair $(\mathcal{L,A})$. These modules are defined as cycles of exact chain complexes with components in $\mathcal{L}$ which remain exact after tensoring by objects in $\mathcal{A} \cap {}^\perp\mathcal{A} = \mathcal{A} \cap \Big( \bigcap_{i \in \mathbb{Z}_{> 0}} {\rm Ker}({\rm Ext}^i_{R^{\rm o}}(-,\mathcal{A})) \Big)$. In the case where $(\mathcal{L,A})$ is product closed and bicomplete (meaning in addition that $\mathcal{L}$ is closed under extensions, (co)products, $R \in \mathcal{L}$, $(\mathcal{A,L})$ is also a duality pair, and $\mathcal{A}$ is the right half of a hereditary complete cotorsion pair) we prove that these relative Gorenstein flat modules are closed under extensions, and that the corresponding Gorenstein flat dimension is well behaved in the sense that it recovers many of the properties and characterizations of its (absolute) Gorenstein flat counterpart (for instance, it can be described in terms of torsion functors). The latter in turn is a consequence of a Pontryagin duality relation that we show between these relative Gorenstein flat modules and certain Gorenstein injective modules relative to $\mathcal{A}$. We also find several hereditary and cofibrantly generated abelian model structures from these Gorenstein flat modules and complexes relative to $(\mathcal{L,A})$. At the level of chain complexes, we find three recollements between the homotopy categories of these model structures, along with several derived adjunctions connecting these recollements.

math.RT↗

arXiv2024

Balanced systems for $\mathrm{Hom}$

From the notion of (co)generator in relative homological algebra, we present the concept of finite balanced system $[(\mathcal{X} , ω); (ν, \mathcal{Y})]$ as a tool to induce balanced pairs $(\mathcal{X} , \mathcal{Y} )$ for the $\mathrm{Hom}$ functor with domain determined by the finiteness of homological dimensions relative to $\mathcal{X}$ and $\mathcal{Y}$. This approach to balance will cover several well known ambients where right derived functors of $\mathrm{Hom}$ are obtained relative to certain classes of objects in an abelian category, such as Gorenstein projective and injective modules and chain complexes, Gorenstein modules relative to Auslander and Bass classes, among others.

math.CT↗

arXiv2023

STB-VMM: Swin Transformer Based Video Motion Magnification

The goal of video motion magnification techniques is to magnify small motions in a video to reveal previously invisible or unseen movement. Its uses extend from bio-medical applications and deepfake detection to structural modal analysis and predictive maintenance. However, discerning small motion from noise is a complex task, especially when attempting to magnify very subtle, often sub-pixel movement. As a result, motion magnification techniques generally suffer from noisy and blurry outputs. This work presents a new state-of-the-art model based on the Swin Transformer, which offers better tolerance to noisy inputs as well as higher-quality outputs that exhibit less noise, blurriness, and artifacts than prior-art. Improvements in output image quality will enable more precise measurements for any application reliant on magnified video sequences, and may enable further development of video motion magnification techniques in new technical fields.

cs.CV↗

arXiv2022

Model structures and relative Gorenstein flat modules and chain complexes

A recent result by J. Šaroch and J. Šťov\'ıček 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 $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by Šaroch and Šťov\'ıček, plus some other arguments coming from model theory, we determine some conditions for $\mathcal{B}$ so that the class of Gorenstein $\mathcal{B}$-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.

math.CT↗

arXiv2022

$m$-periodic Gorenstein objects

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

math.RA↗

arXiv2022

Torsion and torsion-free classes from objects of finite type in Grothendieck categories

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-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 $\text{FP}_n$-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. In 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.

math.CT↗

arXiv2020

Cut cotorsion pairs

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, 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 co-$t$-structures.

math.CT↗

arXiv2019

Locally type $\text{FP}_n$ and $n$-coherent categories

We study finiteness conditions in Grothendieck categories by introducing the concepts of objects of type $\text{FP}_n$ and studying their closure properties with respect to short exact sequences. This allows us to propose a notion of locally type $\text{FP}_n$ 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 $\text{FP}_n$, called $\text{FP}_n$-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 $\text{FP}_n$-injective cotorsion pair is hereditary, and where it is possible to construct (pre)covers by $\text{FP}_n$-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 $\text{FP}_n$-injective modules. We define Gorenstein $\text{FP}_n$-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.

