Resolution quiver and cyclic homology criteria for Nakayama algebras

TL;DR

This paper demonstrates the equivalence of two criteria—resolution quiver and cyclic homology—for determining finite global dimension in Nakayama algebras, unifying previous characterizations and providing new insights.

Contribution

It proves the equivalence of resolution quiver and cyclic homology criteria for Nakayama algebras, simplifying the understanding of their global dimension properties.

Findings

Resolution quiver criterion is equivalent to cyclic homology criterion for Nakayama algebras.

The paper re-proves both existing characterizations through a direct comparison.

It sets the stage for future generalizations to monomial relation algebras.

Abstract

If a Nakayama algebra is not cyclic, it has finite global dimension. For a cyclic Nakayama algebra, there are many characterizations of when it has finite global dimension. In [She17], Shen gave such a characterization using Ringel's resolution quiver. In [IZ92], the second author, with Zacharia, gave a cyclic homology characterization for when a monomial relation algebra has finite global dimension. We show directly that these criteria are equivalent for all Nakayama algebras. Our comparison result also reproves both characterizations. In a separate paper we discuss an interesting example that came up in our attempt to generalize this comparison result to arbitrary monomial relation algebras [HI19].