Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dualit\'e et principe local-global sur des corps locaux de dimension 2   Let $k$ be an algebraically closed field, a finite field or a $p$-adic field.
Let $K_0=k((x,y))$ be the field of Laurent series in two variables over $k$. We
define Tate-Shafarevich groups of a commutative group scheme over $K_0$ via
cohomology classes locally trivial at each completion of $K_0$ coming from a
codimension 1 point of $\text{Spec}\; k[[x,y]]$. We establish duality theorems
between Tate-Shafarevich groups for finite groups schemes and for tori. We
apply these results to the study of the obstruction to the local-global
principle for $K_0$-torsors under a connected linear algebraic group, answering
in that way a question of Colliot-Th\'el\`ene, Parimala and Suresh, and to the
weak approximation for tori over $K_0$.
  Soit $k$ un corps alg\'ebriquement clos, un corps fini, ou encore un corps
$p$-adique. Soit $K_0=k((x,y))$ le corps des s\'eries de Laurent \`a deux
variables sur $k$. On d\'efinit les groupes de Tate-Shafarevich d'un
$K_0$-sch\'ema en groupes commutatif en consid\'erant les classes de
cohomologie qui deviennent triviales sur chaque compl\'et\'e de $K_0$ provenant
d'un point codimension 1 de $\text{Spec}\; k[[x,y]]$. On \'etablit des
th\'eor\`emes de dualit\'e arithm\'etique entre des groupes de Tate-Shafarevich
pour les modules finis et pour les tores. On applique ces r\'esultats \`a
l'\'etude du principe local-global pour les $K_0$-torseurs sous un groupe
lin\'eaire connexe, r\'epondant ainsi \`a une question de Colliot-Th\'el\`ene,
Parimala et Suresh, ainsi qu'\`a l'approximation faible pour les tores sur
$K_0$.

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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: funct-an
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The categories of this paper are: gr-qc