math.CT↗

arXiv2019

$n$-Cotorsion pairs

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 $\mathcal{C}$. Two classes $\mathcal{A}$ and $\mathcal{B}$ of objects of $\mathcal{C}$ form a left $n$-cotorsion pair $(\mathcal{A,B})$ in $\mathcal{C}$ if the orthogonality relation $\mathsf{Ext}^i_{\mathcal{C}}(\mathcal{A,B}) = 0$ is satisfied for indexes $1 \leq i \leq n$, and if every object of $\mathcal{C}$ has a resolution by objects in $\mathcal{A}$ whose syzygies have $\mathcal{B}$-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.

math.RT↗

arXiv2018

On some applications of balanced pairs and their relation with cotorsion triplets

Balanced pairs appear naturally in the realm of Relative Homological Algebra associated to the balance of right derived functors of the $\mathsf{Hom}$ 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 $\mathsf{Hom}$ computed by using flat resolutions which extends the one showed by Enochs in the commutative case.

math.RT↗

arXiv2017

Relative FP-injective and FP-flat complexes and their model structures

In this paper, we introduce the notions of ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes in terms of complexes of type ${\rm FP}_n$. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes. We also introduce and study ${\rm FP}_n$-injective and ${\rm FP}_n$-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 ${\rm FP}_n$-flat cover and an ${\rm FP}_n$-flat pre-envelope, and in the case $n \geq 2$ that any complex has an ${\rm FP}_n$-injective cover and an ${\rm FP}_n$-injective pre-envelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded ${\rm FP}_n$-injective and ${\rm FP}_n$-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.

math.CT↗

arXiv2016

Coends and the tensor product of $\mathcal{C}$-modules

We give an introduction to the concept of Kan extensions, and study its relation with the notions of coend and adjoint functors. We state and prove in detail a well known formula to compute Kan extensions by using coends: a certain colimit related to the concept of copower. Finally, we study the tensor product of functors, and its relation with Kan extensions, in order to represent the tensor product of $\mathcal{C}$-modules as a particular case.

math.CT↗

arXiv2016

Frobenius pairs in abelian categories: correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts

In this work, we revisit Auslander-Buchweitz Approximation Theory and find some relations with cotorsion pairs and model category structures. From the notions of relatives generators and cogenerators in Approximation Theory, we introduce the concept of left Frobenius pairs $(\mathcal{X},ω)$ in an abelian category $\mathcal{C}$. We show how to construct from $(\mathcal{X},ω)$ a projective exact model structure on $\mathcal{X}^\wedge$, as a result of Hovey-Gillespie Correspondence applied to two compatible and complete cotorsion pairs in $\mathcal{X}^\wedge$. These pairs can be regarded as examples of what we call cotorsion pairs relative to a thick subcategory of $\mathcal{C}$. We establish some correspondences between Frobenius pairs, relative cotorsion pairs, exact model structures and Auslander-Buchweitz contexts. Finally, some applications of these results are given in the context of Gorenstein homological algebra by generalizing some existing model structures on the categories of modules over Gorenstein and Ding-Chen rings, and by encoding the stable module category of a ring as a certain homotopy category. We also present some connections with perfect cotorsion pairs, covering classes, and cotilting modules.

math.CT↗

arXiv2015

Finiteness conditions and cotorsion pairs

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 ${\rm Ext}^1_R(F,-)$ and ${\rm Tor}_1^R(F,-)$. We also construct cotorsion pairs from these orthogonal complements, allowing us to provide further characterizations of $n$-coherent rings.

math.RA↗

arXiv2015

Toward formalizing ologs: Linguistic structures, instantiations, and mappings

We define the notion of linguistic structure on a small category, in order to provide a more formal description of ontology logs, also known as ologs, introduced by R. E. Kent and D. I. Spivak in their paper "Ologs: A categorical framework for knowledge representation." In particular, we construct a bicategory $\mathsf{Eng}$, of English noun phrases and verb phrases, endorsed as functional by varying sets of authors. An olog is then defined as a lax functor to $\mathsf{Eng}$. We then present a new notion of linguistic functor, which extends Spivak's notion of meaningful functors. Finally, we discuss the relationship between ologs and databases in this context.

math.CT↗