MORE ON GALOIS COHOMOLOGY, DEFINABILITY, AND DIFFERENTIAL ALGEBRAIC GROUPS

OMAR LEÓN SÁNCHEZ, DAVID MERETZKY, ANAND PILLAY
{"title":"MORE ON GALOIS COHOMOLOGY, DEFINABILITY, AND DIFFERENTIAL ALGEBRAIC GROUPS","authors":"OMAR LEÓN SÁNCHEZ, DAVID MERETZKY, ANAND PILLAY","doi":"10.1017/jsl.2024.25","DOIUrl":null,"url":null,"abstract":"<p>As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.</p><p>Applied to the special case of differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/jsl.2024.25","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.

Applied to the special case of differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable.

关于伽罗瓦同调、可定义性和微分代数群的更多信息
作为第三作者在[5]中工作的延续,我们进一步观察了一般模型论背景下伽罗瓦同调的特征。我们明确了可定义群的形式与在适当的自动群中具有系数的第一同调集之间的联系。然后,我们用扭转同调的方法(受塞尔代数扭转的启发)来描述同调序列中的任意纤维--产生了一个关于同调集的有用的 "有限性 "结果。应用于微分域和科尔琴约束同调的特殊情况,我们通过证明有界、微分大域上的微分代数群的第一约束同调集是可数的,完成了[3]中的结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
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