Relations between Poincaré series for quasi-complete intersection homomorphisms
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910594177695744 |
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| author | Pollitz, Josh Sega, Liana M. |
| author_facet | Pollitz, Josh Sega, Liana M. |
| contents | In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $φ\colon Q \to R$, where $Q$ is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series $\mathrm{P}^Q_M(t)$ of a finitely generated $R$-module $M$ to $\mathrm{P}^R_M(t)$ when the kernel of $φ$ is contained in $\mathfrak{m}\,\mathrm{ann}_Q(M)$. This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras $Q\to E$, with $E$ the Koszul complex on a minimal set of generators for the kernel of $φ.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_17079 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relations between Poincaré series for quasi-complete intersection homomorphisms Pollitz, Josh Sega, Liana M. Commutative Algebra Primary: 13D40. Secondary: 13D02, 13D07, 16E45 In this article we study base change of Poincaré series along a quasi-complete intersection homomorphism $φ\colon Q \to R$, where $Q$ is a local ring with maximal ideal $\mathfrak{m}$. In particular, we give a precise relationship between the Poincaré series $\mathrm{P}^Q_M(t)$ of a finitely generated $R$-module $M$ to $\mathrm{P}^R_M(t)$ when the kernel of $φ$ is contained in $\mathfrak{m}\,\mathrm{ann}_Q(M)$. This generalizes a classical result of Shamash for complete intersection homomorphisms. Our proof goes through base change formulas for Poincaré series under the map of dg algebras $Q\to E$, with $E$ the Koszul complex on a minimal set of generators for the kernel of $φ.$ |
| title | Relations between Poincaré series for quasi-complete intersection homomorphisms |
| topic | Commutative Algebra Primary: 13D40. Secondary: 13D02, 13D07, 16E45 |
| url | https://arxiv.org/abs/2403.17079 |