arXiv subjects

Manuel Cortés-Izurdiaga

Publications and source records attributed to Manuel Cortés-Izurdiaga.

13 recordsLinked to original sources

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 $Σ$-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 $n$-$Σ$-cotorsion rings, as well as rings over which direct sums of cotorsion modules have finite cotorsion dimension (called $n$-$Σ$-cotorsion rings). In the process, we obtain new characterizations of $n$-perfect rings and extend previous results by Guil Asensio and Herzog, and by Šaroch and Šťovíček.

math.RA↗

arXiv2025

Flatness in finitely accessible additive categories

Motivated by some problems proposed by Cuadra and Simson related to flat objects in finitely accessible Grothendieck categories, we study flatness in the more general setting of finitely accessible additive categories. For such category $\mathcal{A}$, we characterize when $\mathcal{A}$ is preabelian and abelian. We prove that if the class of flat objects in $\mathcal A$ is closed under pure subobjects, then every flat object is a direct union of \textit{small} flat subobjects. Finally, we characterize when $\mathcal{A}$ has enough flat and projective objects and we prove that, in this case, the class of flat objects is closed under pure subobjects.

math.CT↗

arXiv2024

Almost free modules, perfect decomposition and Enochs's conjecture

Given a module $X$ and a regular cardinal $κ$ we study various notions of $(κ,\mathrm{Add}(X))$-freeness and $(κ,\mathrm{Add}(X))$-separability. Bearing on appropriate set-theoretic assumptions, we construct a non-trivial $κ^+$-generated, $(κ^+,\mathrm{Add}(X))$-free and $(κ^+,\mathrm{Add}(X))$-separable module. Our construction allows $κ$ to be singular thus extending \cite[Theorem~4.7]{CortesGuilTorrecillas}. Bearing on similar set-theoretic assumptions, we characterize when every module $X$ has a perfect decomposition. As a subproduct we show that Enoch's conjecture for classes $\mathrm{Add}(X)$ is consistent with ZFC -- a fact first proved by Šaroch \cite{Saroch}.

math.RA↗

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↗

arXiv2023

The cotorsion pair generated by the Gorenstein projective modules and $λ$-pure-injective modules

We prove that, if $\textrm{GProj}$ is the class of all Gorenstein projective modules over a ring $R$, then $\mathfrak{GP}=(\textrm{GProj},\textrm{GProj}^\perp)$ is a cotorsion pair. Moreover, $\mathfrak{GP}$ is complete when all projective modules are $λ$-pure-injective for some infinite regular cardinal $λ$ (in particular, if $R$ is right $Σ$-pure-injective); the latter condition is shown to be consistent with the axioms of ZFC modulo the existence of strongly compact cardinals. We also thoroughly study $λ$-pure-injective modules for an arbitrary infinite regular cardinal $λ$, proving along the way that: any cosyzygy module in an injective coresolution of a $λ$-pure-injective module is $λ$-pure-injective; the cotorsion pair cogenerated by a class of $λ$-pure-injective modules is cogenerated by a set and, under an additional technical assumption, generated by a set. Finally, assuming the set-theoretic hypothesis that $0^\sharp$ does not exist, we prove that the category of right $R$-modules has enough $λ$-pure-injective objects if and only if the ring $R$ is right pure-semisimple. This, in turn, follows from a rather surprising result that $λ$-pure-injectivity amounts to pure-injectivity in the absence of $0^\sharp$.

math.RT↗

arXiv2023

Verdier quotients of homotopy categories of rings and Gorenstein-projective precovers

Let $R$ be a ring, $\textrm{Proj}$ be the class of all projective right $R$-modules, $\mathcal K$ be the full subcategory of the homotopy category $\mathbf K(\textrm{Proj})$ whose class of objects consists of all totally acyclic complexes, and $\textrm{Mor}_{\mathcal K}$ be the class of all morphisms in $\mathbf K(\textrm{Proj})$ whose cones belong to $\mathcal K$. We prove that if $\mathbf K(\textrm{Proj})$ has enough $\textrm{Mor}_{\mathcal K}$-injective objects, then the Verdier quotient $\mathbf K(\textrm{Proj})/\mathcal K$ has small Hom-sets, and this last condition implies the existence of Gorenstein-projective precovers in $\textrm{Mod-}R$ and of totally acyclic precovers in $\mathbf C(\textrm{Mod-}R)$.

math.RA↗

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

Strongly exchange rings

Two elements $a,b$ in a ring $R$ form a right coprime pair, written $\langle a,b\rangle$, if $aR+bR=R$. Right coprime pairs have shown to be quite useful in the study of left cotorsion or exchange rings. In this paper, we define the class of strongly right exchange rings in terms of descending chains of them. We show that they are semiregular and that this class of rings contains left injective, left pure-injective, left cotorsion, local and left continuous rings. This allows us to give a unified study of all these classes of rings in terms of the behaviour of descending chains of right coprime pairs.

math.RA↗

arXiv2020

Endomorphism rings via minimal morphisms

We prove that if $u:K \rightarrow M$ is a left minimal extension, then there exists an isomorphism between two subrings, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of $K$ from those of the endomorphism ring of $M$ in certain situations such us when $K$ is invariant under endomorphisms of $M,$ or when $K$ is invariant under automorphisms of $M$.

math.RA↗

arXiv2020

Ziegler Partial Morphisms in additive exact categories

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate partial morphisms with (co-)phantom morphisms and injective approximations and study the existence of such approximations in these exact categories.

math.RA↗

arXiv2017

Maximal ideals in module categories and applications

We study the existence of maximal ideals in preadditive categories defining an order $\preceq$ between objects, in such a way that if there do not exist maximal objects with respect to $\preceq$, then there is no maximal ideal in the category. In our study, it is sometimes sufficient to restrict our attention to suitable subcategories. We give an example of a category $\mathbf C_F$ of modules over a right noetherian ring $R$ in which there is a unique maximal ideal. The category $\mathbf C_F$ is related to an indecomposable injective module $F$, and the objects of $\mathbf C_F$ are the $R$-modules of finite $F$-rank.

math.RA↗

arXiv2017

The cotorsion pair generated by the class of flat Mittag-Leffler modules

Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion modules. In this note we extend this result to the class of all rings $R$ satisfying that each flat left $R$-module is filtered by totally ordered limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these rings, all flat left $R$-modules have finite projective dimension.

math.RA↗

arXiv2015

Product of flat modules and global dimension relative to $\mathcal F$-Mittag-Leffler modules

Let $R$ be any ring. We prove that all direct products of flat right $R$-modules have finite flat dimension if and only if each finitely generated left ideal of $R$ has finite projective dimension relative to the class of all $\mathcal F$-Mittag-Leffler left $R$-modules, where $\mathcal F$ is the class of all flat right $R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $\mathcal X$ is any class of left $R$-modules closed under filtrations that contains all projective modules, then $R$ has finite left global projective dimension relative to $\mathcal X$ if and only if each left ideal of $R$ has finite projective dimension relative to $\mathcal X$. This result contains, as particular cases, the well known results concerning the classical left global, weak and Gorenstein global dimensions.

math.RA↗