arXiv subjects

Sergio Estrada

Publications and source records attributed to Sergio Estrada.

At least 19 recordsLinked to original sources

arXiv2026

Balanced pairs of Cartan-Eilenberg complexes over virtually Gorenstein rings

Let R be a ring and let Ch(R) be the category of complexes of left R-modules. We consider the C-E (Cartan-Eilenberg) exact structure on Ch(R) and show that it is an efficient exact category. We prove that if R is a left virtually Gorenstein ring, the pair (C-E(GProj),C-E(GInj)) is a C-E-admissible balanced pair. Under suitable additional hypotheses, the converse holds. We also consider the C-E version of Tate cohomology and establish C-E analogues of the Avramov-Martsinkovsky exact sequences and balance results for both Ext and Tor.

math.RA↗

arXiv2026

Accessibility and Gorenstein injective envelopes

Let $\mathcal{G}$ be a Grothendieck category. We prove completeness of the Gorenstein injective cotorsion pair whenever $\mathcal{G}$ admits a set of Tate trivial generators, and show that having such generators is necessary for completeness. In this case it must be a perfect cotorsion pair, cogenerated by a set, and equivalent to an injective abelian model structure on $\mathcal{G}$. Examples include Grothendieck categories (possibly without enough projectives) that admit a generating set consisting of objects of finite projective dimension, such as the category of quasi-coherent sheaves on a quasi-compact and semi-separated scheme. More generally, for a given set $\mathcal{S}$, we characterize the completeness of the Gorenstein $\mathcal{B}$-injective cotorsion pair, where $\mathcal{B} = \mathcal{S}^\perp$, in terms of the existence of a set of $\mathcal{B}$-Tate trivial generators for $\mathcal{G}$. The key ingredient to our proof is the fact that any class of the form $\mathcal{B} :=\mathcal{S}^\perp$ is an accessibly embedded, accessible subcategory of $\mathcal{G}$. The general approach allows for further applications such as the existence of Ding injective envelopes and other relative Gorenstein injective envelopes without imposing additional assumptions on $\mathcal{G}$.

math.CT↗

arXiv2026

Balanced pairs, virtually Gorenstein rings, and cotorsion torsion triples

For any ring $R$, we investigate balanced pairs of classes of modules and their relations to cotorsion triples. We characterize the case when a balanced pair generates a tilting cotorsion pair, and dually, when it cogenerates a cotilting cotorsion pair. If $R$ is right noetherian, we prove that the pair consisting of Gorenstein projective modules and Gorenstein injective modules is balanced if and only if $R$ is right virtually Gorenstein. In [4], cotorsion torsion triples in abelian categories were employed in the representation theory of rectangular grids occurring in persistent homology theory. For module categories, we use infinite dimensional tilting theory to completely classify all cotorsion torsion triples by means of $1$-resolving subcategories of $\rfmod R$, and to give an explicit 1-1 correspondence between the formally dual notions of cotorsion torsion triples of right $R$-modules and torsion cotorsion triples of left $R$-modules. This correspondence is bijective in case the underlying ring $R$ is left noetherian, but not in general.

math.RT↗

arXiv2026

Weakly Sigma-cotorsion rings

We study the class of rings $R$ for which every direct sum of injective $R$-modules is cotorsion. We call them weakly $\Sigma$-cotorsion rings. The defining property might be seen as the dual of Chase's characterization of coherence in terms of the flatness of every direct product of projective $R$-modules. More generally, we study rings over which direct sums of injective modules have finite cotorsion dimension and call them weakly $n$-$\Sigma$-cotorsion rings, as well as rings over which direct sums of cotorsion modules have finite cotorsion dimension (called $n$-$\Sigma$-cotorsion rings). In the process, we obtain new characterizations of $n$-perfect rings and extend previous results by Guil Asensio and Herzog, and by \v{S}aroch and \v{S}\v{t}ov\'i\v{c}ek.

math.RA↗

arXiv2025

Acyclic complexes and regular rings

A 2009 paper by Iacob and Iyengar characterizes noetherian regular rings in terms of properties of complexes of projective modules, flat modules, and injective modules. We show that the relevant properties of such complexes are equivalent without reference to regularity of the ring and that they characterize coherent regular rings and von Neumann regular rings.

