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arXiv · 2011.14613

On the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in a reductive group

Alexey Ananyevskiy
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Abstract

We show that for a reductive group $G$ over a field $k$ the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in $G$ is an invertible element of the Grothendieck-Witt ring $\mathrm{GW}(k)$, settling the weak form of a conjecture by Fabien Morel. As an application we obtain a generalized splitting principle which allows one to reduce the structure group of a Nisnevich locally trivial $G$-torsor to the normalizer of a maximal torus.

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Alexey Ananyevskiy. 2020-11-30. On the $\mathbb{A}^1$-Euler characteristic of the variety of maximal tori in a reductive group. https://arxiv.org/abs/2011.14613

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