The Artin-Springer Theorem for quadratic forms over semi-local rings with finite residue fields
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- by Stephen Scully PDF
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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 étale 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 one 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.References
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Additional Information
- Stephen Scully
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton Alberta, Canada T6G 2G1
- MR Author ID: 1016999
- Email: stephenjscully@gmail.com
- Received by editor(s): February 26, 2016
- Published electronically: October 5, 2017
- Additional Notes: The author was supported by a PIMS postdoctoral fellowship held at the University of Alberta.
- Communicated by: Matthew A. Papanikolas
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 1-13
- MSC (2010): Primary 11E81; Secondary 11E08
- DOI: https://doi.org/10.1090/proc/13744
- MathSciNet review: 3723116