Continuity of K-theory: An example in equal characteristics

Dundas, Bjørn

doi:10.1090/S0002-9939-98-04382-2

Continuity of K-theory: An example in equal characteristics
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by Bjørn Ian Dundas PDF

Proc. Amer. Math. Soc. 126 (1998), 1287-1291 Request permission

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 $K(A)\to {\operatorname *{holim}}_{\overleftarrow {n}} K(A/\mathfrak {m}^{n})$ is $4$-connected after profinite completion for $A$ a complete discrete valuation ring with perfect residue field.

References

A. K. Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. MR 551009, DOI 10.1016/0040-9383(79)90018-1
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573
M. Bökstedt and I. Madsen, Algebraic $K$-theory of local number fields: the unramified case, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 28–57. MR 1368652
R. Keith Dennis and Michael R. Stein, $K_{2}$ of discrete valuation rings, Advances in Math. 18 (1975), no. 2, 182–238. MR 437620, DOI 10.1016/0001-8708(75)90157-7
B. I. Dundas, A model for the K-theory of complete extensions, In preparation.
Ofer Gabber, $K$-theory of Henselian local rings and Henselian pairs, Algebraic $K$-theory, commutative algebra, and algebraic geometry (Santa Margherita Ligure, 1989) Contemp. Math., vol. 126, Amer. Math. Soc., Providence, RI, 1992, pp. 59–70. MR 1156502, DOI 10.1090/conm/126/00509
S. M. Gersten, Some exact sequences in the higher $K$-theory of rings, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 211–243. MR 0354660
L. Hesselholt, Topological cyclic homology and local function fields, Aarhus Universitet, Preprint series (31) (December 1993).
Howard L. Hiller, $\lambda$-rings and algebraic $K$-theory, J. Pure Appl. Algebra 20 (1981), no. 3, 241–266. MR 604319, DOI 10.1016/0022-4049(81)90062-1
O. Izhboldin, On $p$-torsion in $K^M_*$ for fields of characteristic $p$, Algebraic $K$-theory, Adv. Soviet Math., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 129–144. MR 1124629
Ch. Kratzer, $\lambda$-structure en $K$-théorie algébrique, Comment. Math. Helv. 55 (1980), no. 2, 233–254 (French). MR 576604, DOI 10.1007/BF02566684
R. McCarthy, Relative algebraic K-theory and topological cyclic homology, To appear in Acta Math.
A. S. Merkur′ev and A. A. Suslin, The group $K_3$ for a field, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 522–545 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 541–565. MR 1072694
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
I. A. Panin, The Hurewicz theorem and $K$-theory of complete discrete valuation rings, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 763–775, 878 (Russian). MR 864175
Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
A. A. Suslin, Algebraic $K$-theory of fields, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 222–244. MR 934225
J. B. Wagoner, Delooping the continuous $K$-theory of a valuation ring, Pacific J. Math. 65 (1976), no. 2, 533–538. MR 444744