A criterion for modules over Gorenstein local rings to have rational Poincaré series
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911484229976064 |
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| author | Gupta, Anjan |
| author_facet | Gupta, Anjan |
| contents | We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincaré 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 Poincaré 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 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 three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03858 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A criterion for modules over Gorenstein local rings to have rational Poincaré series Gupta, Anjan Commutative Algebra 13D02, 13D40, 13H10 We prove that modules over an Artinian Gorenstein local ring $R$ have rational Poincaré 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 Poincaré 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 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 three have rational Poincaré series sharing a common denominator, which is originally due to Avramov, Kustin and Miller. |
| title | A criterion for modules over Gorenstein local rings to have rational Poincaré series |
| topic | Commutative Algebra 13D02, 13D40, 13H10 |
| url | https://arxiv.org/abs/2603.03858 |