Related papers: Hereditary cotorsion pairs on extriangulated subca…

Balanced pairs appear naturally in the realm of Relative Homological Algebra associated to the balance of right derived functors of the $\mathsf{Hom}$ functor. A natural source to get such pairs is by means of cotorsion triplets. In this…

Let $A$, $B$ be two rings and $T=\left(\begin{smallmatrix} A & M 0 & B \end{smallmatrix}\right)$ with $M$ an $A$-$B$-bimodule. Given two complete hereditary cotorsion pairs $(\mathcal{A}_{A},\mathcal{B}_{A})$ and…

We introduce the notion of Gabriel filter for a preadditive category C and we show that there is a bijective correspondence between Gabriel filters of C and hereditary torsion theories in the category of additive functors (C,Ab), obtaining…

In this work we introduce notions in Auslander-Buchweitz theory and cotorsion theory in extriangulated categories which extend the given ones for abelian categories. Although these notions have been already developed for extriangulated…

Let $\mathscr{C}$ be an extriangulated category with enough projectives and injectives. We give the definitions of Wakamatsu-tilting subcategories and Wakamatsu-cotilting subcategories of $\mathscr{C}$ and show that they coincide with each…

Let $(\mathcal B,\mathbb{E},\mathfrak{s})$ be an extriangulated category and $\mathcal S$ be an extension closed subcategory of $\mathcal B$. In this article, we prove that the Gabriel-Zisman localization $\mathcal B/\mathcal S$ can be…

In an abelian category $\mathscr{A}$, we can generate torsion pairs from tilting objects of projective dimension $\leq 1$. However, when we look at tilting objects of projective dimension $2$, there is no longer a natural choice of an…

We define the notion of an infinitely generated tilting object of infinite homological dimension in an abelian category. A one-to-one correspondence between $\infty$-tilting objects in complete, cocomplete abelian categories with an…

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and…

We give a full classification of representation types of the subcategories of representations of an $m \times n$ rectangular grid with monomorphisms (dually, epimorphisms) in one or both directions, which appear naturally in the context of…

We define tilting subcategories in arbitrary exact categories to archieve the following. Firstly: Unify existing definitions of tilting subcategories to arbitrary exact categories. Discuss standard results for tilting subcategories:…

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…

In this paper, we introduce and study relative phantom morphisms in extriangulated categories defined by Nakaoka and Palu. Then using their properties, we show that if $(\C,\E,\s)$ is an extriangulated category with enough injective objects…

Given two (hereditary) complete cotorsion pairs $(\mathcal{X}_1,\mathcal{Y}_1)$ and $(\mathcal{X}_2,\mathcal{Y}_2)$ in an exact category with $\mathcal{X}_1\subseteq \mathcal{Y}_2$, we prove that $\left({\rm Smd}\langle…

Recently, Wang, Wei and Zhang define the recollement of extriangulated categories, which is a generalization of both recollement of abelian categories and recollement of triangulated categories. For a recollement $(\mathcal A ,\mathcal…

Using the concept of prime submodule introduced by Raggi et.al. we extend the notion of reduced rank to the module-theoretic context of $\sigma[M]$. We study the quotient category of $\sigma[M]$ modulo the hereditary torsion theory…

For the module category of an Artin algebra, we generalize the notion of torsion pairs to ideal torsion pairs. Instead of full subcategories of modules, ideals of morphisms of the ambient category are considered. We characterize the…

We introduce the concept of a pseudo-cluster tilting subcategory from the viewpoint of the fact that the quotient of an exact category by a cluster tilting subcategory is an abelian category. We prove that the quotients in the case of…

We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…

In this note, the notion of cotorsion classes is introduced into the higher homological algebra. Our results motivate the definition, showing that this notion of $n$-cotorsion classes satisfies usual properties one could expect. In…