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

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).

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