Correspondences between Valued Division Algebras and Graded Division Algebras

arwadsworth@ucsd.edu, yhwang@semi.korea.ac.kr

Submission: 1998, Aug. 22

If $D$ is a tame central division algebra over a Henselian valued field $F$, then the valuation on $D$ yields an associated graded ring $GD$ which is a graded division ring and is also central and graded simple over $GF$. After proving some properties of graded central simple algebras over a graded field (including a cohomological characterization of its graded Brauer group), it is proved that the map $[D]\mapsto [GD]_g$ yields an index-preserving isomorphism from the tame part of the Brauer group of $F$ to the graded Brauer group of $GF$. This isomorphism is shown to be functorial with respect to field extensions and corestrictions, and using this it is shown that there is a correspondence between $F$-subalgebras of $D$ (with center tame over $F$) and graded $GF$-subalgebras of $GD$.

1991 Mathematics Subject Classification:

Keywords and Phrases:

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