ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$
In this article the $p$ -essential dimension of generic symbols over fields of characteristic $p$ is stud...
| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2017-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509417000111/type/journal_article |
| _version_ | 1827994866472189952 |
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| author | KELLY MCKINNIE |
| author_facet | KELLY MCKINNIE |
| author_sort | KELLY MCKINNIE |
| collection | DOAJ |
| description | In this article the
$p$
-essential dimension of generic symbols over fields of characteristic
$p$
is studied. In particular, the
$p$
-essential dimension of the length
$\ell$
generic
$p$
-symbol of degree
$n+1$
is bounded below by
$n+\ell$
when the base field is algebraically closed of characteristic
$p$
. The proof uses new techniques for working with residues in Milne–Kato
$p$
-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on
$p$
-symbol algebras (i.e, degree 2 symbols) result from this work. The generic
$p$
-symbol algebra of length
$\ell$
is shown to have
$p$
-essential dimension equal to
$\ell +1$
as a
$p$
-torsion Brauer class. The second is a lower bound of
$\ell +1$
on the
$p$
-essential dimension of the functor
$\operatorname{Alg}_{p^{\ell },p}$
. Roughly speaking this says that you will need at least
$\ell +1$
independent parameters to be able to specify any given algebra of degree
$p^{\ell }$
and exponent
$p$
over a field of characteristic
$p$
and improves on the previously established lower bound of 3. |
| first_indexed | 2024-04-10T04:47:35Z |
| format | Article |
| id | doaj.art-d066fd901e904b16989f3025d7486f80 |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:35Z |
| publishDate | 2017-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-d066fd901e904b16989f3025d7486f802023-03-09T12:34:44ZengCambridge University PressForum of Mathematics, Sigma2050-50942017-01-01510.1017/fms.2017.11ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$KELLY MCKINNIE0Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, USA;In this article the $p$ -essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$ -essential dimension of the length $\ell$ generic $p$ -symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$ . The proof uses new techniques for working with residues in Milne–Kato $p$ -cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$ -symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$ -symbol algebra of length $\ell$ is shown to have $p$ -essential dimension equal to $\ell +1$ as a $p$ -torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$ -essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$ . Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.https://www.cambridge.org/core/product/identifier/S2050509417000111/type/journal_article16K20 (primary)20G1013A3513A18 (secondary) |
| spellingShingle | KELLY MCKINNIE ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ Forum of Mathematics, Sigma 16K20 (primary) 20G10 13A35 13A18 (secondary) |
| title | ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ |
| title_full | ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ |
| title_fullStr | ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ |
| title_full_unstemmed | ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ |
| title_short | ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC $p$ |
| title_sort | essential dimension of generic symbols in characteristic p |
| topic | 16K20 (primary) 20G10 13A35 13A18 (secondary) |
| url | https://www.cambridge.org/core/product/identifier/S2050509417000111/type/journal_article |
| work_keys_str_mv | AT kellymckinnie essentialdimensionofgenericsymbolsincharacteristicp |