#### 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 $R∕socle\left(R\right)$ 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).

##### Keywords
Poincaré series, Golod rings, DG algebras, fibre products, connected sums
##### Mathematical Subject Classification 2010
Primary: 13D02, 13D40, 13H10