arXiv 0903.46342009-03math.GRmath.CO

局部域在单代数和Bruhat-Tits建筑上的单纯结构中的嵌入

Embeddings of local fields in simple algebras and simplicial structures on the Bruhat-Tits building

Dr. Daniel Skodlerack

备注：: 18 pages. arXiv admin note: substantial text overlap with arXiv:1004.4030

摘要：

这篇文章回答了一个问题，自然产生的文章Grabitz和Broussous“纯元素和交织类的简单阶层在当地中央单代数“和Broussous和Lemaire“大厦GL（m，D）和Centralizers”。对于非阿基米德局部域F上的Azumaya-代数A，Grabitz和Broussous引入了域嵌入的嵌入不变量，即对（E，a），其中E是F在A中的域扩张，而$a$是由E^x归一化的遗传序。另一方面，如果我们取这样一个域扩张E，并定义B是E在A中的中心化子，则G：=A^x和G_E：=B^x分别是定义在F和E上的约化群的有理点的集合。Broussous和Lemaire定义了一个映射j_E：I^{E^x} \to I_E，其中$I$是$G$的欧几里德建筑，I_E是G_E的欧几里德建筑.我们要解决的问题是把嵌入不变量与映射j_E关于$I$和I_E的单纯结构的行为联系起来。我必须非常感谢柏林洪堡大学的Zink教授，感谢他的有益评论，修改了我的工作，并给了我有趣的任务。

英文摘要：

This article answers a question that naturally arises from the articles by Grabitz and Broussous "Pure elements and intertwining classes of simple strata in local central simple algebras" and Broussous and Lemaire "Buildings of GL(m,D) and Centralizers". For an Azumaya-Algebra A over a non-Archimedean local field F, Grabitz and Broussous have introduced embedding invariants for field embeddings, that is for pairs (E,a), where E is a field extension of F in A, and $a$ is a hereditary order which is normalised by E^x. On the other hand if we take such a field extension E and define B to be the centralizer of E in A, then G:=A^x and G_E:=B^x are sets of rational points of reductive groups defined over F and E respectively. Broussous and Lemaire have defined a map j_E: I^{E^x}\to I_E, where $I$ is the the Euclidean building of $G$, and I_E is the Euclidean building of G_E. The question which we address is to relate the embedding invariants to the behavior of the map j_E with respect to the simplicial structures of $I$ and I_E. I have to thank very much Prof. Zink from Homboldt University Berlin for his helpful remarks, the revision of the work and for giving my the interesting task.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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