arXiv subjects

Alejandro Argudín Monroy

Publications and source records attributed to Alejandro Argudín Monroy.

3 recordsLinked to original sources

arXiv2024

Relative tilting theory in abelian categories I: Auslander-Buchweitz-Reiten approximations theory in subcategories and cotorsion pairs

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible generalization of a part of the Auslander-Buchweitz approximation theory that is useful for developing $n$-$\mathcal{X}$-tilting theory in [4]. With this goal, new concepts as $\mathcal{X}$-complete and $\mathcal{X}$-hereditary pairs are introduced as a generalization of complete and hereditary cotorsion pairs. These pairs appear in a natural way in the study of the category of representations of a quiver in an abelian category [5]. Our main results will include an existence theorem for relative approximations, among other results related with closure properties of relative (co)resolution classes and relative homological dimensions which are essential in the development of $n$-$\mathcal{X}$-tilting theory in [4].

math.RT↗

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.

math.RT↗

arXiv2020

The Yoneda Ext and arbitrary coproducts in abelian categories

There are well known identities that involve the Ext bifunctor, coproducts, and products in Ab4 and Ab4* abelian categories with enough projectives and enough injectives. Namely, for every such category $\mathcal{A}$, the isomorphisms $\operatorname{Ext}^n (\bigoplus_{i\in I}A_{i},X) \cong \prod_{i\in I} \operatorname{Ext}^n(A_{i},X)$ and $\operatorname{Ext}^n (X,\prod_{i\in I}A_{i}) \cong \prod_{i\in I}\operatorname{Ext}^n (X,A_{i})$ always exist. The goal of this paper is to show similar isomorphisms for the Yoneda Ext in Ab4 and Ab4* abelian categories with not necessarily enough projectives nor injectives. The desired isomorphisms are constructed explicitely by using limits and colimits.

math.CT↗