On the algebraic tori over some
function fields

V.I. Andriychuk

Using the fact that the global class field theory can be generalized to the case of an algebraic function field in one variable over a pseudofinite constant field (such a field is called pseudoglobal), we show that some theorems on Tate-Shafarevich groups of algebraic tori proved by J. Tate for tori over global fields hold also for tori over pseudoglobal fields. Besides it is proved that there exists an exact sequence (discovered by V. Voskresenskiy in the case of a global basic field) which connects the Tate-Shafarevich group of a torus defined over a pseudoglobal field and the obstruction to the property of weak approximation of this torus. Some results concerning $R$-equivalence on an algebraic torus over a global field are transferred to the case of a pseudoglobal basic field.

1. J. Ax The elementary theory of finite fields // Ann. Math. V. 88 № 2 – 1968 p. 239–271

2. D.S. Rim, G. Whaples Global norm-residue map over quasi-finite fields // Nagoya Math. J. V. 27 № 1 – 1966 p. 323–329

3. В.E. Воскресенский Алгебраические торы , M.:Наука – 1977

4. Алгебраическая теория чисел. (Под ред. Дж. Касселс, А. Фрелиха). ,addr М.: Мир – 1969

5. Серр Ж.-П. Когомологии Галуа , М.:Мир – 1968

6. В.П. Платонов, А.С. Рапинчук Алгебраические группы и теория чисел , M.:Наука – 1991

7. А. Картан, С. Эйленберг Гомологическая алгебра , М.:ИЛ – 1960

8. Ю.И. Манин Кубические формы , M.:Наука – 1972

9. J. L. Colliot-Thelen, J. J. Sansuc La $R$-equivalence sur les tores// Ann. Scient. Ec. Norm. Sup. V. 4 № 10 -- 1977 p. 175--229

	On the algebraic tori over some function fields
Author	V.I. Andriychuk Department of Mechanics and Mathematics, Lviv University,Universytetska 1, Lviv, 290602, Ukraine.
Abstract	Using the fact that the global class field theory can be generalized to the case of an algebraic function field in one variable over a pseudofinite constant field (such a field is called pseudoglobal), we show that some theorems on Tate-Shafarevich groups of algebraic tori proved by J. Tate for tori over global fields hold also for tori over pseudoglobal fields. Besides it is proved that there exists an exact sequence (discovered by V. Voskresenskiy in the case of a global basic field) which connects the Tate-Shafarevich group of a torus defined over a pseudoglobal field and the obstruction to the property of weak approximation of this torus. Some results concerning $R$-equivalence on an algebraic torus over a global field are transferred to the case of a pseudoglobal basic field.
Keywords	global class field theory, algebraic function field, pseudofinite field, pseudoglobal field, Tate-Shafarevich groups, algebraic tori, exact sequence, weak approximation
DOI	doi:10.30970/ms.12.2.115-126
Reference	1. J. Ax The elementary theory of finite fields // Ann. Math. V. 88 № 2 – 1968 p. 239–271 2. D.S. Rim, G. Whaples Global norm-residue map over quasi-finite fields // Nagoya Math. J. V. 27 № 1 – 1966 p. 323–329 3. В.E. Воскресенский Алгебраические торы , M.:Наука – 1977 4. Алгебраическая теория чисел. (Под ред. Дж. Касселс, А. Фрелиха). ,addr М.: Мир – 1969 5. Серр Ж.-П. Когомологии Галуа , М.:Мир – 1968 6. В.П. Платонов, А.С. Рапинчук Алгебраические группы и теория чисел , M.:Наука – 1991 7. А. Картан, С. Эйленберг Гомологическая алгебра , М.:ИЛ – 1960 8. Ю.И. Манин Кубические формы , M.:Наука – 1972 9. J. L. Colliot-Thelen, J. J. Sansuc La $R$-equivalence sur les tores// Ann. Scient. Ec. Norm. Sup. V. 4 № 10 -- 1977 p. 175--229
Pages	115-126
Volume	12
Issue	2
Year	1999
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On the algebraic tori over some function fields