arXiv subjects

Octavio Mendoza

Publications and source records attributed to Octavio Mendoza.

At least 19 recordsLinked to original sources

arXiv2026

Tilting objects in the extended heart of a $t$-structure

Building on the recent work of Adachi, Enomoto and Tsukamoto on a generalization of the Happel-Reiten-Smalø tilting process, we study extended tilting objects in extriangulated categories with negative first extension. These objects coincide with the 1-tilting objects in abelian categories as in the work of Parra, Saor{í}n and Virili. We will be particularly interested in the case where the extriangulated category in question is the heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ of an interval of $t$-structures $[\mathbf{t}_{1},\mathbf{t}_{2}]$. Our main results consist of a characterization of the extended tilting objects of a heart $\mathcal{H}_{[\mathbf{t}_{1},\mathbf{t}_{2}]}$ for the case when $\text{\ensuremath{\mathbf{t}}}_{2}\leqΣ^{-1}\mathbf{t}_{1}$, and another one for the case when $Σ^{-2}\mathbf{t}_{1}<\mathbf{t}_{2}$. In the first one, we give conditions for these tilting objects to coincide with the quasi-tilting objects of the abelian category $\mathcal{H}_{[\mathbf{t}_{1},Σ^{-1}\mathbf{t}_{1}]}$. In the second one, it is given conditions for these to coincide with projective generators in the extriangulated category $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$

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arXiv2025

Recollements, coproducts and products in extriangulated categories

We introduce a notion similar to the AB4 (resp. AB4{*}) condition for abelian categories but in the context of extriangulated categories. We will refer to this notion as AET4 (resp. AET4{*}). One of our main results shows equivalent statements for AET4 (resp. AET4{*}), which generalize statements commonly used in homological constructions in abelian categories. As an application, we will give conditions for a recollement $(\mathcal{A},\mathcal{B},\mathcal{C})$ of extriangulated categories with $\mathcal{B}$ AET4 (resp. AET4{*}) to imply that the categories $\mathcal{A}$ and $\mathcal{C}$ are AET4 (resp. AET4{*}); and we will show a relation between the $n$-smashing (resp. $n$-co-smashing) condition for a $t$-structure and the AET4 (resp. AET4{*}) condition of the extended hearts of the $t$-structure. It is also included an appendix where we study in detail the properties of adjoint pairs between extriangulated categories which are necessary for the development of the paper, including some special properties for higher extension groups.

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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.

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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.

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arXiv2024

Corrigendum to "$m$-Periodic Gorenstein objects" [J. Algebra 621 (2023)]

Let $(\mathcal{A,B})$ be a GP-admissible pair and $(\mathcal{Z,W})$ be a GI-admissible pair of classes of objects in an abelian category $\mathcal{C}$, and consider the class $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ of $1$-periodic $(ω,\mathcal{B})$-Gorenstein projective objects, where $ω:= \mathcal{A} \cap \mathcal{B}$ and $ν:= \mathcal{Z} \cap \mathcal{W}$. We claimed in \cite[Lem. 8.1]{HMP2023m} that the $(\mathcal{Z,W})$-Gorenstein injective dimension of $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ is bounded by the $(\mathcal{Z,W})$-Gorenstein injective dimension of $ω$, provided that: (1) $ω$ is closed under direct summands, (2) $\mathrm{Ext}^1(π\mathcal{GP}_{(ω,\mathcal{B},1)},ν) = 0$, and (3) every object in $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ admits a $\mathrm{Hom}(-,ν)$-acyclic $ν$-coresolution. These conditions are their duals are part of what we called ``Setup 1''. Moreover, if we replace $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ by the class $\mathcal{GP}_{(\mathcal{A,B})}$ of $(\mathcal{A,B})$-Gorenstein projective objects, the resulting inequality is claimed to be true under a set of conditions named ``Setup 2''. The proof we gave for the claims $\mathrm{Gid}_{(\mathcal{Z,W})}(π\mathcal{GP}_{(ω,\mathcal{B},1)}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(ω)$ and $\mathrm{Gid}_{(\mathcal{Z,W})}(\mathcal{GP}_{(\mathcal{A,B})}) \leq \mathrm{Gid}_{(\mathcal{Z,W})}(ω)$ is incorrect, and the purpose of this note is to exhibit a corrected proof of the first inequality, under the additional assumption that every object in $π\mathcal{GP}_{(ω,\mathcal{B},1)}$ has finite injective dimension relative to $\mathcal{Z}$. Setup 2 is no longer required, and as a result the second inequality was removed. We also fix those results in §\ 8 of \cite{HMP2023m} affected by Lemma 8.1, and comment some applications and examples.

