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Adams' operation and the symbol for the normed residue

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Abstract

We study the possibility of a meaningful extension of the ring of cohomologies of a point using algebraic extensions of local fields. It appears that complex K-theory is a good model problem for this. We prove that the effect of the Adams operation on the cohomologies of a point in extended K-theory coincides with the symbol for the local Artin reciprocity law. K-theories contained in a fixed K-theory with extended ring of cohomologies of a point are defined by higher branching groups.

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Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 433–441, October, 1972.

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Golo, V.L. Adams' operation and the symbol for the normed residue. Mathematical Notes of the Academy of Sciences of the USSR 12, 693–698 (1972). https://doi.org/10.1007/BF01093676

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Keywords

  • Local Field
  • Model Problem
  • Algebraic Extension
  • Normed Residue
  • Meaningful Extension