Adams' operation and the symbol for the normed residue

V. L. Golo

M. V. Lomonosov Moscow State University

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.

UDC: 513.83

Received: 10.03.1971

 English version: Mathematical Notes, 1972, 12:4, 693–698

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