arXiv subjects

Carlos E. Parra

Publications and source records attributed to Carlos E. Parra.

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 $\mathcal{H}_{[\mathbf{t}_{1},Σ\mathbf{t}_{2}]}$

math.RT↗

arXiv2025

Locally finitely presented Grothendieck categories with a flat generator

A problem raised by Cuadra and Simson in 2007 asks whether any locally finitely presented Grothendieck category with enough flat objects also has enough projectives. In this paper, we start from a key observation: a locally finitely presented Grothendieck category has enough flat objects if, and only if, it has exact products. This enables several equivalent reformulations of the problem, allowing us to identify a counterexample (thus providing a negative solution to the problem), while also connecting it to a classical ring-theoretical question posed by Miller in 1975, and even to the Telescope Conjecture for compactly generated triangulated categories. Moreover, we describe several classes of Grothendieck categories where the problem can be answered affirmatively. For example, we show that a locally finitely presented Grothendieck category whose category of finitely presented objects is Krull--Schmidt has enough flats if, and only if, it is generated by a family of finitely generated projectives.

math.CT↗

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.

math.CT↗

arXiv2025

Puntajes máximos en el juego de dominó

In this work, we study the maximum scores that can be achieved in a team-based domino game. Specifically, we show that if the game ends because it is blocked, then the maximum score that can be obtained under this assumption is 107. -- En este trabajo estudiamos los puntajes máximos que se pueden obtener en una partida, por equipos, en el juego del dominó. Concretamente; nosotros mostramos que, si la partida termina porque el juego esta trancado, entonces el puntaje máximo que se puede obtener bajo este supuesto es de 107.

math.HO↗

arXiv2024

TTF classes generated by silting modules

We study the conditions under which a TTF class in a module category over a ring is silting. Using the correspondence between idempotent ideals over a ring and TTF classes in the module category, we focus on finding the necessary and sufficient conditions for $R/I$ to be a silting $R$-module, and hence for the TTF class $\mathbf{Gen}(R/I)$ to be silting, where $I$ is an idempotent two-sided ideal of $R$. In our main result, we show that $R/I$ is a silting module whenever $I$ is the trace of a projective $R$-module. Furthermore, we demonstrate that the converse holds for a broad class of rings, including semiperfect rings.

math.RA↗

arXiv2023

Universal co-Extensions of torsion abelian groups

In [16], a theory of universal extensions in abelian categories is developed; in particular, the notion of Ext-universal object is presented. In the present paper, we show that an Ab3 abelian category which is Ext-small satisfies the Ab4 condition if, and only if, each one of its objects is Ext-universal. We also give a characterization of the co-Ext-universal objects of the category of torsion abelian groups. In particular, we show that such groups are the ones admitting a decomposition $Q\oplus R$, in which $Q$ is injective and $R$ is a reduced group on which each $p$-component is bounded.

math.GR↗

arXiv2022

Locally finitely presented and coherent hearts

Starting with a Grothendieck category $\mathcal{G}$ and a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal G$, we study the local finite presentability and local coherence of the heart $\mathcal{H}_{\mathbf{t}}$ of the associated Happel-Reiten-Smalø $t$-structure in the derived category $\mathrm{Der} (\mathcal{G})$. We start by showing that, in this general setting, the torsion pair $\mathbf t$ is of finite type, if and only if it is quasi-cotilting, if and only if it is cosilting. We then proceed to study those $\mathbf t$ for which $\mathcal{H}_{\mathbf{t}}$ is locally finitely presented, obtaining a complete answer under some additional assumptions on the ground category $\mathcal{G}$, which are general enough to include all locally coherent categories, all categories of modules and several categories of quasi-coherent sheaves over schemes. The third problem that we tackle is that of local coherence. In this direction we characterize those torsion pairs $\mathbf t=(\mathcal T,\mathcal F)$ in a locally finitely presented $\mathcal G$ for which $\mathcal{H}_{\mathbf{t}}$ is locally coherent in two cases: when the tilted t-structure in $\mathcal{H}_{\mathbf{t}}$ is assumed to restrict to finitely presented objects, and when $\mathcal F$ is cogenerating. In the last part of the paper we concentrate on the case when $\mathcal G$ is a category of modules over a small preadditive category, giving several examples and obtaining very neat (new) characterizations even in this more classical setting, also underlying connections with the notion of an elementary cogenerator.

