Resolution quiver and cyclic homology criteria for Nakayama 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].

10 pages, v3: final version, v2: references added and updated

References in corpus (5)

• Total positivity, Grassmannians, and networks
• Cluster X-varieties, amalgamation and Poisson-Lie groups
• A counterexample to the $\varphi$-dimension conjecture
• Retractions and Gorenstein homological properties
• A Legendrian Turaev torsion via generating families

Cited by in corpus (1)

• A counterexample to the $\varphi$-dimension conjecture