Rationally isotropic quadratic spaces are locally isotropic. III

Authors:
I. Panin and K. Pimenov

Original publication:
Algebra i Analiz, tom **27** (2015), nomer 6.

Journal:
St. Petersburg Math. J. **27** (2016), 1029-1034

MSC (2010):
Primary 13H05

DOI:
https://doi.org/10.1090/spmj/1433

Published electronically:
September 30, 2016

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a regular semilocal domain containing a field such that all the residue fields are infinite. Let be the fraction field of . Let be a quadratic space over such that the quadric is smooth over . If the quadratic space over is isotropic over , then there is a unimodular vector such that . If , then in the case of even the assumption on is equivalent to the fact that is a nonsingular quadratic space and in the case of odd this assumption on is equivalent to the fact that is a semiregular quadratic space.

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Additional Information

**I. Panin**

Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

Email:
panin@pdmi.ras.ru

**K. Pimenov**

Affiliation:
Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskiĭ pr. 28, Petergof, 198504 St. Petersburg, Russia

Email:
kip302002@yahoo.com

DOI:
https://doi.org/10.1090/spmj/1433

Keywords:
Quadratic form,
regular local ring,
isotropic vector,
Grothendieck--Serre conjecture

Received by editor(s):
June 15, 2015

Published electronically:
September 30, 2016

Additional Notes:
Theorem 3 was proved with the support of the Russian Science Foundation (grant no. 14-11-00456). The research of the second author was partially supported by RFBR grant 12-01-33057 “Motivic homotopic cohomology theories on algebraic varieties” and by RFBR grant 13-01-00429 “Cohomological, classical, and motivic approach to algebraic numbers and algebraic varieties”

Dedicated:
Dedicated to Professor S. V. Vostokov with great respect

Article copyright:
© Copyright 2016
American Mathematical Society