Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Continuity of K-theory:
An example in equal characteristics


Author: Bjørn Ian Dundas
Journal: Proc. Amer. Math. Soc. 126 (1998), 1287-1291
MSC (1991): Primary 11S70; Secondary 13J05, 19D45, 19D50
DOI: https://doi.org/10.1090/S0002-9939-98-04382-2
MathSciNet review: 1452802
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If $k$ is a perfect field of characteristic $p>0$, we show that the Quillen K-groups $K_{i}(k[[t]])$ are uniquely $p$-divisible for $i=2,3$. In fact, the Milnor K-groups $K^{M}_{n}(k((t)))$ are uniquely $p$-divisible for all $n>1$. This implies that $K(A)\to {\operatornamewithlimits{holim}}_{\overleftarrow {n}} K(A/\mathfrak{m}^{n})$ is $4$-connected after profinite completion for $A$ a complete discrete valuation ring with perfect residue field.


References [Enhancements On Off] (What's this?)

  • [B] A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), 257-281. MR 80m:55006
  • [BK] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Springer Lecture Notes in Math., vol. 304, 1972. MR 51:1825
  • [BM] M. Bökstedt and I. Madsen, Algebraic K-theory of local number fields: the unramified case, Ann. of Math. Stud. Princeton University Press 138 (1995), 28-57. MR 97e:19004
  • [DS] R. K. Dennis and M. R. Stein, $K_{2}$ of discrete valuation rings, Adv. in Math. 18 (1975), 182-238. MR 55:10544
  • [D] B. I. Dundas, A model for the K-theory of complete extensions, In preparation.
  • [Ga] O. Gabber, K-theory of Henselian local rings and Henselian pairs, Contemp. Math. 126 (1992), 59-70. MR 93c:19005
  • [Ge] S. M. Gersten, Some exact sequences in the higher K-theory of rings, Springer Lecture Notes in Math. 341 (1972), 211-243. MR 50:7138
  • [He] L. Hesselholt, Topological cyclic homology and local function fields, Aarhus Universitet, Preprint series (31) (December 1993).
  • [Hi] H. L. Hiller, $\lambda $-rings and algebraic K-theory, J. Pure Appl. Alg. 20 (1981), 241-266. MR 82e:18016
  • [I] O. Izhboldin, On $p$-torsion in $K^{M}_{*}$ for fields of characteristic $p$, Adv. Soviet Math. 4, 129-144. MR 92f:11165
  • [K] C. Kratzer, $\lambda $-structure en K-théorie algébrique, Comment. Math. Helv 55 (1980), 233-254. MR 81m:18011
  • [Mc] R. McCarthy, Relative algebraic K-theory and topological cyclic homology, To appear in Acta Math.
  • [MS] A. S. Merkur'ev and A. A. Suslin, The group $K_{3}$ for a field, Math. USSR Izv. 36 (1991), 541-565. MR 91g:19002
  • [M] J. Milnor, Introduction to algebraic K-theory, Ann. of Math. Stud., vol. 72, Princeton University Press, 1971. MR 50:2304
  • [P] I. A. Panin, On a theorem of Hurewicz and K-theory of complete discrete valuation rings, Math. USSR Izv. 29 (1987), 81-99. MR 88a:18021
  • [Se] J. P. Serre, Corps locaux, Actualités scientifiques et industrielles. 1296. Publications de l'Institut de Mathématique de l'Université de Nacango VIII, Hermann, Paris 1968. MR 50:7096
  • [Su] A. A. Suslin, Algebraic K-theory of fields, Proc. Int. Congr. Math., Berkeley/Calif. 1986 1 (1987), 222-244. MR 89k:12010
  • [W] J. B. Wagoner, Delooping the continuous K-theory of a valuation ring, Pacific J. Math. 65 (1976), 533-538. MR 56:3093

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 11S70, 13J05, 19D45, 19D50

Retrieve articles in all journals with MSC (1991): 11S70, 13J05, 19D45, 19D50


Additional Information

Bjørn Ian Dundas
Affiliation: Department of Mathematical Sciences, Section Gløshaugen, The Norwegian University of Science and Technology, N-7034 Trondheim, Norway
Email: dundas@math.ntnu.no

DOI: https://doi.org/10.1090/S0002-9939-98-04382-2
Keywords: Continuity of K-theory, complete discrete valuation ring, ring of formal power series, Milnor K-theory
Received by editor(s): October 17, 1996
Additional Notes: The author was supported by the Danish research academy.
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society