On wild division algebras over fields of power series

Abstract

In this paper we study some special classes of division algebras over a Laurent series field with arbitrary residue field. We call the algebras from these classes as splittable and good splittable division algebras. It is shown that theses classes contain the group of tame division algebras. For the class of good division algebras a decomposition theorem is given. This theorem is a generalization of the decomposition theorems for tame division algebras given by Jacob and Wadsworth in [6]. For both clases we introduce a notion of a δ -map and develop a technique of δ -maps for division algebras from these classes. Using this technique we reprove several old well known results of Saltman and get the positive answer on the period-index conjecture of M.Artin: the exponent of A is equal to its index for any division algebra A over a C 2 -field F , when F = F 1 ((t 2 )) , where F 1 is a C 1 -field (see [10], 3.4.5.). The paper includes also some other results about splittable division algebras, which, we hope, will be useful for the further investigation of wild division algebras.

On the Brauer $p$-dimension of Henselian discrete valued fields of residual characteristic $p > 0$

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Representations of Higher Adelic Groups and Arithmetic

“…Vostokov and his school, see surveys [17,16,28,66] • adelic resolutions for sheaves, intersection theory, Chern classes, Lefschetz formula for coherent sheaves [55,76,26,27,21,22] • algebraic groups over local fields, buildings, Hecke algebras [56,60,34,18,19,20,7] • restricted adelic complexes and the Krichever correspondence [59,46,48,40,41] • relations with non-commutative algebra [57,78].…”