Tilting pairs in extriangulated categories

Extriangulated categories were introduced by Nakaoka and Palu to give a unification of properties in exact categories and extension-closed subcategories of triangulated categories. A notion of tilting pairs in an extriangulated category is introduced in this paper. We give a Bazzoni characterization of tilting pairs in this setting. We also obtain the Auslander–Reiten correspondence of tilting pairs which classifies finite $\mathcal {C}$ -tilting subcategories for a certain self-orthogonal subcategory $\mathcal {C}$ with some assumptions. This generalizes the known results given by Wei and Xi for the categories of finitely generated modules over Artin algebras, thereby providing new insights in exact and triangulated categories.

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.

τ-tilting modules over triangular matrix artin algebras

Let [Formula: see text] and [Formula: see text] be artin algebras and [Formula: see text] the triangular matrix algebra with [Formula: see text] a finitely generated ([Formula: see text])-bimodule. We construct support [Formula: see text]-tilting modules and ([Formula: see text]-)tilting modules in [Formula: see text] from that in [Formula: see text] and [Formula: see text], and give the converse constructions under some condition.

Resolving resolution dimension of recollements of abelian categories

In this paper, we investigate the resolving resolution dimension with respect to the recollements of abelian categories. Let [Formula: see text] be a recollement of abelian categories such that [Formula: see text] and [Formula: see text] have enough projective objects and let [Formula: see text], [Formula: see text], [Formula: see text] be resolving subcategories of [Formula: see text], [Formula: see text] and [Formula: see text], respectively. We establish some upper and lower bounds of [Formula: see text]-resolution dimension of [Formula: see text] in terms of the [Formula: see text]-resolution dimension of [Formula: see text] and [Formula: see text]-resolution dimension of [Formula: see text]. Based on these upper and lower bounds, we study the Gorensteinness of abelian categories involved in [Formula: see text]. Under some suitable assumptions, we show that if [Formula: see text] and [Formula: see text] are Gorenstein, then [Formula: see text] is Gorenstein. As applications, we apply our results to ring theory and the triangular matrix artin algebras, we study the quasi-Frobenius and Gorenstein hereditary properties of the ring [Formula: see text] and [Formula: see text], where [Formula: see text] is an idempotent element of [Formula: see text]. We also investigate Gorensteinness of the triangular matrix artin algebras, some known results are obtained as corollaries. At the end of this paper, we give two examples to illustrate our results.

A characterization of right 4-Nakayama artin algebras

In this paper, we study the category of finitely generated modules over a class of right [Formula: see text]-Nakayama artin algebras. This class of algebras appear naturally in the study of representation-finite artin algebras. First, we give a characterization of right [Formula: see text]-Nakayama artin algebras. Then, we classify finitely generated indecomposable right modules over right [Formula: see text]-Nakayama artin algebras. We also compute almost split sequences for the class of right [Formula: see text]-Nakayama artin algebras.

Gabriel–Roiter Measure, Representation Dimension and Rejective Chains

The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu$, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

Integer sequences arising from Auslander–Reiten quivers of some hereditary artin algebras

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type.

Simple reflexive modules over Artin algebras

Let [Formula: see text] be an Artin algebra. It is well known that [Formula: see text] is selfinjective if and only if every finitely generated [Formula: see text]-module is reflexive. In this paper, we pose and motivate the question whether an algebra [Formula: see text] is selfinjective if and only if every simple module is reflexive. We give a positive answer to this question for large classes of algebras which include for example all Gorenstein algebras and all QF-3 algebras.