Abstract:
Let $R$ be a regular semilocal domain containing a field such that all the residue fields are infinite. Let $K$ be the fraction field of $R$. Let $(R^n,q\colon R^n\to R)$ be a quadratic space over $R$ such that the quadric $\{q=0\}$ is smooth over $R$. If the quadratic space $(R^n,q\colon R^n\to R)$ over $R$ is isotropic over $K$, then there is a unimodular vector $v\in R^n$ such that $q(v)=0$. If $char(R)=2$, then in the case of even $n$ our assumption on $q$ is equivalent to the fact that $q$ is a nonsingular quadratic space and in the case of odd $n>2$ our assumption on $q$ is equivalent to the fact that $q$ is a semiregular quadratic space.
Keywords:
quadratic form, regular local ring, isotropic vector, Grothendieck–Serre conjecture.
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”.
This publication is cited in the following 4 articles:
I. A. Panin, D. N. Tyurin, “Pfister forms and a conjecture due to Colliot–Thélène in the mixed characteristic case”, Izv. Math., 88:5 (2024), 977–987
Kȩstutis Česnavičius, “Problems About Torsors over Regular Rings”, Acta Math Vietnam, 47:1 (2022), 39
I. Panin, “A Purity Theorem for Quadratic Spaces”, J Math Sci, 261:4 (2022), 567
I. Panin, “A purity theorem for quadratic spaces”, Algebra i teoriya chisel. 3, Zap. nauchn. sem. POMI, 490, POMI, SPb., 2020, 98–103