伽罗瓦上同调的转移原理与Serre猜想2

IF 1.5 1区数学 Q1 MATHEMATICS

Advances in Mathematics Pub Date : 2025-09-16 DOI:10.1016/j.aim.2025.110532

Diego Izquierdo , Giancarlo Lucchini Arteche

{"title":"伽罗瓦上同调的转移原理与Serre猜想2","authors":"Diego Izquierdo , Giancarlo Lucchini Arteche","doi":"10.1016/j.aim.2025.110532","DOIUrl":null,"url":null,"abstract":"<div><div>In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field <em>K</em> with finite cohomological dimension <em>δ</em>, the two main ones allow to:<ul><li><span>-</span><span><div>construct totally ramified extensions of <em>K</em> with cohomological dimension <span><math><mo>≤</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> when <em>K</em> is a complete discrete valuation field;</div></span></li><li><span>-</span><span><div>construct algebraic extensions of <em>K</em> with cohomological dimension <span><math><mo>≤</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> and satisfying a norm condition.</div></span></li></ul> We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic 0 fields implies Serre's conjecture II for positive characteristic fields.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"480 ","pages":"Article 110532"},"PeriodicalIF":1.5000,"publicationDate":"2025-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Transfer principles for Galois cohomology and Serre's conjecture II\",\"authors\":\"Diego Izquierdo , Giancarlo Lucchini Arteche\",\"doi\":\"10.1016/j.aim.2025.110532\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field <em>K</em> with finite cohomological dimension <em>δ</em>, the two main ones allow to:<ul><li><span>-</span><span><div>construct totally ramified extensions of <em>K</em> with cohomological dimension <span><math><mo>≤</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> when <em>K</em> is a complete discrete valuation field;</div></span></li><li><span>-</span><span><div>construct algebraic extensions of <em>K</em> with cohomological dimension <span><math><mo>≤</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span> and satisfying a norm condition.</div></span></li></ul> We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic 0 fields implies Serre's conjecture II for positive characteristic fields.</div></div>\",\"PeriodicalId\":50860,\"journal\":{\"name\":\"Advances in Mathematics\",\"volume\":\"480 \",\"pages\":\"Article 110532\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2025-09-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S000187082500430X\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S000187082500430X","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

本文证明了场的上同调维数的若干传递原理。给定一个有限上同调维数δ的固定域K，当K是一个完全离散估值域时，两种主要的方法允许构造上同调维数≤δ−1的K的全分支扩展；-构造上同调维数≤δ−1且满足范数条件的K的代数扩展。然后，我们将这些结果应用于Serre猜想II以及受Kato和Kuzumaki猜想启发的任何上同维域的一些变体。特别地，我们证明了特征0场的Serre猜想II蕴涵了正特征场的Serre猜想II。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Transfer principles for Galois cohomology and Serre's conjecture II

In this article, we prove several transfer principles for the cohomological dimension of fields. Given a fixed field K with finite cohomological dimension δ, the two main ones allow to:

-
construct totally ramified extensions of K with cohomological dimension $\leq δ - 1$ when K is a complete discrete valuation field;
-
construct algebraic extensions of K with cohomological dimension $\leq δ - 1$ and satisfying a norm condition.

We then apply these results to Serre's conjecture II and to some variants for fields of any cohomological dimension that are inspired by conjectures of Kato and Kuzumaki. In particular, we prove that Serre's conjecture II for characteristic 0 fields implies Serre's conjecture II for positive characteristic fields.

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

Advances in Mathematics 数学-数学

CiteScore

2.80

自引率

5.90%

发文量

497

审稿时长

7.5 months

期刊介绍： Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.