Strong A 1 ${\mathbb {A}}^1$ -invariance of A 1 ${\mathbb {A}}^1$ -connected components of reductive algebraic groups

IF 0.8 2区数学 Q2 MATHEMATICS

Journal of Topology Pub Date : 2023-05-27 DOI:10.1112/topo.12298

Chetan Balwe, Amit Hogadi, Anand Sawant

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引用次数: 0

Abstract

We show that the sheaf of $A^{1}$ -connected components of a reductive algebraic group over a perfect field is strongly $A^{1}$ -invariant. As a consequence, torsors under such groups give rise to $A^{1}$ -fiber sequences. We also show that sections of $A^{1}$ -connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $R$ -equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

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还原代数群的A1$｛\mathbb｛A｝｝^1$连通分量的强A1$｛

我们证明了完美域上的归约代数群的A1$｛\mathbb｛A｝｝^1$连通分量的sheaf是强A1$｛。因此，这类群下的扭转子产生A1$｛\mathbb｛a｝｝^1$纤维序列。我们还证明了任意域上各向异性、半单、单连通代数群的A1$｛\mathbb｛A｝｝^1$连通分量的截面与它们的R$R$等价类一致，从而消除了先前已知的关于用Nisnevich局部平凡扭体的仿射同伦不变性表征各向同性的结果中的完全性假设。

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来源期刊

Journal of Topology 数学-数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal. The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.