math.CT↗

arXiv2022

Local Coherence of Hearts Associated with Thomason Filtrations

Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [Hrb20,HHZ21]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [SS20], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a useful recursive characterisation in case of a finite length filtration. Low length cases involve hereditary torsion classes of finite type of the ring, and even their Happel-Reiten-Smalo hearts; in these cases, the relevant characterisations are given by few module-theoretic conditions.

math.RT↗

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↗

arXiv2021

Tilting preenvelopes and cotilting precovers in general Abelian categories

We consider an arbitrary Abelian category $\mathcal{A}$ and a subcategory $\mathcal{T}$ closed under extensions and direct summands, and characterize those $\mathcal{T}$ that are (semi-)special preenveloping in $\mathcal{A}$; as a byproduct, we generalize to this setting several classical results for categories of modules. For instance, we get that the special preenveloping subcategories $\mathcal{T}$ of $\mathcal{A}$ closed under extensions and direct summands are precisely those for which $(_{}^{\perp_1}\mathcal{T},\mathcal{T})$ is a right complete cotorsion pair, where $_{}^{\perp_1}\mathcal{T}:=\text{Ker} (\text{Ext}_{\mathcal{A}}^1(-,\mathcal{T}))$. Particular cases appear when $\mathcal{T}=V^{\perp_1}:=\text{Ker}(\text{Ext}_{\mathcal{A}}^1(V,-))$, for an $\text{Ext}^1$-universal object $V$ such that $\text{Ext}_{\mathcal{A}}^1(V,-)$ vanishes on all (existing) coproducts of copies of $V$. For many choices of $\mathcal{A}$, we show that these latter examples exhaust all the possibilities. We then show that, when $\mathcal{A}$ has an epi-generator, the (semi-)special preenveloping torsion classes $\mathcal{T}$ given by (quasi-)tilting objects are exactly those for which any object $T\in\mathcal{T}$ is the epimorphic image of some object in $_{}^{\perp_1}\mathcal{T}$ (and the subcategory $\mathcal{B}:=\text{Sub}(\mathcal{T})$ of subobjects of objects in $\mathcal{T}$ is reflective) and they are, in turn, the right constituents of complete cotorsion pairs in $\mathcal{A}$ (resp., $\mathcal{B}$). In a final section, we apply the results when $\mathcal{A}=\mathrm{mod}\text{-}R$ is the category of finitely presented modules over a right coherent ring $R$, something that gives new results and raises new questions even at the level of classical tilting theory in categories of modules.

math.RT↗

arXiv2020

The HRS tilting process and Grothendieck hearts of t-structures

In this paper we revisit the problem of determining when the heart of a t-structure is a Grothendieck category, with special attention to the case of the Happel-Reiten-Smalø (HSR) t-structure in the derived category of a Grothendieck category associated to a torsion pair in the latter. We revisit the HRS tilting process deriving from it a lot of information on the HRS t-structures which have a projective generator or an injective cogenerator, and obtain several bijections between classes of pairs $(\mathcal{A},\mathbf{t})$ consisting of an abelian category and a torsion pair in it. We use these bijections to re-prove, by different methods, a recent result of Tilting Theory and the fact that if $\mathbf{t}=(\mathcal{T},\mathcal{F})$ is a torsion pair in a Grothendieck category $\mathcal{G}$, then the heart of the associated HRS t-structure is itself a Grothendieck category if, and only if, $\mathbf{t}$ is of finite type. We survey this last problem and recent results after its solution.

