arXiv subjects

Manuel Saorín

Publications and source records attributed to Manuel Saorín.

At least 19 recordsLinked to original sources

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

A poisonous example to explicit resolutions of unbounded complexes

We show that various methods for explicitly building resolutions of unbounded complexes in fact fail when applied to a rather simple and explicit complex. We show that one way to rescue these methods is to assume Roos (Ab.4$^*$)-$k$ axiom, which we adapt to encompass also resolutions in the framework of relative homological algebra. In the end we discuss the existence of model structures for relative homological algebra for unbounded complex under the relative (Ab.4$^*$)-$k$ condition, and present a variety of examples where our results apply.

math.RA↗

arXiv2024

Hearts of set-generated t-structures have a set of generators

We show that if $α$ is a regular cardinal, $\mathcal{D}$ is an $α$-compactly generated triangulated category, in the sense of Neeman \cite{N}, and $τ$ is a t-structure in $\mathcal{D}$ generated by a set of $α$-compact objects, then the heart of $τ$ is a locally $α$-presentable (not necessarily Ab5) abelian category. As a consequence, in a well-generated triangulated category any t-structure generated by a set of objects has a heart with a set of generators.

math.CT↗

arXiv2023

$t$-Structures with Grothendieck hearts via functor categories

We study when the heart of a t-structure in a triangulated category $\mathcal{D}$ with coproducts is AB5 or a Grothendieck category. If $\mathcal{D}$ satisfies Brown representability, a t-structure has an AB5 heart with an injective cogenerator and coproduct-preserving associated homological functor if, and only if, the coaisle has a pure-injective t-cogenerating object. If $\mathcal{D}$ is standard well generated, such a heart is automatically a Grothendieck category. For compactly generated t-structures (in any ambient triangulated category with coproducts), we prove that the heart is a locally finitely presented Grothendieck category. We use functor categories and the proofs rely on two main ingredients. Firstly, we express the heart of any t-structure in any triangulated category as a Serre quotient of the category of finitely presented additive functors for suitable choices of subcategories of the aisle or the co-aisle that we, respectively, call t-generating or t-cogenerating subcategories. Secondly, we study coproduct-preserving homological functors from $\mathcal{D}$ to complete AB5 abelian categories with injective cogenerators and classify them, up to a so-called computational equivalence, in terms of pure-injective objects in $\mathcal{D}$. This allows us to show that any standard well generated triangulated category $\mathcal{D}$ possesses a universal such coproduct-preserving homological functor, to develop a purity theory and to prove that pure-injective objects always cogenerate t-structures in such triangulated categories.

math.CT↗

arXiv2022

Reflective and coreflective subcategories

Given any additive category $\mathcal{C}$ with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory $\mathcal{B}$ is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in $\mathcal{B}$ has a pseudocokernel in $\mathcal{C}$ that belongs to $\mathcal{B}$. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if $\mathcal{C}$ is pretriangulated with split idempotents, then a subcategory $\mathcal{B}$ is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively. By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Peña stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.

math.CT↗

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↗

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↗

arXiv2021

$t$-Structures on stable derivators and Grothendieck hearts

We prove that given any strong, stable derivator and a $t$-structure on its base triangulated category $\cal D$, the $t$-structure canonically lifts to all the (coherent) diagram categories and each incoherent diagram in the heart uniquely lifts to a coherent one. We use this to show that the $t$-structure being compactly generated implies that the coaisle is closed under directed homotopy colimit which in turns implies that the heart is an (Ab.$5$) Abelian category. If, moreover, $\cal D$ is a well generated algebraic or topological triangulated category, then the heart of any accessibly embedded (in particular, compactly generated) $t$-structure has a generator. As a consequence, it follows that the heart of any compactly generated $t$-structure of a well generated algebraic or topological triangulated category is a Grothendieck category.

