On the Rees algebra and the conductor of an ideal
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916317728079872 |
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| author | Gasanova, Oleksandra Herzog, Jürgen Kling, Filip Jonsson Moradi, Somayeh |
| author_facet | Gasanova, Oleksandra Herzog, Jürgen Kling, Filip Jonsson Moradi, Somayeh |
| contents | For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the symmetric algebra of $I$ by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_06922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Rees algebra and the conductor of an ideal Gasanova, Oleksandra Herzog, Jürgen Kling, Filip Jonsson Moradi, Somayeh Commutative Algebra 13A30 (Primary) 05E40 (Secondary) For an ideal $I$ in a Noetherian ring $R$, we introduce and study its conductor as a tool to explore the Rees algebra of $I$. The conductor of $I$ is an ideal $C(I)\subset R$ obtained from the defining ideals of the Rees algebra and the symmetric algebra of $I$ by a colon operation. Using this concept we investigate when adding an element to an ideal preserves the property of being of linear type. In this regard, a generalization of a result by Valla in terms of the conductor ideal is presented. When the conductor of a graded ideal in a polynomial ring is the graded maximal ideal, a criteria is given for when the Rees algebra and the symmetric algebra have the same Krull dimension. Finally, noting the fact that the conductor of a monomial ideal is a monomial ideal, the conductor of some families of monomial ideals, namely bounded Veronese ideals and edge ideals of graphs, are determined. |
| title | On the Rees algebra and the conductor of an ideal |
| topic | Commutative Algebra 13A30 (Primary) 05E40 (Secondary) |
| url | https://arxiv.org/abs/2407.06922 |