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Essential Dimension, Symbol Length and $p$-rank

Part of: Linear algebraic groups and related topics Higher algebraic $K$-theory Division rings and semisimple Artin rings General commutative ring theory

Published online by Cambridge University Press:  04 February 2020

Adam Chapman  and
Kelly McKinnie
Adam Chapman
Affiliation:
Department of Computer Science, Tel-Hai College, Upper Galilee, 12208, Israel Email: adam1chapman@yahoo.com
Kelly McKinnie
Affiliation:
Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, USA Email: kelly.mckinnie@mso.umt.edu
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Abstract

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ 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.

Keywords

essential dimensionsymbol lengthp-rankfields of positive characteristicBrauer groupcentral simple algebraKato–Milne cohomology

MSC classification

Primary: 16K20: Finite-dimensional
Secondary: 13A35: Characteristic $p$ methods (Frobenius endomorphism) and reduction to characteristic $p$; tight closure 19D45: Higher symbols, Milnor $K$-theory 20G10: Cohomology theory

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 4 , December 2020 , pp. 882 - 890
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Copyright
© Canadian Mathematical Society 2020

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