math.CT↗

arXiv2021

Lifting of recollements and gluing of partial silting sets

This paper focuses on recollements and silting theory in triangulated categories. It consists of two main parts. In the first part a criterion for a recollement of triangulated subcategories to lift to a torsion torsion-free triple (TTF triple) of ambient triangulated categories with coproducts is proved. As a consequence, lifting of TTF triples is possible for recollements of stable categories of repetitive algebras or self-injective finite length algebras and recollements of bounded derived categories of separated Noetherian schemes. When, in addition, the outer subcategories in the recollement are derived categories of small linear categories the conditions from the criterion are sufficient to lift the recollement to a recollement of ambient triangulated categories up to equivalence. In the second part we use these results to study the problem of constructing silting sets in the central category of a recollement generating the t-structure glued from the silting t-structures in the outer categories. In the case of a recollement of bounded derived categories of Artin algebras we provide an explicit construction for gluing classical silting objects.

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↗

arXiv2019

Torsion pairs in categories of modules over a preadditive category

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.

math.RA↗

arXiv2019

Degenerating $0$ in Triangulated Categories

In previous work, based on work of Zwara and Yoshino, we defined and studied degenerations of objects in triangulated categories analogous to degeneration of modules. In triangulated categories it is surprising that the zero object may degenerate. We study this systematically. In particular we show that the degeneration of the zero object actually induces all other degenerations by homotopy pullback, that degeneration of $0$ is closely linked, but not equivalent, to having zero image in the Grothendieck group.

math.RT↗

arXiv2017

Contractibility of the stability manifold for silting-discrete algebras

We show that any bounded t-structure in the bounded derived category of a silting-discrete algebra is algebraic, i.e. has a length heart with finitely many simple objects. As a corollary, we obtain that the space of Bridgeland stability conditions for a silting-discrete algebra is contractible.

math.RT↗

arXiv2017

Dg algebras with enough idempotents, their dg modules and their derived categories

We develop the theory dg algebras with enough idempotents and their dg modules and show their equivalence with that of small dg categories and their dg modules. We introduce the concept of dg adjunction and show that the classical covariant tensor-Hom and contravariant Hom-Hom adjunctions of modules over associative unital algebras are extended as dg adjunctions between categories of dg bimodules. The corresponding adjunctions of the associated triangulated functors are studied, and we investigate when they are one-sided parts of bifunctors which are triangulated on both variables. We finally show that, for a dg algebra with enough idempotents, the perfect left and right derived categories are dual to each other.

math.RT↗

arXiv2017

On locally coherent hearts

We show that, under particular conditions, if a t-structure in the unbounded derived category of a locally coherent Grothendieck category restricts to the bounded derived category of its category of finitely presented objects, then its heart is itself a locally coherent Grothendieck category. Those particular conditions are always satisfied when the Grothendieck category is arbitrary and one considers the t-structure associated to a torsion pair in the category of finitely presented objects. They are also satisfied when one takes any compactly generated t-structure in the derived category of a commutative noetherian ring which restricts to the bounded derived category of finitely generated modules. As a consequence, any t-structure in this latter bounded derived category has a heart which is equivalent to the category of finitely presented objects of some locally coherent Grothendieck category.

math.CT↗

arXiv2016

Symmetry of the Definition of Degeneration in Triangulated Categories

Module structures of an algebra on a fixed finite dimensional vector space form an algebraic variety. Isomorphism classes correspond to orbits of the action of an algebraic group on this variety and a module is a degeneration of another if it belongs to the Zariski closure of the orbit. Riedtmann and Zwara gave an algebraic characterisation of this concept in terms of the existence of short exact sequences. Jensen, Su and Zimmermann, as well as independently Yoshino, studied the natural generalisation of the Riedtmann-Zwara degeneration to triangulated categories. The definition has an intrinsic non-symmetry. Suppose that we have a triangulated category in which idempotents split and either for which the endomorphism rings of all objects are artinian, or which is the category of compact objects in an algebraic compactly generated triangulated K-category. Then we show that the non-symmetry in the algebraic definition of the degeneration is inessential in the sense that the two possible choices which can be made in the definition lead to the same concept.

math.RT↗

arXiv2016

Derived equivalences induced by nonclassical tilting objects

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $Hom$ sets and arbitrary products.

math.RT↗

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↗