{"title":"论纤维的布劳尔群","authors":"Yanshuai Qin","doi":"10.1007/s00209-024-03487-8","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\({\\mathcal {X}}\\rightarrow C\\)</span> be a flat <i>k</i>-morphism between smooth integral varieties over a finitely generated field <i>k</i> such that the generic fiber <i>X</i> is smooth, projective and geometrically connected. Assuming that <i>C</i> is a curve with function field <i>K</i>, we build a relation between the Tate-Shafarevich group of <span>\\(\\textrm{Pic}^0_{X/K}\\)</span> and the geometric Brauer groups of <span>\\({\\mathcal {X}}\\)</span> and <i>X</i>, generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.</p>","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":null,"pages":null},"PeriodicalIF":16.4000,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Brauer groups of fibrations\",\"authors\":\"Yanshuai Qin\",\"doi\":\"10.1007/s00209-024-03487-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\({\\\\mathcal {X}}\\\\rightarrow C\\\\)</span> be a flat <i>k</i>-morphism between smooth integral varieties over a finitely generated field <i>k</i> such that the generic fiber <i>X</i> is smooth, projective and geometrically connected. Assuming that <i>C</i> is a curve with function field <i>K</i>, we build a relation between the Tate-Shafarevich group of <span>\\\\(\\\\textrm{Pic}^0_{X/K}\\\\)</span> and the geometric Brauer groups of <span>\\\\({\\\\mathcal {X}}\\\\)</span> and <i>X</i>, generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.</p>\",\"PeriodicalId\":1,\"journal\":{\"name\":\"Accounts of Chemical Research\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":16.4000,\"publicationDate\":\"2024-04-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Accounts of Chemical Research\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00209-024-03487-8\",\"RegionNum\":1,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"CHEMISTRY, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00209-024-03487-8","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
摘要
让 \({\mathcal {X}}\rightarrow C\) 是有限生成域 k 上光滑积分 varieties 之间的平 k 形，使得泛函纤维 X 是光滑的、投影的和几何连接的。假设 C 是有函数域 K 的曲线，我们在 \(\textrm{Pic}^0_{X/K}\) 的 Tate-Shafarevich 群和\({mathcal {X}}\) 与 X 的几何布劳尔群之间建立了一种关系，将阿尔廷和格罗登第克关于纤维曲面的定理推广到了更高的相对维度。
Let \({\mathcal {X}}\rightarrow C\) be a flat k-morphism between smooth integral varieties over a finitely generated field k such that the generic fiber X is smooth, projective and geometrically connected. Assuming that C is a curve with function field K, we build a relation between the Tate-Shafarevich group of \(\textrm{Pic}^0_{X/K}\) and the geometric Brauer groups of \({\mathcal {X}}\) and X, generalizing a theorem of Artin and Grothendieck for fibered surfaces to higher relative dimensions.
期刊介绍：
Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance.
Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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