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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S. Leng, Algebraic Number Theory, Addison-Wesley (1970).
V. Bukhshtaber, “Representative spaces for a K-functor with coefficients,” AN SSSR,186, No. 3, 499–502 (1969).
S. Leng, Algebra, Addison-Wesley (1965).
M. At'ya (Atiya), “Characters and cohomology of finite groups,” Publ. Math. Inst. des Hautes Etudes,9, 23–64 (1961).
M. At'ya, K-Theory [in Russian], Moscow (1968).
J. Lubin and J. Tate, “Formal complex multiplication in local fields,” Ann. Math.,81, 380–387 (1965).
G. H. Whitehead, “Generalized cohomology theories,” Trans. Amer. Math. Soc,102, 227–283 (1962).
D. Sullivan, “Geometric topology, Part I,” Massachusetts Inst. of Techn., Preprint (1970).
J. P. Serre, Corps Locaux, Hermann, Paris (1968).
V. Bukhshtaber and A. Mishchenko, “The K-theory of infinite complexes,” Izv. Akad. Nauk SSSR, Ser. Matem.,32, No. 3, 560–604 (1968).
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
- Local Field
- Model Problem
- Algebraic Extension
- Normed Residue
- Meaningful Extension