横截群的局部-整体原理
Local-Global Principle for Transvection Groups
摘要：
本文将Suslin关于一般线性群、辛群和正交群（n > 2）的初等横射子群的局部-整体原理推广到关于自同构群Aut（P）的初等横射子群的局部-整体原理，Aut（P）是一个整体秩> 0且常局部秩> 2的投射模P，或者是全局双曲秩> 0和常数局部双曲秩> 2的非奇异辛模或正交模P。在Suslin的结果中，局部秩和全局秩是相同的，因为他只关心自由模。我们假设整体（双曲）秩> 0用于定义基本平射子群。进一步证明了Aut（P）中的初等横射子群ET（P）是正规的，ET（P）= T（P），其中T（P）表示Aut（P）的全横射子群，以及不稳定K_1-群K_1（Aut（P））= Aut（P）/ET（P）= Aut（P）/T（P）是交换幂零群，只要R具有有限稳定维数.最后一个结果推广了Bak和Hazrat对上述典型群的结果.本文结果的一个重要应用可以在最后两位作者的工作中找到，他们研究了完美C_1-域上的非奇异仿射代数上的经典模的内射稳定性的下降。我们建议读者参阅那篇文章以了解更多细节。
英文摘要：
In this article we extend the validity Suslin's Local-Global Principle for the elementary transvection subgroup of the general linear group, the symplectic group, and the orthogonal group, where n > 2, to a Local-Global Principle for the elementary transvection subgroup of the automorphism group Aut(P) of either a projective module P of global rank > 0 and constant local rank > 2, or of a nonsingular symplectic or orthogonal module P of global hyperbolic rank > 0 and constant local hyperbolic rank > 2. In Suslin's results, the local and global ranks are the same, because he is concerned only with free modules. Our assumption that the global (hyperbolic) rank > 0 is used to define the elementary transvection subgroups. We show further that the elementary transvection subgroup ET(P) is normal in Aut(P), that ET(P) = T(P) where the latter denotes the full transvection subgroup of Aut(P), and that the unstable K_1-group K_1(Aut(P)) = Aut(P)/ET(P) = Aut(P)/T(P) is nilpotent by abelian, provided R has finite stable dimension. The last result extends previous ones of Bak and Hazrat for the above mentioned classical groups. An important application to the results in the current paper can be found in the work of last two named authors where they have studied the decrease in the injective stabilization of classical modules over a non-singular affine algebra over perfect C_1-fields. We refer the reader to that article for more details.
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