多重范数一环面的上同调性质
Cohomological properties of multinorm-one tori
AI总结：
本文研究了全局域上多重范数一环面的Tate-Shafarevich群，通过建立上同调群间的函子映射获得了基本结构结果，推广并统一了多项先前研究的结论，同时改进了该群的显式描述。
摘要：
本文研究了全局域$k$上多重范数一环面$T$的Tate--Shafarevich群Sha^1(k, T)。我们建立了上同调群之间的若干函子映射，并探讨了它们的关系。利用这些性质和关系，我们得到了Sha^1(k, T)的一些基本结构结果，并将Bayer-Fluckiger--Lee--Parimala [Adv. in Math., 2019]的若干结果推广到更一般的多重范数一环面。我们还对Demarche--Wei关于Sha^1(k, T)消失判据的一个结果，以及Pollio [Pure App. Math. Q., 2014]在所讨论的étale $k$-代数为两个阿贝尔扩张的乘积情形下的主要结果，给出了统一的证明。此外，我们通过移除一个交集条件，改进了Lee [J. Pure Appl. Alg., 2022]中对Sha^1(k, T)的显式描述。
英文摘要：
In this paper we investigate the Tate--Shafarevich group Sha^1(k, T) of a multinorm-one torus $T$ over a global field $k$. We establish a few functorial maps among cohomology groups and explore their relations. Using these properties and relations we obtain a few basic structural results for Sha^1(k, T) and extend a few results of Bayer-Fluckiger--Lee--Parimala [Adv. in Math., 2019] to some more general multinorm-one tori. We also give a uniform proof of a result of Demarche--Wei for a criterion of the vanishing of Sha^1(k, T), and of the main result of Pollio [Pure App. Math. Q., 2014] for the case where the étale $k$-algebra in question is a product of two abelian extensions. Moreover, we improve the explicit description of Sha^1(k, T) in Lee [J. Pure Appl. Alg., 2022] by removing an intersection condition.
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