Erdmann, Karin Holm, Thorsten Iyama, Osamu Schröer, Jan
Published in
Advances in Mathematics

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.

Green, Edward L. Hartman, Gregory Marcos, Eduardo N. Solberg, Øyvind
Published in
Archiv der Mathematik

In this paper we show that if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda = \mathop \coprod \limits_{i \geqq 0} \Lambda _i $$\end{document} is a Koszul alge...

Jue, Brian
Published in
Open Mathematics

Let $$\mathbb{K}$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda = {{\mathbb{K}\Gamma } \mathord{\left/ {\vphantom {{\mathbb{K}\Gamma } I}} \right. \kern-\nulldelimiterspace} I}$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path...

Herzog, Ivo
Published in
Advances in Mathematics

A morphism f : M → N of left R-modules is a phantom morphism if for any morphism g : A → M , with A finitely presented, the composition fg factors through a projective module. Equivalently, Tor 1 ( X , f ) = 0 for every right R-module X. It is proved that every R-module possesses a phantom cover, whose kernel is pure injective. If mod ̲ - R is the ...

Huang, Chong-hui Huang, Zhao-yong
Published in
Science in China Series A: Mathematics

In this paper, we first introduce the notion of generalized κ-syzygy modules, and then give an equivalent characterization that the class of generalized κ-syzygy modules coincides with that of ω-κ-torsionfree modules. We further study the extension closure of the category consisting of generalized κ-syzygy modules. Some known results are obtained a...

Lafont, Yves
Published in
Applied Categorical Structures

We present various results of the last 20 years converging towards a homotopical theory of computation. This new theory is based on two crucial notions: polygraphs (introduced by Albert Burroni) and polygraphic resolutions (introduced by François Métayer). There are two motivations for such a theory: Providing invariants of computational systems to...

Mao, Li-xin Ding, Nan-qing
Published in
Science in China Series A: Mathematics

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom...

Happel, Dieter Zacharia, Dan
Published in
Mathematische Zeitschrift

Let Λ be a finite dimensional algebra over a field k. We will show here that Λ is piecewise hereditary if and only if its strong global dimension is finite.

Lü, Jia-feng He, Ji-wei Lu, Di-ming
Published in
Science in China Series A: Mathematics

It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are, in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in term...

Wei, Jiaqun
Published in
Advances in Mathematics

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radi...