{"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}
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.