Finitely generated saturated multi-Rees algebras
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arXiv
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866913191913586688 |
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| author | Das, Suprajo Roy, Sudeshna |
| author_facet | Das, Suprajo Roy, Sudeshna |
| contents | We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14587 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Finitely generated saturated multi-Rees algebras Das, Suprajo Roy, Sudeshna Commutative Algebra Primary 13A30, 13H15, 13D45, Secondary 16W50, 13E05, 05E40 We study the question of finite generation of saturated multi-Rees algebras and investigate the asymptotic behaviour of related length functions. In the setup of excellent local domains, we show that the saturated multi-Rees algebra of a finite collection of ideals is finitely generated when the analytic spread is not maximal and the associated length function eventually agrees with a polynomial. Similar results are obtained when we restrict to two-dimensional local UFDs with no restrictions on the analytic spread. We further prove that the saturated multi-Rees algebra of finitely many monomial ideals in a polynomial ring modulo an irreducible monomial ideal, is always finitely generated. In this case, the corresponding length function is shown to exhibit piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a theorem of Amao. |
| title | Finitely generated saturated multi-Rees algebras |
| topic | Commutative Algebra Primary 13A30, 13H15, 13D45, Secondary 16W50, 13E05, 05E40 |
| url | https://arxiv.org/abs/2112.14587 |