Rationally isotropic quadratic spaces are locally isotropic. III

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Rationally isotropic quadratic spaces are locally isotropic. III. / Panin, I.; Pimenov, K.

In: АЛГЕБРА И АНАЛИЗ, Vol. 27, No. 6, 2015, p. 234-241.

Research output: Contribution to journal › Article

BibTeX

@article{efc2c9afc85e4101a1012f43841de9d7,

title = "Rationally isotropic quadratic spaces are locally isotropic. III",

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 (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. ",

keywords = "QUADRATIC SPACES, Quadratic form, Regular local ring, Isotropic vector, Grothendieck–Serre",

author = "I. Panin and K. Pimenov",

note = "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",

year = "2015",

language = "English",

volume = "27",

pages = "234--241",

journal = "АЛГЕБРА И АНАЛИЗ",

issn = "0234-0852",

publisher = "Издательство {"}Наука{"}",

number = "6",

}

RIS

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 -

ID: 5808280