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Central simple algebras of prime exponent and divided power operations

Published online by Cambridge University Press:  15 January 2013

A.S. Sivatski
A.S. Sivatski*
Affiliation:
Departamento de Matemática, Universidade Federal do Ceará, Fortaleza, Brasilalexander.sivatski@gmail.com
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Abstract

Let p be a prime and F a field of characteristic different from p. Suppose all p-primary roots of unity are contained in F. Let α ∈ pBr(F) which has a cyclic splitting field. We prove that γi(α) = 0 for all i ≥ 2, where γi : pBr(F) → K2i(F)/pK2i(F) are the divided power operations of degree p. We also show that if char F ≠ 2, √−1 ∈ F*. D ∈2 Br(F), indD = 8 and a ∈ F* such that ind DF(√a) = 4, then γ3(D) = {a,s}γ2(D) for some s ∈ F*. Consequently, we prove that if D, considered as a division algebra, has a subfield of degree 4 of certain type, then γ3(D) = 0. At the end of the paper we pose a few open questions.

Keywords

Cyclic algebrasMilnor K-groupsDivided power operationsNorm residue isomorphism16K5019D45
Type
Research Article
Information
Journal of K-Theory , Volume 11 , Issue 1 , February 2013 , pp. 113 - 123
Copyright
Copyright © ISOPP 2013

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References

ART.Amitsur, S.A., Rowen, L.H., Tignol, J.-P., Division algebras of degree 4 and 8 with involution. Israel J. Math. 33 (1979), 133–148.Google Scholar
BM.Baek, S., Merkurjev, A.S., Invariants of simple algebras. Manuscripta Math. 129 (2009) 409–421.Google Scholar
EL.Elman, R., Lam, T.Y., Quadratic forms under algebraic extensions. Math. Ann. 219 (1976), 21–42.Google Scholar
ELW.Elman, R., Lam, T.Y., Wadsworth, A.R., Amenable fields and Pfister extensions. Queen's Papers Pure Appl. Math. 46 (1976), 445–492.Google Scholar
GS.Gille, P., Szamuely, T., Central simple algebras and Galois cohomology. Cambridge studies in advanced mathematics 101, 2006.Google Scholar
K.Kahn, B., Comparison of some field invariants. J. Algebra 220(2) (2000), 485–492.CrossRefGoogle Scholar
Ka.Karpenko, N.A., Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), 305–330.Google Scholar
MS.Merkurjev, A.S., Suslin, A.A., K-cohomology of Severi-Brauer varieties and the norm residue homomorphism (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046.Google Scholar
P.Pierce, R., Associative algebras. Graduates texts in mathematics 88, Springer-Verlag New York Inc. 1982.Google Scholar
RST.Rost, M., Serre, J.-P., Tignol, J.-P., La forme trace d'une algebre simple centrale de degre 4. C.R. Acad. Sci. Paris, Ser. I 342(2) (2006), 83–87.Google Scholar
S1.Sivatski, A.S., On indecomposable algebras of exponent 2. Israel J. Math 164 (2008), 365–379.CrossRefGoogle Scholar
S2.Sivatski, A.S., Nonexcellence of certain field extensions. Journal of Mathematical Sciences 145(1) (2007), 4811–4817.CrossRefGoogle Scholar
Vi.Vial, C., Operations in Milnor K-theory J. Pure Appl. Algebra 213(7) (2009), 1325–1345.Google Scholar
V.Voevodsky, V., On motivic cohomology with ℤ/l- coefficients. Ann. of Math. 174 (2011), 401–438.Google Scholar
W.Weibel, C., The norm residue isomorphism theorem. Journal of Topology 2(2) (2009), 346–372.CrossRefGoogle Scholar