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Abstract
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We prove that modules over an Artinian Gorenstein local ring
have rational Poincaré series sharing a common denominator if
is a Golod
ring. If
is a Gorenstein local ring with square of the maximal ideal being
generated by at most two elements, we show that modules over
have
rational Poincaré series sharing a common denominator. By a result of Şega, it follows
that
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
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Mathematical Subject Classification 2010
Primary: 13D02, 13D40, 13H10
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Milestones
Received: 30 January 2018
Revised: 21 September 2019
Accepted: 2 October 2019
Published: 17 March 2020
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