TY - JOUR

T1 - Rationally isotropic quadratic spaces are locally isotropic. III

AU - Panin, I.

AU - Pimenov, K.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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 (Rn, q: Rn → R) be a quadratic space over R such that the quadric (q = 0) is smooth over R. If the quadratic space (Rn, q: Rn → R) over R is isotropic over K, then there is a unimodular vector v ∈ Rn such that q(v) = 0. If char(R) = 2, then in the case of even n the assumption on q is equivalent to the fact that q is a nonsingular quadratic space and in the case of odd n > 2 this assumption on q is equivalent to the fact that q is a semiregular quadratic space.

AB - 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 (Rn, q: Rn → R) be a quadratic space over R such that the quadric (q = 0) is smooth over R. If the quadratic space (Rn, q: Rn → R) over R is isotropic over K, then there is a unimodular vector v ∈ Rn such that q(v) = 0. If char(R) = 2, then in the case of even n the assumption on q is equivalent to the fact that q is a nonsingular quadratic space and in the case of odd n > 2 this assumption on q is equivalent to the fact that q is a semiregular quadratic space.

KW - Grothendieck-Serre conjecture

KW - Isotropic vector

KW - Quadratic form

KW - Regular local ring

UR - http://www.scopus.com/inward/record.url?scp=84999143170&partnerID=8YFLogxK

U2 - 10.1090/spmj/1433

DO - 10.1090/spmj/1433

M3 - Article

AN - SCOPUS:84999143170

VL - 27

SP - 1029

EP - 1034

JO - St. Petersburg Mathematical Journal

JF - St. Petersburg Mathematical Journal

SN - 1061-0022

IS - 6

ER -