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arXiv2023

Categories of quiver representations and relative cotorsion pairs

We study the category $\operatorname{Rep}(Q,\mathcal{C})$ of representations of a quiver $Q$ with values in an abelian category $\mathcal{C}$. For this purpose we introduce the mesh and the cone-shape cardinal numbers associated to the quiver $Q$ and we use them to impose conditions on $\mathcal{C}$ that allow us to prove interesting homological properties of $\operatorname{Rep} (Q,\mathcal{C})$ that can be constructed from $\mathcal{C}.$ For example, we compute the global dimension of $\operatorname{Rep} (Q,\mathcal{C})$ in terms of the global one of $\mathcal{C}.$ We also review a result of H. Holm and P. Jørgensen which states that (under certain conditions on $\mathcal{C}$) every hereditary complete cotorsion pair $(\mathcal{A},\mathcal{B})$ in $\mathcal{C}$ induces the hereditary complete cotorsion pairs $(\operatorname{Rep}(Q,\mathcal{A}),\operatorname{Rep}(Q,\mathcal{A})^{\bot_{1}})$ and $(^{\bot_{1}}Ψ(\mathcal{B}),Ψ(\mathcal{B}))$ in $\operatorname{Rep}(Q,\mathcal{C})$, and then we obtain a strengthened version of this and others related results. Finally, we will apply the above developed theory to study the following full abelian subcategories of $\operatorname{Rep}(Q,\mathcal{C}),$ finite-support, finite-bottom-support and finite-top-support representations. We show that the above mentioned cotorsion pairs in $\operatorname{Rep}(Q,\mathcal{C})$ can be restricted nicely on the aforementioned subcategories and under mild conditions we also get hereditary complete cotorsion pairs.

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arXiv2023

Quotient categories with exact structure from $(n+2)$-rigid subcategories in extriangulated categories

In this work we introduce the notion of higher $\mathbb{E}$-extension groups for an extriangulated category $\mathcal{C}$ and study the quotients $\mathcal{X}_{n+1}^{\vee}/[\mathcal{X}]$ and $\mathcal{X}_{n+1}^{\wedge}/[\mathcal{X}]$ when $\mathcal{X}$ is an $(n+2)$-rigid subcategory of $\mathcal{C}$. We also prove (under mild conditions) that each one is equivalent to a suitable subcategory of the category of functors of the stable category of $\mathcal{X}_{n}^{\vee}$ and the co-stable category of $\mathcal{X}_{n}^{\wedge}$, respectively. Moreover, it can be induced an exact structure through these equivalences and we analyze when such quotients are weakly idempotent complete, Krull-Schmidt or abelian. The above discussion is also considered in the particular case of an $(n+2)$-cluster tilting subcategory of $\mathcal{C}$ since in this case we know that $\mathcal{X}_{n+1}^{\vee}=\mathcal{C}=\mathcal{X}_{n+1}^{\wedge}.$ Finally, by considering the category of conflations of a exact category, we show that it is possible to get an abelian category from these quotients.

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arXiv2022

$n$-term silting complexes in $K^b(proj(Λ))$

Let $Λ$ be an Artin algebra and $K^b(proj(Λ))$ be the triangulated category of bounded co-chain complexes in $proj(Λ).$ It is well known that two-terms silting complexes in $K^b(proj(Λ))$ are described by the $τ$-tilting theory. The aim of this paper is to give a characterization of certain $n$-term silting complexes in $K^b(proj(Λ))$ which are induced by $Λ$-modules. In order to do that, we introduce the notions of $τ_n$-rigid, $τ_n$-tilting and $τ_{n,m}$-tilting $Λ$-modules. The latter is both a generalization of $τ$-tilting and tilting in $mod(Λ).$ It is also stated and proved some variant, for $τ_n$-tilting modules, of the well known Bazzoni's characterization for tilting modules. We give some connections between $n$-terms presilting complexes in $K^b(proj(Λ))$ and $τ_n$-rigid $Λ$-modules. Moreover, a characterization is given to know when a $τ_n$-tilting $Λ$-module is $n$-tilting. We also study more deeply the properties of the $τ_{n,m}$-tilting $Λ$-modules and their connections of being $m$-tilting in some quotient algebras. We apply the developed $τ_{n,m}$-tilting theory to the finitistic dimension of $Λ.$ Finally, at the end of the paper we discuss and state some open questions (conjectures) that we consider crucial for the future develop of the $τ_{n,m}$-tilting theory.

