Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-04T19:11:48.216Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuity of the norm map on Milnor K-theory

Published online by Cambridge University Press:  21 November 2011

Matthew Morrow
Matthew Morrow
Affiliation:
University of Chicago, 5734 S. University Ave., Chicago, IL, 60637, [email protected]://math.uchicago.edu/~mmorrow/
Get access

Abstract

The norm map on the Milnor K-groups of a finite extension of complete, discrete valuation fields is continuous with respect to the unit group filtrations. The only proof in the literature, due to K. Kato, uses semi-global methods. Here we present an elementary algebraic proof.

Keywords

Milnor K-theorynorm maphigher local class field theory
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 565 - 577
Copyright
Copyright © ISOPP 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Bass, H., and Tate, J. The Milnor ring of a global field. In Algebraic K-theory, II: “Classical” algebraic K-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972). Lecture Notes in Math. 342. Springer, Berlin, 1973, 349–446.Google Scholar
2.Fesenko, I. B., and Vostokov, S. V. Local fields and their extensions, second ed., Translations of Mathematical Monographs 121. American Mathematical Society, Providence, RI, 2002. With a foreword by Shafarevich, I. R..Google Scholar
3.Fukaya, T.Explicit reciprocity laws for p-divisible groups over higher dimensional local fields. J. Reine Angew. Math. 531, (2001), 61–119.Google Scholar
4.Gille, P., and Szamuely, T.Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics 101. Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
5.Kato, K.A generalization of local class field theory by using K-groups. I. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26, 2 (1979), 303–376.Google Scholar
6.Kato, K.A generalization of local class field theory by using K-groups. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27, 3 (1980), 603–683.Google Scholar
7.Kato, K.A generalization of local class field theory by using K-groups. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29, 1 (1982), 31–43.Google Scholar
8.Kato, K. Residue homomorphisms in Milnor K-theory. In Galois groups and their representations (Nagoya, 1981), Adv. Stud. Pure Math 2. North-Holland, Amsterdam, 1983, pp. 153–172.Google Scholar
9.Kato, K. Generalization of class field theory [translation of Sūgaku 40 (1988), no. 4, 289–311; MR1002305 (90g:11080)]. In Selected papers on number theory and algebraic geometry, Amer. Math. Soc. Transl. 172. Ser. 2. Amer. Math. Soc., Providence, RI, 1996, pp. 31–59.Google Scholar
10.Kurihara, M.The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law. J. Reine Angew. Math. 498, (1998), 201–221.CrossRefGoogle Scholar
11.Merkurjev, A. S. On the torsion in K 2 of local fields. Ann. of Math. (2) 118, 2 (1983), 375–381.CrossRefGoogle Scholar
12.Milnor, J. Algebraic K-theory and quadratic forms. Invent. Math. 9, (1970), 318–344.CrossRefGoogle Scholar
13.Rosset, S., and Tate, J. A reciprocity law for K 2-traces. Comment. Math. Helv. 58, 1 (1983), 38–47.CrossRefGoogle Scholar
14.Suslin, A. A. Mennicke symbols and their applications in the K-theory of fields. In Algebraic K-theory, Part I (Oberwolfach, 1980), Lecture Notes in Math. 966. Springer, Berlin, 1982, pp. 334–356.Google Scholar
15.Tate, J. On the torsion in K 2 of fields. In Algebraic number theory (Kyoto Internat. Sympos., Res. Inst. Math. Sci., Univ. Kyoto, Kyoto, 1976). Japan Soc. Promotion Sci., Tokyo, 1977, pp. 243–261.Google Scholar