Skip to main content Accessibility help
×
Home

Central simple algebras of prime exponent and divided power operations

  • A.S. Sivatski (a1)

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*. D2 Br(F), indD = 8 and aF* 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.

Copyright

References

Hide All
ART.Amitsur, S.A., Rowen, L.H., Tignol, J.-P., Division algebras of degree 4 and 8 with involution. Israel J. Math. 33 (1979), 133148.
BM.Baek, S., Merkurjev, A.S., Invariants of simple algebras. Manuscripta Math. 129 (2009) 409421.
EL.Elman, R., Lam, T.Y., Quadratic forms under algebraic extensions. Math. Ann. 219 (1976), 2142.
ELW.Elman, R., Lam, T.Y., Wadsworth, A.R., Amenable fields and Pfister extensions. Queen's Papers Pure Appl. Math. 46 (1976), 445492.
GS.Gille, P., Szamuely, T., Central simple algebras and Galois cohomology. Cambridge studies in advanced mathematics 101, 2006.
K.Kahn, B., Comparison of some field invariants. J. Algebra 220(2) (2000), 485492.
Ka.Karpenko, N.A., Codimension 2 cycles on Severi-Brauer varieties. K-Theory 13 (1998), 305330.
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), 10111046.
P.Pierce, R., Associative algebras. Graduates texts in mathematics 88, Springer-Verlag New York Inc. 1982.
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), 8387.
S1.Sivatski, A.S., On indecomposable algebras of exponent 2. Israel J. Math 164 (2008), 365379.
S2.Sivatski, A.S., Nonexcellence of certain field extensions. Journal of Mathematical Sciences 145(1) (2007), 48114817.
Vi.Vial, C., Operations in Milnor K-theory J. Pure Appl. Algebra 213(7) (2009), 13251345.
V.Voevodsky, V., On motivic cohomology with ℤ/l- coefficients. Ann. of Math. 174 (2011), 401438.
W.Weibel, C., The norm residue isomorphism theorem. Journal of Topology 2(2) (2009), 346372.

Keywords

Related content

Powered by UNSILO

Central simple algebras of prime exponent and divided power operations

  • A.S. Sivatski (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.