math.RT↗

arXiv2018

tCG Torsion Pairs

We investigate conditions for when the $t$-structure of Happel-Reiten-Smalø associated to a torsion pair is a compactly generated $t$-structure. The concept of a $t$CG torsion pair is introduced and for any ring $R$, we prove that $\mathbf{t}=(\mathcal{T},\mathcal{F})$ is a $t$CG torsion pair in $R\text{-Mod}$ if, and only if, there exists, $\{T_λ\}$ a set of finitely presented $R$-modules in $\mathcal{T}$, such that $\mathcal{F}=\bigcap \text{Ker}({\text{Hom}}_{R}(T_λ,?))$. We also show that every $t$CG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the $t$CG torsion pairs over Noetherian rings and von Neumman regular rings.

math.CT↗

arXiv2018

Torsion pairs over $n$-Hereditary rings

We study the notions of $n$-hereditary rings and its connection to the classes of finitely $n$-presented modules, FP$_n$-injective modules, FP$_n$-flat modules and $n$-coherent rings. We give characterizations of $n$-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of $n$-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FP$_n$-injective modules and the FP$_n$-flat modules over $n$-hereditary rings. We also provide an example of a Bézout ring which is 2-hereditary, but not 1-hereditary, such that the torsion pairs over this ring are not trivial.

math.RA↗

arXiv2017

Properties of abelian categories via recollements

A recollement is a decomposition of a given category (abelian or triangulated) into two subcategories with functorial data that enables the glueing of structural information. This paper is dedicated to investigating the behaviour under glueing of some basic properties of abelian categories (well-poweredness, Grothendieck's axioms AB3, AB4 and AB5, existence of a generator) in the presence of a recollement. In particular, we observe that in a recollement of a Grothendieck abelian category the other two categories involved are also Grothendieck abelian and, more significantly, we provide an example where the converse does not hold and explore multiple sufficient conditions for it to hold.

math.CT↗

arXiv2015

Hearts of t-structures in the derived category of a commutative Noetherian ring

Let $R$ be a commutative Noetherian ring and let $\mathcal D(R)$ be its (unbounded) derived category. We show that all compactly generated t-structures in $\mathcal D(R)$ associated to a left bounded filtration by supports of Spec$(R)$ have a heart which is a Grothendieck category. Moreover, we identify all compactly generated t-structures in $\mathcal D(R)$ whose heart is a module category. As geometric consequences for a compactly generated t-structure $(\mathcal{U},\mathcal{U}^\perp [1])$ in the derived category $\mathcal{D}(\mathbb{X})$ of a Noetherian scheme $\mathbb{X}$, we get the following: 1) If the sequence $(\mathcal{U}[-n]\cap\mathcal{D}^{\leq 0}(\mathbb{X}))_{n\in\mathbb{N}}$ is stationnary, then the heart $\mathcal{H}$ is a Grothendieck category; 2) If $\mathcal{H}$ is a module category, then $\mathcal{H}$ is always equivalent to $\text{Qcoh}(\mathbb{Y})$, for some affine subscheme $\mathbb{Y}\subseteq\mathbb{X}$; 3) If $\mathbb{X}$ is connected, then: a) when $\bigcap_{k\in\mathbb{Z}}\mathcal{U}[k]=0$, the heart $\mathcal{H}$ is a module category if, and only if, the given t-structure is a translation of the canonical t-estructure in $\mathcal{D}(\mathbb{X})$; b) when $\mathbb{X}$ is irreducible, the heart $\mathcal{H}$ is a module category if, and only if, there are an affine subscheme $\mathbb{Y}\subseteq\mathbb{X}$ and an integer $m$ such that $\mathcal{U}$ consists of the complexes $U\in\mathcal{D}(\mathbb{X})$ such that the support of $H^j(U)$ is in $\mathbb{X}\setminus\mathbb{Y}$, for all $j>m$.

