Essential dimension of simple algebras in positive characteristic

Baek, Sanghoon

doi:10.1016/j.crma.2011.03.014

Algebra

Essential dimension of simple algebras in positive characteristic
[Dimension essentielle des algèbres simples en caractéristique positive]

Présenté par : the Editorial Board

Baek, Sanghoon ¹

¹ Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada

Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 375-378.

Résumé
Abstract

Soit p un nombre premier. Pour toutes nombres entiers $1 ⩽ s ⩽ r$ , on note $_{p^{r}, p^{s}}$ la classe des algèbres simples centrales de degré $p^{r}$ et dʼexposant au plus $p^{s}$ . Pour tous $s < r$ , nous trouvons une borne inférieure pour la p-dimension essentielle de $_{p^{r}, p^{s}}$ . De plus, nous calculons une borne supérieure pour $_{8, 2}$ sur un corps de caractéristique 2. En conséquence, on montre que ${ed}_{2} (_{4, 2}) = ed (_{4, 2}) = 3$ et $3 ⩽ ed (_{8, 2}) ⩽ 10$ sur un corps de caractéristique 2.

Let p be a prime integer. For any integers $1 ⩽ s ⩽ r$ , $_{p^{r}, p^{s}}$ denotes the class of central simple algebras of degree $p^{r}$ and exponent dividing $p^{s}$ . For any $s < r$ , we find a lower bound for the essential p-dimension of $_{p^{r}, p^{s}}$ . Furthermore, we compute an upper bound for $_{8, 2}$ over a field of characteristic 2. As a result, we show ${ed}_{2} (_{4, 2}) = ed (_{4, 2}) = 3$ and $3 ⩽ ed (_{8, 2}) ⩽ 10$ over a field of characteristic 2.

Reçu le : 2011-01-18
Accepté le : 2011-03-15
Publié le : 2011-04-01

DOI : 10.1016/j.crma.2011.03.014

Affiliations des auteurs :

Baek, Sanghoon ¹

¹ Department of Mathematics and Statistics, University of Ottawa, 585 King Edward, Ottawa, ON K1N6N5, Canada

@article{CRMATH_2011__349_7-8_375_0,
     author = {Baek, Sanghoon},
     title = {Essential dimension of simple algebras in positive characteristic},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {375--378},
     publisher = {Elsevier},
     volume = {349},
     number = {7-8},
     year = {2011},
     doi = {10.1016/j.crma.2011.03.014},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2011.03.014/}
}

TY  - JOUR
AU  - Baek, Sanghoon
TI  - Essential dimension of simple algebras in positive characteristic
JO  - Comptes Rendus. Mathématique
PY  - 2011
SP  - 375
EP  - 378
VL  - 349
IS  - 7-8
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2011.03.014/
DO  - 10.1016/j.crma.2011.03.014
LA  - en
ID  - CRMATH_2011__349_7-8_375_0
ER  -

%0 Journal Article
%A Baek, Sanghoon
%T Essential dimension of simple algebras in positive characteristic
%J Comptes Rendus. Mathématique
%D 2011
%P 375-378
%V 349
%N 7-8
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2011.03.014/
%R 10.1016/j.crma.2011.03.014
%G en
%F CRMATH_2011__349_7-8_375_0

Baek, Sanghoon. Essential dimension of simple algebras in positive characteristic. Comptes Rendus. Mathématique, Tome 349 (2011) no. 7-8, pp. 375-378. doi : 10.1016/j.crma.2011.03.014. http://www.numdam.org/articles/10.1016/j.crma.2011.03.014/

Bibliographie
Cité par

[1] S. Baek, Essential dimension of simple algebras with involutions, preprint, . | arXiv

[2] Baek, S.; Merkurjev, A. Invariants of simple algebras, Manuscripta Math., Volume 129 (2009) no. 4, pp. 409-421

[3] S. Baek, A. Merkurjev, Essential dimension of central simple algebras, Acta Math., in press.

[4] Berhuy, G.; Favi, G. Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math., Volume 8 (2003), pp. 279-330

[5] de Jong, A.J. The period-index problem for the Brauer group of an algebraic surface, Duke Math. J., Volume 123 (2004), pp. 71-94

[6] Garibaldi, R.; Merkurjev, A.; Serre, J.-P. Cohomological Invariants in Galois Cohomology, American Mathematical Society, Providence, RI, 2003

[7] Herstein, I.N. Noncommutative Rings, Mathematical Association of America, Washington, DC, 1994

[8] Kanzaki, T. Note on quaternion algebras over a commutative ring, Osaka J. Math., Volume 13 (1976) no. 3, pp. 503-512

[9] Lieblich, M. Twisted sheaves and the period-index problem, Compos. Math., Volume 144 (2008) no. 1, pp. 1-31

[10] Merkurjev, A.S. Essential dimension, Quadratic Forms—Algebra, Arithmetic, and Geometry, Contemp. Math., vol. 493, American Mathematical Society, Providence, RI, 2009, pp. 299-325

[11] Merkurjev, A.S. Essential p-dimension of $PGL (p^{2})$ , J. Amer. Math. Soc., Volume 23 (2010), pp. 693-712

[12] Pierce, R.S. Associative Algebras, Springer-Verlag, New York, 1982

[13] Reichstein, Z. On the notion of essential dimension for algebraic groups, Transform. Groups, Volume 5 (2000) no. 3, pp. 265-304

[14] Rouzzi, A. Essential p-dimension of ${PGL}_{n}$ , J. Algebra, Volume 328 (2011) no. 1, pp. 488-494

[15] Rowen, L. Division algebras of exponent 2 and characteristic 2, J. Algebra, Volume 90 (1984) no. 1, pp. 71-83

[16] Saltman, D.J. Lectures on Division Algebras, American Mathematical Society, Providence, RI, 1999

[17] Tignol, J.-P. Sur les classes de similitude de corps à involution de degré 8, C. R. Acad. Sci. Paris Sér. A–B, Volume 286 (1978) no. 20, p. A875-A876

Cité par Sources :