A criterion for modules over Gorenstein local rings to have rational Poincar\'e series

TL;DR

This paper establishes criteria under which modules over Gorenstein local rings have rational Poincaré series sharing a common denominator, extending known results and providing new proofs for specific classes of rings.

Contribution

It introduces new conditions ensuring rational Poincaré series for modules over Gorenstein rings and offers alternative proofs for existing results.

Findings

Modules over Artinian Gorenstein rings with certain conditions have rational Poincaré series.

Modules over Gorenstein rings with specific maximal ideal conditions share a common denominator in their Poincaré series.

New proofs are provided for rationality of Poincaré series in known classes of Gorenstein rings.

Abstract

We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincar\'e series sharing a common denominator if $R/\soc(R)$ is a Golod ring. If $R$ is a Gorenstein local ring with square of the maximal ideal being generated by at most two elements, we show that modules over $R$ have rational Poincar\'e series sharing a common denominator. By a result of \c Sega, it follows that $R$ satisfies the Auslander-Reiten conjecture. We provide a different proof of a result of Rossi and \c Sega concerning rationality of Poincar\'e series of modules over compressed Gorenstein local rings. We also give a new proof of the fact that modules over Gorenstein local rings of codepth at most three have rational Poincar\'e series sharing a common denominator, which is originally due to Avramov, Kustin and Miller.