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Betti numbers under small perturbations
- Source :
- Journal of Algebra. 594:138-153
- Publication Year :
- 2022
- Publisher :
- Elsevier BV, 2022.
-
Abstract
- We study how Betti numbers of ideals in a local ring change under small perturbations. Given $p\in\mathbb N$ and given an ideal $I$ of a Noetherian local ring $(R,\mathfrak m)$, our main result states that there exists $N>0$ such that if $J$ is an ideal with $I\equiv J\bmod \mathfrak m^N$ and with the same Hilbert function as $I$, then the Betti numbers $\beta_i^R(R/I)$ and $\beta_i^R(R/J)$ coincide for $0\le i\le p$. Moreover, we present several cases in which an ideal $J$ such that $I \equiv J \bmod \mathfrak m^N$ is forced to have the same Hilbert function as $I$, and therefore the same Betti numbers.<br />Comment: 13 pages
Details
- ISSN :
- 00218693
- Volume :
- 594
- Database :
- OpenAIRE
- Journal :
- Journal of Algebra
- Accession number :
- edsair.doi.dedup.....3a497f2efaad86ca29aa3c778098bc03