Multiplicity of powers of squarefree monomial ideals

IF 0.5 4区 数学 Q3 MATHEMATICS
Phan Thi Thuy, Thanh Vu
{"title":"Multiplicity of powers of squarefree monomial ideals","authors":"Phan Thi Thuy,&nbsp;Thanh Vu","doi":"10.1007/s00013-025-02116-y","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>I</i> be an arbitrary nonzero squarefree monomial ideal of dimension <i>d</i> in a polynomial ring <span>\\(S = \\textrm{k}[x_1,\\ldots ,x_n]\\)</span>. Let <span>\\(\\mu \\)</span> be the number of associated primes of <i>S</i>/<i>I</i> of dimension <i>d</i>. We prove that the multiplicity of powers of <i>I</i> is given by </p><div><div><span>$$\\begin{aligned} e_0(S/I^s) = \\mu \\left( {\\begin{array}{c}n-d+s-1\\\\ s-1\\end{array}}\\right) \\end{aligned}$$</span></div></div><p>for all <span>\\(s \\ge 1\\)</span>. Consequently, we compute the multiplicity of all powers of path ideals of cycles.</p></div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"125 1","pages":"9 - 15"},"PeriodicalIF":0.5000,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-025-02116-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

Let I be an arbitrary nonzero squarefree monomial ideal of dimension d in a polynomial ring \(S = \textrm{k}[x_1,\ldots ,x_n]\). Let \(\mu \) be the number of associated primes of S/I of dimension d. We prove that the multiplicity of powers of I is given by

$$\begin{aligned} e_0(S/I^s) = \mu \left( {\begin{array}{c}n-d+s-1\\ s-1\end{array}}\right) \end{aligned}$$

for all \(s \ge 1\). Consequently, we compute the multiplicity of all powers of path ideals of cycles.

无平方单项式理想的幂的多重性
假设是多项式环中任意一个d维的非零无平方单项式理想\(S = \textrm{k}[x_1,\ldots ,x_n]\)。设\(\mu \)为维数d的S/I的关联素数。我们证明了对于所有\(s \ge 1\), I的幂次的多重性由$$\begin{aligned} e_0(S/I^s) = \mu \left( {\begin{array}{c}n-d+s-1\\ s-1\end{array}}\right) \end{aligned}$$给出。因此,我们计算了环的路径理想的所有幂次的多重性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Archiv der Mathematik
Archiv der Mathematik 数学-数学
CiteScore
1.10
自引率
0.00%
发文量
117
审稿时长
4-8 weeks
期刊介绍: Archiv der Mathematik (AdM) publishes short high quality research papers in every area of mathematics which are not overly technical in nature and addressed to a broad readership.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:604180095
Book学术官方微信