Vol. 305, No. 1, 2020

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A criterion for modules over Gorenstein local rings to have rational Poincaré series

Anjan Gupta

Vol. 305 (2020), No. 1, 165–187
Abstract

We prove that modules over an Artinian Gorenstein local ring R have rational Poincaré series sharing a common denominator if Rsocle(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 Poincaré series sharing a common denominator. By a result of Şega, it follows that R satisfies the Auslander–Reiten conjecture. We provide a different proof of a result of Rossi and Şega (Adv. Math.259 (2014), 421–447) concerning rationality of Poincaré 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 3 have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller (J. Algebra118:1 (1988), 162–204).

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Keywords
Poincaré series, Golod rings, DG algebras, fibre products, connected sums
Mathematical Subject Classification 2010
Primary: 13D02, 13D40, 13H10
Milestones
Received: 30 January 2018
Revised: 21 September 2019
Accepted: 2 October 2019
Published: 17 March 2020
Authors
Anjan Gupta
Department of Mathematics
Indian Institute of Science Education and Research Bhopal
Bhopal
India