$$\mathbb{P}^{1}_{mathbb{Z}}$ 上向量束的雷德梅斯特扭转

Pub Date : 2024-08-20 DOI:10.1134/s008154382403012x
V. M. Polyakov
{"title":"$$\\mathbb{P}^{1}_{mathbb{Z}}$ 上向量束的雷德梅斯特扭转","authors":"V. M. Polyakov","doi":"10.1134/s008154382403012x","DOIUrl":null,"url":null,"abstract":"<p>We consider vector bundles of rank <span>\\(2\\)</span> with trivial generic fiber on the projective line over <span>\\(\\mathbb{Z}\\)</span>. For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to <span>\\(\\mathcal{O}^{2}\\)</span> in the fiber over <span>\\(\\mathbb{Q}\\)</span> and are isomorphic to <span>\\(\\mathcal{O}^{2}\\)</span> or <span>\\(\\mathcal{O}(-1)\\oplus\\mathcal{O}(1)\\)</span> over each closed point of Spec<span>\\((\\mathbb{Z})\\)</span>, we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Reidemeister Torsion for Vector Bundles on $$\\\\mathbb{P}^{1}_{\\\\mathbb{Z}}$$\",\"authors\":\"V. M. Polyakov\",\"doi\":\"10.1134/s008154382403012x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We consider vector bundles of rank <span>\\\\(2\\\\)</span> with trivial generic fiber on the projective line over <span>\\\\(\\\\mathbb{Z}\\\\)</span>. For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to <span>\\\\(\\\\mathcal{O}^{2}\\\\)</span> in the fiber over <span>\\\\(\\\\mathbb{Q}\\\\)</span> and are isomorphic to <span>\\\\(\\\\mathcal{O}^{2}\\\\)</span> or <span>\\\\(\\\\mathcal{O}(-1)\\\\oplus\\\\mathcal{O}(1)\\\\)</span> over each closed point of Spec<span>\\\\((\\\\mathbb{Z})\\\\)</span>, we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.\\n</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1134/s008154382403012x\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s008154382403012x","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

我们考虑的是秩为 \(2\) 的向量束,它在\(\mathbb{Z}\) 上的投影线上具有微不足道的一般纤维。对于这样的束,我们构建了一个新的不变量--雷德梅斯特扭转(Reidemeister torsion),它是拓扑学中经典的雷德梅斯特扭转的类似物。对于具有微不足道的泛函纤维和高度为 1 的秩为 2 的向量束,即的纤维上与\(\mathbb{Q}\)的\(\mathcal{O}^{2}\)同构,并且在 Spec\((\mathbb{Z})\) 的每个闭合点上与\(\mathcal{O}^{2}\)或\(\mathcal{O}(-1)\oplus\mathcal{O}(1)\)同构的束、我们计算了这个不变量,并证明它与束的判别式一起完全决定了这样一个束。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

摘要图片

查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
Reidemeister Torsion for Vector Bundles on $$\mathbb{P}^{1}_{\mathbb{Z}}$$

We consider vector bundles of rank \(2\) with trivial generic fiber on the projective line over \(\mathbb{Z}\). For such bundles, a new invariant is constructed — the Reidemeister torsion, which is an analog of the classical Reidemeister torsion from topology. For vector bundles of rank 2 with trivial generic fiber and jumps of height 1, that is, for the bundles that are isomorphic to \(\mathcal{O}^{2}\) in the fiber over \(\mathbb{Q}\) and are isomorphic to \(\mathcal{O}^{2}\) or \(\mathcal{O}(-1)\oplus\mathcal{O}(1)\) over each closed point of Spec\((\mathbb{Z})\), we calculate this invariant and show that it, together with the discriminant of the bundle, completely determines such a bundle.

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
×
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
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 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 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1