math.CT↗

arXiv2015

Addendum to 'Direct limits in the heart of a t-structure: The case of a torsion pair'

Let $\mathcal{G}$ be a Grothendieck category, let $\mathbf{t}=(\mathcal{T},\mathcal{F})$ be a torsion pair in $\mathcal{G}$ and let $(\mathcal{U}_\mathbf{t},\mathcal{W}_\mathbf{t})$ be the associated Happel-Reiten-Smal$ø$ t-structure in the derived category $\mathcal{D}(\mathcal{G})$. We prove that the heart of this t-structure is a Grothendieck category if, and only if, the torsionfree class $\mathcal{F}$ is closed under taking direct limits in $\mathcal{G}$.

math.CT↗

arXiv2015

On hearts which are module categories

Given a torsion pair $\mathbf{t} = (\mathcal{T} ;\mathcal{F})$ in a module category $R$-Mod we give necessary and sufficient conditions for the associated Happel-Reiten-Smalø$\text{ }$ t-structure in $\mathcal{D}(R)$ to have a heart $\mathcal{H}_{\mathbf{t}}$ which is a module category. We also study when such a pair is given by a 2-term complex of projective modules in the way described by Hoshino-Kato-Miyachi ([HKM]). Among other consequences, we completely identify the hereditary torsion pairs $\mathbf{t}$ for which $\mathcal{H}_{\mathbf{t}}$ is a module category in the following cases: i) when $\mathbf{t}$ is the left constituent of a TTF triple, showing that $\mathbf{t}$ need not be HKM; ii) when $\mathbf{t}$ is faithful; iii) when $\mathbf{t}$ is arbitrary and the ring $R$ is either commutative, semi-hereditary, local, perfect or Artinian. We also give a systematic way of constructing non-tilting torsion pairs for which the heart is a module category generated by a stalk complex at zero

math.RT↗

arXiv2014

Hearts of t-structures which are Grothendieck or module categories

This thesis deals with the general problem of determining when the heart $\mathcal{H}$ of a t-structure in a triangulated category $\mathcal{D}$ is a Grothendieck or a module category. As preliminaries, we study Grothendieck conditions AB3-AB5 for $\mathcal{H}$ in a very general setting. We then concentrate on two familiar examples of smashing t-structures. First, we consider that $\mathcal{D}=\mathcal{D}(\mathcal{G})$ is the (unbounded) derived category of a Grothendieck category $\mathcal{G}$ and that the t-structure is the one associated to a torsion pair $\mathbf{t}=(\mathcal{T},\mathcal{F})$ in $\mathcal{G}$, usually known as Happel-Reiten-Smal$\emptyset$ t-structure. In the second example studied, we assume that $\mathcal{D}=\mathcal{D}(R)$ is the derived category of a commutative Noetherian ring $R$ and that the t-structure is compactly generated. On what concern the Happel-Reiten-Smal$\emptyset$ example, we show that if $\mathcal{H}=\mathcal{H}_\mathbf{t}$ is AB5, then $\mathcal{F}$ is closed under taking direct limits in $\mathcal{G}$. Moreover, the converse is true, even implying that $\mathcal{H}_\mathbf{t}$ is a Grothendieck category, for a wide class of torsion pairs in $\mathcal{G}$ which includes the hereditary, tilting and cotilting ones. When $\mathcal{G}=R-\text{Mod}$ is a module category, we are able to identify the hereditary torsion pairs $\mathbf{t}$ in $R-\text{Mod}$ for which $\mathcal{H}_\mathbf{t}$ is a module category. When $R$ is a commutative noetherian ring, we show that all compactly generated t-structures in $\mathcal{D}(R)$ whose associated filtration by supports is left bounded have a heart $\mathcal{H}$ which is a Grothendieck category. This is used to identify all compactly generated t-structures in $\mathcal{D}(R)$ whose heart is a module category.

math.CT↗