TY - JOUR

T1 - Rationally isotropic quadratic spaces are locally isotropic. III

AU - Panin, I.

AU - Pimenov, K.

N1 - I. Panin, K. Pimenov, “Rationally isotropic quadratic spaces are locally isotropic. III”, Алгебра и анализ, 27:6 (2015), 234–241; St. Petersburg Math. J., 27:6 (2016), 1029–1034

PY - 2015

Y1 - 2015

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 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.

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 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.

KW - QUADRATIC SPACES

KW - Quadratic form

KW - Regular local ring

KW - Isotropic vector

KW - Grothendieck–Serre

UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=aa&paperid=1474&option_lang=rus

M3 - Article

VL - 27

SP - 234

EP - 241

JO - АЛГЕБРА И АНАЛИЗ

JF - АЛГЕБРА И АНАЛИЗ

SN - 0234-0852

IS - 6

ER -