Essential Dimension, Symbol Length and $p$-rank

Chapman, Adam ; McKinnie, Kelly

Canadian mathematical bulletin, Tome 63 (2020) no. 4, pp. 882-890

Résumé

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.

Détail

DOI : 10.4153/S0008439520000119

Mots-clés : essential dimension, symbol length, p-rank, fields of positive characteristic, Brauer group, central simple algebra, Kato–Milne cohomology

Bibliographie

[AJ018] Aravire, R., Jacob, B., and O’Ryan, M., The de Rham Witt complex, cohomological kernels and p m-extensions in characteristic p. J. Pure Appl. Algebra 222(2018), 3891–3945. https://doi.org/10.1016/j.jpaa.2018.02.013 Google Scholar | DOI

[Alb68] Albert, A., Structure of algebras. Colloquium Publications, 24, American Mathematical Society, Providence, RI, 1968. Google Scholar

[Bae11] Baek, S., Essential dimension of simple algebras in positive characteristic. C. R. Math. Acad. Sci. Paris 349(2011), no. 7–8, 375–378. https://doi.org/10.1016/j.crma.2011.03.014 Google Scholar | DOI

[BM09] Baek, S. and Merkurjev, A., Invariants of simple algebras. Manuscripta Math. 129(2009), no. 4, 409–421. https://doi.org/10.1007/s00229-009-0265-4 Google Scholar | DOI

[BM12] Baek, S. and Merkurjev, A., Essential dimension of central simple algebras. Acta Math. 209(2012), no. 1, 1–27. https://doi.org/10.1007/s11511-012-0080-8 Google Scholar | DOI

[Bou90] Bourbaki, N., Algebra. II. Chapters 4–7. Elements of Mathematics (Berlin), Translated from the French by P. M. Cohn and J. Howie, Springer-Verlag, Berlin, 1990. Google Scholar

[CM18] Chapman, A. and Mckinnie, K., Kato–Milne cohomology and polynomial forms. J. Pure Appl. Algebra 222(2018), 3547–3559. https://doi.org/10.1016/j.jpaa.2017.12.022 Google Scholar | DOI

[CM19] Chapman, A. and Mckinnie, K., The u n-invariant and the symbol length of H n(F). Proc. Amer. Math. Soc. 147(2019), 513–521. https://doi.org/10.1090/proc/14308 Google Scholar | DOI

[Flo13] Florence, M., On the symbol length of p-algebras. Compos. Math. 149(2013), 1353–1363. https://doi.org/10.1112/S0010437X13007070 Google Scholar | DOI

[GS17] Gille, P. and Szamuely, T., Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics, 165, Cambridge University Press, Cambridge, 2017. Google Scholar | DOI

[Izh96] Izhboldin, O. T., On the cohomology groups of the field of rational functions. In: Mathematics in St. Petersburg, Amer. Math. Soc. Transl. Ser. 2, 174, American Mathematical Society, Providence, RI, 1996, pp. 21–44. https://doi.org/10.1090/trans2/174/03 Google Scholar

[IZh00] Izhboldin, O., p-primary part of the Milnor K-groups and Galois cohomologies of fields of characteristic p. In: Invitation to higher local fields (Münster, 1999), Geom. Topol. Monogr., 3, Geom. Topol. Publ., Coventry, 2000, pp. 19–41. https://doi.org/10.2140/gtm.2000.3.19 Google Scholar

[Jac89] Jacobson, N., Basic algebra. II. Second ed., W. H. Freeman and Company, New York, 1989. Google Scholar

[Kar95] Karpenko, N. A., Torsion in CH2of Severi–Brauer varieties and indecomposability of generic algebras. Manuscripta Math. 88(1995), 109–117. https://doi.org/10.1007/BF02567809 Google Scholar | DOI

[Kar98] Karpenko, N. A., Codimension 2 cycles on Severi–Brauer varieties. K-Theory 13(1998), 305–330. https://doi.org/10.1023/A:1007705720373 Google Scholar | DOI

[Led04] Ledet, A., On the essential dimension of p-groups. In: Galois theory and modular forms, Dev. Math., 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 159–172. https://doi.org/10.1007/978-1-4613-0249-0_8 Google Scholar | DOI

[Mat16] Matzri, E., Symbol length in the Brauer group of a field. Trans. Amer. Math. Soc. 368(2016), 413–427. https://doi.org/10.1090/tran/6326 Google Scholar | DOI

[McK08] Mckinnie, K., Indecomposable p-algebras and Galois subfields in generic abelian crossed products. J. Algebra 320(2008), 1887–1907. https://doi.org/10.1016/j.jalgebra.2008.05.028 Google Scholar | DOI

[McK17] Mckinnie, K., Essential dimension of generic symbols in characteristic p. Forum Math. Sigma 5(2017), e14, 30. https://doi.org/10.1017/fms.2017.11 Google Scholar | DOI

[Mer13] Merkurjev, A. S., Essential dimension: a survey. Transform. Groups 18(2013), 415–481. https://doi.org/10.1007/s00031-013-9216-y Google Scholar | DOI

[MM91] Mammone, P. and Merkurjev, A., On the corestriction of p n-symbol. Israel J. Math. 76(1991), 73–79. https://doi.org/10.1007/BF02782844 Google Scholar | DOI

[MTW91] Mammone, P., Tignol, J.-P., and Wadsworth, A., Fields of characteristic 2 with prescribed u-invariants. Math. Ann. 290(1991), 109–128. https://doi.org/10.1007/BF01459240 Google Scholar | DOI

[Row84] Rowen, L. H., Division algebras of exponent 2 and characteristic 2. J. Algebra 90(1984), 71–83. https://doi.org/10.1016/0021-8693(84)90199-6 Google Scholar | DOI