Essential dimension, symbol length and p-rank

Adam Chapman, Kelly McKinnie

Research output: Contribution to journalArticlepeer-review

Abstract

We prove that the essential dimension of central simple algebras of degree pℓm and exponent pm over fields F containing a base-field k of characteristic p is at least ℓ + 1 when k is perfect. We do this by observing that the p-rank of F bounds the symbol length in Brpm (F) and that there exist indecomposable p-algebras of degree pℓm and exponent pm. We also prove that the symbol length of the Kato-Milne cohomology group Hnp+m1(F) is bounded from above by (nr) where r is the p-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

Original languageEnglish
Pages (from-to)882-890
Number of pages9
JournalCanadian Mathematical Bulletin
Volume63
Issue number4
DOIs
StatePublished - Dec 1 2020

Keywords

  • Brauer group
  • Central simple algebra
  • Essential dimension
  • Fields of positive characteristic
  • Kato-Milne cohomology
  • P-rank
  • Symbol length

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