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

Transfer of homological objects in exact categories via adjoint triples. Applications to functor categories

For a given family $\{(\mathrm{q}_i, \mathrm{t}_i, \mathrm{p_i} )\}_{i \in I}$ of adjoint triples between exact categories $\mathcal{C}$ or $\mathcal{D}$, we show that any cotorsion pair in $\mathcal{C}$ and $\mathcal{D}$ yield two canonical cotorsion pairs providing a concrete description of objects without using any injectives/projectives object hypothesis. We firstly apply this result for the evaluation functor on the functor category $\operatorname{Add}(\mathcal{A}, R \mbox{-Mod})$ equipped with an exact structure $\mathcal{E}$. Under mild conditions on $\mathcal{A}$, we introduce the stalk functor at any object of $\mathcal{A}$, and subsequently, we investigate cotorsion pairs induced by stalk functors. Finally, we use them to present an intrinsic characterization of projective/injective objects in $(\mbox{Add}(\mathcal{A}, R\mbox{-Mod}); \mathcal{E})$.

math.CT↗

arXiv2024

Homotopy equivalences and Grothendieck duality over rings with finite Gorenstein weak global dimension

Let $R$ be a ring with Gwgldim$(R)<\infty$. We obtain a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{GProj})\simeq \mathrm{K}(R\text{-}\mathrm{GInj})$ which restricts to a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{Proj})$ $\simeq \mathrm{K}(R\text{-}\mathrm{Inj})$. This class of rings includes, among others, (left) Gorenstein rings, Ding-Chen rings and the more general Gorenstein $n$-coherent rings ($n\in \mathbb{N}\cup \{\infty\}, n\geq 2$). As application, we establish some triangle-equivalences of Grothendieck duality over Ding-Chen rings and Gorenstein $n$-coherent rings.

math.RA↗

arXiv2023

One-sided Gorenstein rings

Distinctive characteristics of Iwanaga--Gorenstein rings are typically understood through their intrinsic symmetry. We show that several of those that pertain to the Gorenstein global dimensions carry over to the one-sided situation, even without the noetherian hypothesis. Our results yield new relations among homological invariants related to the Gorenstein property, not only Gorenstein global dimensions but also the suprema of projective/injective dimensions of injective/projective modules and finitistic dimensions.

math.RA↗

arXiv2023

K-flatness in Grothendieck categories: Application to quasi-coherent sheaves

Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of $K(\mathcal{G})$ by the K-flat complexes is always a well generated triangulated category. Under the further assumption that $\mathcal{G}$ has a set of $\otimes$-flat generators we can show more: (i) The category is in recollement with the $\otimes$-pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of $\otimes$-pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.

math.AG↗

arXiv2023

Gorenstein projective precovers and finitely presented modules

The existence of the Gorenstein projective precovers over arbitrary rings is an open question. It is known that if the ring has finite Gorenstein global dimension, then every module has a Gorenstein projective precover. We prove here a "reduction" property - we show that, over any ring, it suffices to consider finitely presented modules: if there exists a nonnegative integer $n$ such that every finitely presented module has Gorenstein projective dimension $\le n$, then the class of Gorenstein projective modules is special precovering.

math.AC↗

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

Five theorems on Gorenstein global dimensions

We expand on two existing characterizations of rings of Gorenstein (weak) global dimension zero and give two new characterizations of rings of finite Gorenstein (weak) global dimension. We also include the answer to a question of Y.~Xiang on Gorenstein weak global dimension of group rings.

math.RA↗

arXiv2022

The singularity category of an exact category applied to characterize Gorenstein schemes

We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with this Buchweitz's classic equivalence of three categories over Gorenstein local rings has been generalized to schemes, a project started by Murfet and Salarian more than ten years ago. As another application we use the framework to characterize rings of finite finitistic dimension.

math.KT↗

arXiv2021

Quillen equivalences inducing Grothendieck duality for unbounded chain complexes of sheaves

Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated scheme $\mathbb{X}$. With this we also show that the Gorenstein injective quasi-coherent sheaves are the fibrant objects of a cofibrantly generated abelian model structure for any semiseparated Noetherian scheme $\mathbb{X}$. Finally, we consider both the injective and (mock) projective approach to Tate cohomology of quasi-coherent sheaves. They agree whenever $\mathbb{X}$ is a semiseparated Gorenstein scheme of finite Krull dimension.

math.AT↗

arXiv2021

Gorenstein weak global dimension is symmetric

We study the Gorenstein weak global dimension of associative rings and its relation to the Gorenstein global dimension. In particular, we prove the conjecture that the Gorenstein weak global dimension is a left-right symmetric invariant -- just like the (absolute) weak global dimension.

math.RA↗

arXiv2020

A refinement of Gorenstein flat dimension via the flat--cotorsion theory

We introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring--the Gorenstein flat-cotorsion dimension--and prove that it, unlike the Gorenstein flat dimension, behaves as one expects of a homological dimension without extra assumptions on the ring. Crucially, we show that it coincides with the Gorenstein flat dimension for complexes where the latter is finite, and for complexes over right coherent rings--the setting where the Gorenstein flat dimension is known to behave as expected.

math.RA↗

arXiv2020

The stable category of Gorenstein flat sheaves on a noetherian scheme

For a semi-separated noetherian scheme, we show that the category of cotorsion Gorenstein flat quasi-coherent sheaves is Frobenius and a natural non-affine analogue of the category of Gorenstein projective modules over a noetherian ring. We show that this coheres perfectly with the work of Murfet and Salarian that identifies the pure derived category of F-totally acyclic complexes of flat quasi-coherent sheaves as the natural non-affine analogue of the homotopy category of totally acyclic complexes of projective modules.

math.AC↗