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 [17], Shen gave such a characterization using Ringel's resolution quiver. In [11], 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 [8].

Introduction

A Nakayama algebra is an algebra whose quiver consists of a single oriented cycle with a finite number of monomial relations or a single path, with or without relations. These are called cyclic and linear Nakayama algebras respectively. There are many results about these algebras. In [14], Ringel introduced the “resolution quiver” $R(\mathrm{\Lambda })$ of a Nakayama algebra Λ and used it to characterize when Λ is Gorenstein and to give a formula for its Gorenstein dimension. In [16], Shen showed that every component of the resolution quiver of a Nakayama algebra has the same weight and, in [17], Shen used this to obtain the following result.

Theorem A Shen [17]

A Nakayama algebra has finite global dimension if and only if its resolution quiver has exactly one component and that component has weight 1.

It is important for the purpose of induction that this includes linear Nakayama algebras.

In [11], the second author and Zacharia showed that a monomial relation algebra has finite global dimension if and only if the relative cyclic homology of its radical is equal to zero. For the case of a cyclic Nakayama algebra, they constructed a finite simplicial complex $L(\mathrm{\Lambda })$, which we will call the “relation complex” of Λ, and show that the reduced homology of $L(\mathrm{\Lambda })$ gives the lowest degree term in the relative cyclic homology of the radical of Λ. They further simplified their results to obtain the following.

Theorem B Igusa-Zacharia [11]

A Nakayama algebra Λ has finite global dimension if and only if the Euler characteristic of the relation complex $L(\mathit{\Lambda })$ is equal to one.

In fact, only cyclic Nakayama algebras were considered in [11]. However, if Λ is a linear Nakayama algebra, it is easy to see that $L(\mathrm{\Lambda })$ is still defined and is contractible. Thus it has Euler characteristic one.

In this paper we will review these definitions and statements and prove the following statement showing that the two results above are equivalent.

Theorem C

For any cyclic Nakayama algebra Λ the Euler characteristic of the relation complex $L(\mathit{\Lambda })$ is equal to the number of components of its resolution quiver $R(\mathit{\Lambda })$ with weight 1.

We prove Theorem C using an idea that comes from [15]. In fact, our argument proves Theorem A, Theorem B, Theorem C simultaneously. (See Remark 3.4.) We also obtain a new version of Theorem B:

Theorem B' Corollary 3.14

A Nakayama algebra Λ has finite global dimension if and only if its relation complex $L(\mathit{\Lambda })$ is contractible.

These results arose in our study, joint with Gordana Todorov, of the concept of “amalgamation” [5] and the reverse process which we call “unamalgamation”. Amalgamation is used in [2] to construct plabic diagrams which are used in [1] to construct new invariants in contact topology.

Since the result of [11] holds for any monomial relation algebra, we also attempted to generalize the definition of the resolution quiver to such algebras. This lead us to consider examples which eventually led to a counterexample to the ϕ-dimension conjecture. See [8] for more details about this story.