On the Tate-Shafarevich groups of finite modules over $n$-dimensional pseudolocal fields |
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| Author |
Faculty of Mechanics and Mathematics, Lviv National University
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| Abstract |
Let $k$ be an $n$-dimensional local field over a pseudofinite residue
field $k_0$, $p$ be a prime number, $p \ne char k_0,$
and $X$ be a complete, smooth, absolutely irreducible curve over
$k$ with successive good reductions. Suppose that $1 \le n \le 3,$
and that $k$ contains the group $\mu_p,$ of $p$-th roots of unity.
Let $K$ be the function field on $X.$
It is proved that the Tate-Shafarevich groups $Ш^{n+2} (\mu_p)$
and $Ш^1(\mu_p)$ are dual one another.
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| Keywords |
n-dimensional local field, pseudofinite residue field, successive good reductions, function field
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| DOI |
doi:10.30970/ms.17.1.109-112
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Reference |
1. Ax J. The elementary theory of finite field, Ann. Math. 88 (1968), № 2, 239–271.
2. Bloch S. Algebraic $K$-theory and class-field theory for arithmetic surfaces, Ann. Math. 114 (1981), 229--265. 3. Andriychuk V. I. Algebraic curves over $n$-dimensional general local fields, Математичні студії, 15 (2001), № 2, 209--214. 4. Douai J.-C. Le théorème de Tate-Poitou pour les corps de fonctions definies sur les corps locaux de dimension $N$, J. Algebra 125 (1989), № 1, 181--196. 5. Милн Дж. Этальные когомологии, М.: Мир, 1983, 392 c. |
| Pages |
109-112
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| Volume |
17
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| Issue |
1
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| Year |
2002
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| Journal |
Matematychni Studii
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