The embedded deformation problem for monomial ideals
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909647077638144 |
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| author | Briggs, Benjamin Grifo, Eloísa Pollitz, Josh |
| author_facet | Briggs, Benjamin Grifo, Eloísa Pollitz, Josh |
| contents | This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_10827 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The embedded deformation problem for monomial ideals Briggs, Benjamin Grifo, Eloísa Pollitz, Josh Commutative Algebra Rings and Algebras Primary: 13D09. Secondary: 13C15, 13D02, 13D07, 13H10, 14M10, 16E45 This article is concerned with homological properties of local or graded rings whose defining relations are monomials on some regular sequence. The main result of the article positively answers a question of Avramov for such a ring $R$. More precisely, we establish that an embedded deformation of $R$ corresponds exactly to a degree two central element in the homotopy Lie algebra of $R$, as well as a free summand of the conormal module of $R$. A major input in the proof is an analysis of cohomological support varieties. Other main results include establishing a lower bound for the dimension of the cohomological support variety of any complex over such rings, and classifying all possible subvarieties of affine $n$-space that are the cohomological support of rings defined by $n$ monomial relations where $n$ is five or less. |
| title | The embedded deformation problem for monomial ideals |
| topic | Commutative Algebra Rings and Algebras Primary: 13D09. Secondary: 13C15, 13D02, 13D07, 13H10, 14M10, 16E45 |
| url | https://arxiv.org/abs/2506.10827 |