Extremal Behavior of ideals of minors
Fuente:
arXiv
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915695782002688 |
|---|---|
| author | Chau, Trung DeBellevue, Michale Dey, Souvik Ganapathy, K. Javadekar, Omkar |
| author_facet | Chau, Trung DeBellevue, Michale Dey, Souvik Ganapathy, K. Javadekar, Omkar |
| contents | Let $(R,\mathfrak m,\mathsf k)$ be either a fiber product or an artinian stretched Gorenstein ring, with $\operatorname{ch}(\mathsf k)\neq 2$ in the latter case. We prove that the ideals of minors of the minimal free resolution of any finitely generated $R$-module are eventually 2-periodic. Moreover, if the embedding dimension of $R$ is at least 3, eventually the ideals of minors become the powers of the maximal ideal, yielding the 1-periodicity. These are analogs of results obtained over complete intersections and Golod rings by Brown, Dao, and Sridhar. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring $(R,\mathfrak m)$, if $x\in \mathfrak m$ is a super-regular element and $M$ is an $R/(x)$ module whose ideals of minors are asymptotically the powers of the maximal ideal over $R/(x)$, then the same holds for the ideals of minors of $M$ over $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_05225 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal Behavior of ideals of minors Chau, Trung DeBellevue, Michale Dey, Souvik Ganapathy, K. Javadekar, Omkar Commutative Algebra 13D02, 13H10, 13D10 Let $(R,\mathfrak m,\mathsf k)$ be either a fiber product or an artinian stretched Gorenstein ring, with $\operatorname{ch}(\mathsf k)\neq 2$ in the latter case. We prove that the ideals of minors of the minimal free resolution of any finitely generated $R$-module are eventually 2-periodic. Moreover, if the embedding dimension of $R$ is at least 3, eventually the ideals of minors become the powers of the maximal ideal, yielding the 1-periodicity. These are analogs of results obtained over complete intersections and Golod rings by Brown, Dao, and Sridhar. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring $(R,\mathfrak m)$, if $x\in \mathfrak m$ is a super-regular element and $M$ is an $R/(x)$ module whose ideals of minors are asymptotically the powers of the maximal ideal over $R/(x)$, then the same holds for the ideals of minors of $M$ over $R$. |
| title | Extremal Behavior of ideals of minors |
| topic | Commutative Algebra 13D02, 13H10, 13D10 |
| url | https://arxiv.org/abs/2507.05225 |