On the structure of Milnor K-groups of certain complete discrete valuation fields
Journal de Théorie des Nombres de Bordeaux, Tome 16 (2004) no. 2, pp. 377-401.

Pour un exemple typique de corps de valuation discrète complet K de type II au sens de [12], nous déterminons les quotients gradués gr i K 2 (K) pour tout i>0. Dans l’appendice, nous décrivons les K-groupes de Milnor d’un certain anneau local à l’aide de modules de différentielles, qui sont liés à la théorie de la cohomologie syntomique.

For a typical example of a complete discrete valuation field K of type II in the sense of [12], we determine the graded quotients gr i K 2 (K) for all i>0. In the Appendix, we describe the Milnor K-groups of a certain local ring by using differential modules, which are related to the theory of syntomic cohomology.

@article{JTNB_2004__16_2_377_0,
     author = {Masato Kurihara},
     title = {On the structure of {Milnor} $K$-groups of certain complete discrete valuation fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {377--401},
     publisher = {Universit\'e Bordeaux 1},
     volume = {16},
     number = {2},
     year = {2004},
     doi = {10.5802/jtnb.452},
     zbl = {1079.11058},
     mrnumber = {2143560},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.452/}
}
Masato Kurihara. On the structure of Milnor $K$-groups of certain complete discrete valuation fields. Journal de Théorie des Nombres de Bordeaux, Tome 16 (2004) no. 2, pp. 377-401. doi : 10.5802/jtnb.452. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.452/

[1] S. Bloch, Algebraic K-theory and crystalline cohomology. Publ. Math. IHES 47 (1977), 187–268. | Numdam | MR 488288 | Zbl 0388.14010

[2] S. Bloch, K. Kato, p-adic etale cohomology. Publ. Math. IHES 63 (1986), 107–152. | Numdam | MR 849653 | Zbl 0613.14017

[3] M. Demazure, Lectures on p-divisible groups. Lecture Notes in Math. 302, Springer (1972). | MR 344261 | Zbl 0247.14010

[4] J.-M. Fontaine, W. Messing, p-adic periods and p-adic étale cohomology. Contemporary Math. 67 (1987), 179–207. | MR 902593 | Zbl 0632.14016

[5] J. Graham,Continuous symbols on fields of formal power series, Algebraic K-theory II. Lecture Notes in Math. 342, Springer-Verlag (1973), 474–486. | MR 364187 | Zbl 0272.18008

[6] L. Illusie, Complexes de de Rham Witt et cohomologie crystalline. Ann. Sci. Éc. Norm. Super. 4 e série t. 12 (1979), 501–661. | Numdam | MR 565469 | Zbl 0436.14007

[7] K. Kato, Residue homomorphisms in Milnor K-theory, in Galois groups and their representations. Adv. St. in Pure Math. 2 (1983), 153–172. | MR 732467 | Zbl 0586.12011

[8] K. Kato, A generalization of local class field theory by using K-groups I. J. Fac. Sci. Univ. Tokyo 26 (1979), 303–376, II, ibid 27 (1980), 603–683, III, ibid 29 (1982), 31–43. | MR 550688 | Zbl 0428.12013

[9] K. Kato, On p-adic vanishing cycles (applications of ideas of Fontaine-Messing). Adv. St. in Pure Math. 10 (1987), 207–251. | MR 946241 | Zbl 0645.14009

[10] K. Kato, The explicit reciprocity law and the cohomology of Fontaine-Messing. Bull. Soc. Math. France 119 (1991), 397–441. | Numdam | MR 1136845 | Zbl 0752.14015

[11] M. Kolster, K 2 of non-commutative local rings. J. Algebra 95 (1985), 173–200. | Zbl 0588.16019

[12] M. