Anick resolution and Koszul algebras of finite global dimension

ABSTRACT

We show how to study a certain associative algebra recently discovered by Iyudu and Shkarin using the Anick resolution. This algebra is a counterexample to the conjecture of Positselski on Koszul algebras of finite global dimension.

KEYWORDS:

• Anick resolution
• global dimension
• Koszul duality
• Koszulness

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16S37
• Secondary: 13P10, 16E05, 16Z05, 18G10

Acknowledgments

We are grateful to Leonid Positselski for drawing our attention to the fact that the list of algebras in [7] contains a counterexample to his conjecture. Ivan Yudin informed us that it is also possible to establish the claim on Koszulness by a careful examination of the two Anick resolutions for the orderings x>y>z and y>x>z for the original algebra; his argument utilizes Ufnarovski’s graph for generating Anick chains [10, 9]; we are very thankful for that remark.