局部同调的一些推广研究

IF 0.2 Q4 MATHEMATICS
Yanping Liu
{"title":"局部同调的一些推广研究","authors":"Yanping Liu","doi":"10.11648/j.pamj.20231202.12","DOIUrl":null,"url":null,"abstract":": Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R -module M and a finitely generated R -module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R -modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.","PeriodicalId":46057,"journal":{"name":"Italian Journal of Pure and Applied Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.2000,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Study of Some Generalizations of Local Homology\",\"authors\":\"Yanping Liu\",\"doi\":\"10.11648/j.pamj.20231202.12\",\"DOIUrl\":null,\"url\":null,\"abstract\":\": Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R -module M and a finitely generated R -module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R -modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.\",\"PeriodicalId\":46057,\"journal\":{\"name\":\"Italian Journal of Pure and Applied Mathematics\",\"volume\":\"3 1\",\"pages\":\"\"},\"PeriodicalIF\":0.2000,\"publicationDate\":\"2023-07-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Italian Journal of Pure and Applied Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.11648/j.pamj.20231202.12\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Italian Journal of Pure and Applied Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.11648/j.pamj.20231202.12","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

研究了经典局部上同的重要推广——Tate局部上同和Gorenstein局部上同理论。已经发现它们具有这样的消失性质和长精确序列。然而,对于局部同调,对偶呢?本文讨论了Tate局部同调和Gorenstein局部同调。本文第一部分将局部同调推广为Tate局部同调,并研究了Tate局部同调模的消失性、收敛性和某些精确序列。我们发现,对于具有有限Gorenstein投影维数的任意R模M和有限生成的任意R模N, M和N关于理想I的Tate局部同调模也是任意模。在论文的第二部分,我们将Gorenstein局部同调模作为Gorenstein版本来考虑。讨论了Gorenstein局部同调模的消失性质和一些精确序列,得到了一个连接Gorenstein、Tate和广义局部同调的精确序列。最后,作为连接这些局部同调模的精确序列的一个应用,我们发现对于分别具有有限投影维数和允许Gorenstein投影适当分辨的有限生成R -模,在某些情况下,Gorenstein局部同调与广义局部同调重合。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Study of Some Generalizations of Local Homology
: Tate local cohomology and Gorenstein local cohomology theory, which are important generalizations of the classical local cohomology, has been investigated. It has been found that they have such vanishing properties and long exact sequences. However, for local homology, what about the duality? In this paper we are concerned with Tate local homology and Gorenstein local homology. In the first part of the paper we generalize local homology as Tate local homology, and study such vanishing properties, artinianness and some exact sequence of Tate local homology modules. We find that for an artianian R -module M and a finitely generated R -module N with finite Gorenstein projective dimension, the Tate local homology module of M and N with respect to an ideal I is also an artinian module. In the second part of the paper we consider Gorenstein local homology modules as Gorenstein version. We discuss vanishing properties and some exact sequences of Gorenstein local homology modules and obtain an exact sequence connecting Gorenstein, Tate and generalized local homology. Finally, as an applicaton of the exact sequence connecting these local homology modules, we find that for finitely generated R -modules with finite projective dimension and admitting Gorenstein projective proper resolution respectively, Gorenstein local homology coincides with generalized local homology in certain cases.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
0.60
自引率
0.00%
发文量
2
期刊介绍: The “Italian Journal of Pure and Applied Mathematics” publishes original research works containing significant results in the field of pure and applied mathematics.
×
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:481959085
Book学术官方微信