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arXiv2022

Cut notions in extriangulated categories related to Auslander-Buchweitz theory and cotorsion theory

In this work we introduce notions in Auslander-Buchweitz theory and cotorsion theory in extriangulated categories which extend the given ones for abelian categories. Although these notions have been already developed for extriangulated categories in remarkable works, in this paper, we tackle them in a relative sense by considering subcategories of objects. This approach not only covers the existing theory given on abelian, exact and triangulated categories, but it also shows how to get similar results with an appropriate treatment of local properties.

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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.

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arXiv2021

Stratifying systems and $g$-vectors

In this paper we study the Cartan matrix associated to the Ext-projective stratifying system induced by a basic and $τ$-rigid object $M$ in mod$(A)$ by means of the $g$-vectors of the indecomposable direct summands of $M$. In particular we show that the Cartan group of a stratifying system associated to a $τ$-rigid module can be calculated directly using these vectors. Moreover we characterise the stratifying systems coming from $τ$-rigid modules that have a diagonal Cartan matrix.

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arXiv2021

Relative Torsion Classes, relative tilting and relative silting modules

Let $Λ$ be an Artin algebra. In 2014, T. Adachi, O. Iyama and I. Reiten proved that the torsion funtorially finite classes in $\mathrm{mod}\,(Λ)$ can be described by the $τ$-tilting theory. The aim of this paper is to introduce the notion of $F$-torsion class in $\mathrm{mod}\,(Λ)$, where $F$ is an additive subfunctor of $\mathrm{Ext}^1_Λ,$ and to characterize when these clases are preenveloping and $F$-preenveloping. In order to do that, we introduce the notion of $F$-presilting $Λ$-module. The latter is both a generalization of $τ$-rigid and $F$-tilting in $\mathrm{mod}(Λ).$

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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.

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arXiv2020

Stratifying systems through $τ$-tilting theory

In this paper we first show that every non-zero $τ$-rigid $A$-module induces at least one stratifying system in the module category of $A$. Moreover, we show that each of these stratifying systems can be seen as a signed $τ$-exceptional sequence.

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arXiv2020

Generalised Igusa-Todorov functions and Lat-Igusa-Todorov algebras

In this paper we study a generalisation of the Igusa-Todorov functions which gives rise to a vast class of algebras satisfying the finitistic dimension conjecture. This class of algebras is called Lat-Igusa-Todorov and includes, among others, the Igusa-Todorov algebras (defined by J. Wei) and the self-injective algebras which in general are not Igusa-Todorov algebras. Finally, some applications of the developed theory are given in order to relate the different homological dimensions which have been discussed through the paper.

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arXiv2019

Relative Gorenstein objects in abelian categories

Let $\mathcal{A}$ be an abelian category. For a pair $(\mathcal{X},\mathcal{Y}$ of classes of objects in $\mathcal{A},$ we define the weak and the $(\mathcal{X},\mathcal{Y})$-Gorenstein relative projective objects in $\mathcal{A}$. We point out that such objects generalize the usual Gorenstein projective objects and others generalizations appearing in the literature as Ding-projective, Ding-injective, $\mathcal{X}$-Gorenstein projective, Gorenstein AC-projective and $G_C$-projective modules and Cohen-Macaulay objects in abelian categories. We show that the principal results on Gorenstein projective modules remains true for the weak and the $(\mathcal{X},\mathcal{Y}$-Gorenstein relative objects. Furthermore, by using Auslander-Buchweitz approximation theory, a relative version of Gorenstein homological dimension is developed. Finally, we introduce the notion of $\mathcal{W}$-cotilting pair in the abelian category $\mathcal{A}$, which is very strong connected with the cotorsion pairs related with relative Gorenstein objects in $\mathcal{A}$. It is worth mentioning that the $\mathcal{W}$-cotilting pairs generalize the notion of cotilting objects in the sense of L. Angeleri Hügel and F. Coelho.

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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.

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arXiv2018

Cokernels of the Cartan Matrix and Stratifying Systems

We study the cokernel of the application given by the Cartan Matrix $C_Λ$ of a finite dimensional $k$-algebra $Λ.$ This produces a finitely generated abelian group, the Cartan group $G_Λ,$ which is invariant under derived equivalences. We are interested in the case when $G_Λ$ is finite. For a standardly stratified algebra, it is shown that this group is always finite and some interesting connections with the standard modules are found. As a consequence, it is got that $G_Λ$ can be seen as a measure of how far is a standardly stratified algebra $Λ$ to be quasi-hereditary. Finally, it is also shown that any finite abelian group can be realized as the Cartan group of some standardly stratified algebra.

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