Reduced norms of division algebras over complete discrete valuation fields of local-global type

Abstract

Let [Formula: see text] be a complete discrete valuation field whose residue field [Formula: see text] is a global field of positive characteristic [Formula: see text]. Let [Formula: see text] be a central division [Formula: see text]-algebra of [Formula: see text]-power degree. We prove that the subgroup of [Formula: see text] consisting of reduced norms of [Formula: see text] is exactly the kernel of the cup product map [Formula: see text], if either [Formula: see text] is tamely ramified or of period [Formula: see text]. This gives a [Formula: see text]-torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to-[Formula: see text] degree.