Diveris, Kosmas
Published in
Algebras and Representation Theory

We introduce the finitistic extension degree of a ring and investigate rings for which it is finite. The Auslander–Reiten Conjecture is proved for rings of finite finitistic extension degree and these rings are also shown to have finite finitistic dimension. We apply these results to better understand a generalized version of the Auslander–Reiten C...

Liu, Zeng Feng Huang, Zhao Yong
Published in
Acta Mathematica Sinica, English Series

Recently Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case ov...

Zhu, Xiaosheng
Published in
Algebras and Representation Theory

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{A}$\end{document} be an abelian category. A subcategory \documentclass[12pt]{minimal} \usepackage{amsmath} \...

Huang, ChongHui Huang, ZhaoYong
Published in
Science China Mathematics

Let R be a left and right Noetherian ring and n, k be any non-negative integers. R is said to satisfy the Auslander-type condition Gn(k) if the right flat dimension of the (i + 1)-th term in a minimal injective resolution of RR is at most i + k for any 0 ⩽ i ⩽ n − 1. In this paper, we prove that R is Gn(k) if and only if so is a lower triangular ma...

Zhou, Xiao-Yan
Published in
Acta Mathematica Hungarica

Let H be a weak Hopf algebra, A a right weak H-comodule algebra and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. We prove that A/B is a separable extension if H is semisimple. Using this, we show that the global dimension and weak dimension of A are less than those of B. As an application, we obta...

Chen, Yuan
Published in
Frontiers of Mathematics in China

In this paper, we consider the module-relative-Hochschild homology and cohomology of tensor products of algebras and relate them to those of the factor algebras. Moreover, we show that the tensor product is formally smooth if and only if one of its factor algebras is formally smooth and the other is separable.

Zhang, XiaoJin
Published in
Science China Mathematics

We build a connection between iterated tilted algebras with trivial cluster tilting subcategories and tilted algebras of finite type. Moreover, all tilted algebras with cluster tilting subcategories are determined in terms of quivers. As a result, we draw the quivers of Auslander’s 1-Gorenstein algebras with global dimension 2 admitting trivial clu...

Liu, Ling Shen, Bing-liang Wang, Shuan-hong
Published in
Algebras and Representation Theory

Let H be a weak Hopf algebra in the sense of Böhm et al. (J Algebra 221:385–438, 1999) measuring an algebra A. Let A#σH be a weak crossed product with σ invertible. Then in this paper we first give some conditions for A#σH to be a weak Hopf algebra. Next the spectral sequence for Ext will be constructed which yields an estimate for the global dimen...

D’Este, Gabriella
Published in
ANNALI DELL'UNIVERSITA' DI FERRARA

We construct rather short partial tilting complexes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot T}$$\end{document} such that quite different indecomposable ri...

Wei, Jiaqun
Published in
Mathematische Zeitschrift

We studied the properties of tilting complexes and proved that derived equivalences preserve the validity of the Auslander–Reiten conjecture.