Cohomologie non ramifi\'ee en degr\'e trois d'une vari\'et\'e de Severi-Brauer

TL;DR

This paper extends a known result about unramified cohomology in degree three from function fields of certain surfaces to Severi-Brauer varieties associated with central simple algebras of prime index over finite fields.

Contribution

It generalizes the vanishing of unramified cohomology in degree three to Severi-Brauer varieties linked to central simple algebras of prime index over finite fields.

Findings

Unramified cohomology in degree three vanishes for these Severi-Brauer varieties.

The result extends previous work on function fields of surfaces to more general algebraic varieties.

Provides new insights into the cohomological properties of Severi-Brauer varieties.

Abstract

Soit K le corps des fractions d'une surface projective et lisse, g\'eom\'etriquement int\`egre, d\'efinie sur un corps fini F. Soit C/K une conique. Parimala et Suresh ont montr\'e que le groupe de cohomologie non ramifi\'ee en degr\'e trois de K(C) \`a coefficients de torsion est nul. Dans cette note on \'etend leur r\'esultat aux vari\'et\'es de Severi-Brauer associ\'ees \`a une alg\`ebre centrale simple dont d'indice l est premier et diff\'erent de char F.