Periodic dimensions and some homological properties of eventually periodic algebras

TL;DR

This paper investigates the homological properties of eventually periodic algebras, introducing the concept of periodic dimension, providing bounds, computation methods, and characterizations related to Gorenstein projective dimensions.

Contribution

It introduces the notion of periodic dimension for eventually periodic modules, establishes bounds, and characterizes eventually periodic Gorenstein algebras using bimodule Gorenstein projective dimensions.

Findings

Bound for the periodic dimension of modules with finite Gorenstein projective dimension

Method to compute Gorenstein projective dimension under certain conditions

Characterization of eventually periodic Gorenstein algebras via bimodule Gorenstein projective dimensions

Abstract

For an eventually periodic module, we have the degree and the period of its first periodic syzygy. This paper studies the former under the name \lq\lq periodic dimension\rq\rq. We give a bound for the periodic dimension of an eventually periodic module with finite Gorenstein projective dimension. We also provide a method of computing the Gorenstein projective dimension of an eventually periodic module under certain conditions. Besides, motivated by recent results of Dotsenko, G\'elinas and Tamaroff and of the author, we determine the bimodule periodic dimension of an eventually periodic Gorenstein algebra. Another aim of this paper is to obtain some of the basic homological properties of eventually periodic algebras. We show that a lot of homological conjectures hold for this class of algebras. As an application, we characterize eventually periodic Gorenstein algebras in terms of bimodules Gorenstein projective dimensions.