Hu, Yonggang Yao, Hailou
Published in
Frontiers of Mathematics in China

Let E be a proper class of triangles in a triangulated category C; and let (A, B, C) be a recollement of triangulated categories. Based on Beligiannis's work, we prove that A and C have enough E-projective objects whenever B does. Moreover, in this paper, we give the bounds for the E-global dimension of B in a recollement (A, B, C) by controlling t...

Zhang, Chao
Published in
Science China Mathematics

Bongartz (2013) and Ringel (2011) proved that there is no gaps in the sequence of lengths of indecomposable modules for the finite-dimensional algebras over algebraically closed fields. The present paper mainly studies this "no gaps" theorem as to cohomological length for the bounded derived category Db(A) of a gentle algebra A: if there is an inde...

Xing, Jianmin Zhao, Tiwei Li, Yunxia Hu, Jiangsheng
Published in
Bulletin of the Iranian Mathematical Society

In this paper, we introduce and study a notion of Gorenstein AC-injective dimension for complexes of left modules over associative rings. We show first that the class of complexes with finite Gorenstein AC-injective dimension is exactly the class of complexes admitting a complete AC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasy...

Becerril, Víctor Mendoza, Octavio Pérez, Marco A. Santiago, Valente
Published in
Journal of Homotopy and Related Structures

We revisit Auslander–Buchweitz approximation theory and find some relations with cotorsion pairs and model category structures. From the notion of relative generators, we introduce the concept of left Frobenius pairs (X,ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{a...

Huang, Chonghui Zheng, Lijing
Published in
Open Mathematics

In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of ExtΛ1(S,Λ) $\begin{array}{} \text{Ext}_{{\it\Lambda}}^{1}(S,\ {\it\Lambda}) \end{array} $ is equal to 1 for any simple Λ-module S with gr S = 1. As a result, we give s...

Chan, Aaron Marczinzik, René
Published in
Algebras and Representation Theory

Gendo-symmetric algebras were introduced by Fang and Koenig (Trans. Amer. Math. Soc., 7:5037–5055, 2016) as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendo-symmetric biserial algebras. ...

Li, Ruixin Liu, Miantao Gao, Nan
Published in
Frontiers of Mathematics in China

We introduce the Gorenstein algebraic K-theory space and the Gorenstein algebraic K-group of a ring, and show the relation with the classical algebraic K-theory space, and also show the ‘resolution theorem’ in this context due to Quillen. We characterize the Gorenstein algebraic K-groups by two different algebraic K-groups and by the idempotent com...

Gao, Nan Psaroudakis, Chrysostomos
Published in
Applied Categorical Structures

In this paper we characterize when a recollement of compactly generated triangulated categories admits a ladder of some height going either upwards or downwards. As an application, we show that the derived category of the preprojective algebra of Dynkin type An\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfo...

Suarez, Pamela
Published in
Algebras and Representation Theory

Let A be the one point extension of an algebra B by a projective B-module. We prove that the extension of a given support τ-tilting B-module is a support τ-tilting A-module; and, conversely, the restriction of a given support τ-tilting A-module is a support τ-tilting B-module. Moreover, we prove that there exists a full embedding of quivers between...

Zhang, Chun Xia Liang, Li
Published in
Acta Mathematica Sinica, English Series

For two classes of right R-modules W, X such that P ⊂ W ⊂ X, where P is the class of projective right R-modules, we show that there is an Avramov–Martsinkovsky type exact sequence with generalized Tate homology functor Tor^X.W\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepacka...