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arXiv · 1602.07739

The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields

Stephen Scully
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Abstract

Let $R$ be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over $R$ remain anisotropic after base change to any odd-degree finite \'{e}tale extension of $R$. This generalization of the classical Artin-Springer theorem (concerning the situation where $R$ is a field) was previously established in the case where all residue fields of $R$ are infinite by I. Panin and U. Rehmann. The more general result presented here permits to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic $\neq 2$ to the case where the ring has at least one residue field which is finite.

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Stephen Scully. 2016-02-24. The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields. https://arxiv.org/abs/1602.07739

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