Periodic dimensions and some homological properties of eventually periodic algebras

Satoshi Usui

Published 2023-01-16Version 1

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 periodic dimension\rq. We give a bound for the periodic dimension of an eventually periodic module with finite Gorenstein projective dimension. This bound tells us that the two dimensions are almost equal. Moreover, making use of the bound, we determine the bimodule periodic dimension of a finite dimensional eventually periodic Gorenstein algebra. Another aim of this paper is to obtain some of the basic homological properties of finite dimensional eventually periodic algebras. We show that a lot of homological conjectures hold for this class of algebras. Further, we use this result to characterize finite dimensional eventually periodic Gorenstein algebras. This characterization explains why we consider their bimodule periodic dimensions.