克劳森关于理想类群的观点的非对易类比
A non-commutative analogue of Clausen's view on the idèle class group
摘要：
Clausen预言数域$F$的Chevalley的idèle类群是局部紧$F$-向量空间范畴的第一个$F$-群。这已被证明是正确的，甚至在适当的意义上推广到更高的$K$-组。我们用一个半单$\mathbb{Q}$-代数代替$F$，并以类似的方式得到Fröhlich的非交换的理想类群，模约化的范数为1的元素。即使在数域的情况下，我们的证明比现有的一个简单，并基于局部化定理的子范畴。最后，使用类域理论作为输入，我们解释希尔伯特的互易定律（以及一个非交换的变种）在我们的结果。
英文摘要：
Clausen predicted that Chevalley's idèle class group of a number field $F$ appears as the first $K$-group of the category of locally compact $F$-vector spaces. This has turned out to be true, and even generalizes to the higher $K$-groups in a suitable sense. We replace $F$ by a semisimple $\mathbb{Q}$-algebra, and obtain Fröhlich's non-commutative idèle class group in an analogous fashion, modulo the reduced norm one elements. Even in the number field case our proof is simpler than the existing one, and based on the localization theorem for percolating subcategories. Finally, using class field theory as input, we interpret Hilbert's reciprocity law (as well as a noncommutative variant) in terms of our results.
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