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

Department of Mathematics, Southern University of Science and Technology, No. 1088, Xueyuan Blvd., Nanshan District, 518055 Shenzhen, Guangdong, P. R. China

E-mail Address: huy@sustech.edu.cn

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https://doi.org/10.1142/S0219498820502175Cited by:1 (Source: Crossref)

Abstract

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$ . Let $D$ be a central division $F$ -algebra of $p$ -power degree. We prove that the subgroup of $F^{*}$ consisting of reduced norms of $D$ is exactly the kernel of the cup product map $λ \in F^{*} \mapsto (D) \cup (λ) \in H^{3} (F, ℚ_{p} / ℤ_{p} (2))$ , if either $D$ is tamely ramified or of period $p$ . This gives a $p$ -torsion counterpart of a recent theorem of Parimala, Preeti and Suresh, where the same result is proved for division algebras of prime-to- $p$ degree.

Communicated by L. Rowen

Keywords:

AMSC: 11E72, 17A35, 11R52, 16K50

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