On General fiber product rings, Poincaré series and their structure
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916304535945216 |
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| author | Freitas, T. H. Lima, J. A. |
| author_facet | Freitas, T. H. Lima, J. A. |
| contents | The present paper deals with the investigation of the structure of general fiber product rings $R\times_TS$, where $R$, $S$ and $T$ are local rings with common residue field. We show that the Poincaré series of any $R$-module over the fiber product ring $R\times_TS$ is bounded by a rational function. In addition, we give a description of ${\rm depth}(R\times_TS)$, which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over $R\times_TS$ obtained, we provide certain cases of the Cohen-Macaulayness of $R\times_TS$ and, in particular, we show that $R\times_TS$ is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R\times_TS$ are also established. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On General fiber product rings, Poincaré series and their structure Freitas, T. H. Lima, J. A. Commutative Algebra 13D02 The present paper deals with the investigation of the structure of general fiber product rings $R\times_TS$, where $R$, $S$ and $T$ are local rings with common residue field. We show that the Poincaré series of any $R$-module over the fiber product ring $R\times_TS$ is bounded by a rational function. In addition, we give a description of ${\rm depth}(R\times_TS)$, which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over $R\times_TS$ obtained, we provide certain cases of the Cohen-Macaulayness of $R\times_TS$ and, in particular, we show that $R\times_TS$ is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R\times_TS$ are also established. |
| title | On General fiber product rings, Poincaré series and their structure |
| topic | Commutative Algebra 13D02 |
| url | https://arxiv.org/abs/2402.12125 |