在全局域上约化群的伽罗瓦上同调上是否存在群结构？

IF 0.5 4区数学 Q3 MATHEMATICS

Archiv der Mathematik Pub Date : 2025-04-28 DOI:10.1007/s00013-025-02118-w

Mikhail Borovoi

{"title":"在全局域上约化群的伽罗瓦上同调上是否存在群结构？","authors":"Mikhail Borovoi","doi":"10.1007/s00013-025-02118-w","DOIUrl":null,"url":null,"abstract":"<div>Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets \\(\\textrm{H}^1\\hspace{-0.8pt}(K,G)\\) for all reductive K-groups G in a functorial way.</div>","PeriodicalId":8346,"journal":{"name":"Archiv der Mathematik","volume":"124 6","pages":"583 - 589"},"PeriodicalIF":0.5000,"publicationDate":"2025-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00013-025-02118-w.pdf","citationCount":"0","resultStr":"{\"title\":\"Is there a group structure on the Galois cohomology of a reductive group over a global field?\",\"authors\":\"Mikhail Borovoi\",\"doi\":\"10.1007/s00013-025-02118-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div>Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets \\\\(\\\\textrm{H}^1\\\\hspace{-0.8pt}(K,G)\\\\) for all reductive K-groups G in a functorial way.</div>\",\"PeriodicalId\":8346,\"journal\":{\"name\":\"Archiv der Mathematik\",\"volume\":\"124 6\",\"pages\":\"583 - 589\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2025-04-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s00013-025-02118-w.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archiv der Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00013-025-02118-w\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archiv der Mathematik","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00013-025-02118-w","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

摘要

设K为全局域，即数字域或全局函数域。我们知道，当K没有真实嵌入时，题目K上的问题的答案是“是”。我们表明，否则答案是“否”。即，我们证明了当K是一个允许实嵌入的数域时，对于所有约化K群G，不可能用泛函的方式定义第一伽罗瓦上同集\(\textrm{H}^1\hspace{-0.8pt}(K,G)\)上的群结构。

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

Is there a group structure on the Galois cohomology of a reductive group over a global field?

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets \(\textrm{H}^1\hspace{-0.8pt}(K,G)\) for all reductive K-groups G in a functorial way.

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

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.