Strictly Stratified Algebras Revisited

Abstract

In [1], Ágoston, Dlab, and Lukács introduced the notion of strictly stratified algebras. These algebras are stratified in the sense of Cline, Parshall, and Scott (see [5]) and contain the well-known class of standardly stratified algebras. The latter is well described from a homological point of view. Indeed, many homological conjectures such as the finitistic dimension conjectures (see [2]), the Cartan determinant conjecture (see [10]) and the strong no loop conjecture (see [9]) hold true for standardly stratified algebras. In this article, we shall try to extend these results to strictly stratified algebras. The key idea is to show that the filtration condition of a strictly stratified algebra behaves well with respect to the extension groups. As main results, we establish the finitistic injective dimension conjecture, verify the Cartan determinant conjecture and its converse, and prove a weaker version of the strong no loop conjecture for strictly stratified algebras.

Key Words:

• Cartan determinant conjecture
• Finitistic dimension conjecture
• Strictly stratified algebras
• Strong no loop conjecture

2000 Mathematics Subject Classification:

• 16E05
• 16E10
• 16E30

ACKNOWLEDGMENT

The author gratefully acknowledges the support of the Natural Sciences and Engineering Research Council of Canada. He also would like to thank professor Shiping Liu for his comments and suggestions in the preparation of this article.

Notes

Communicated by D. Zacharia.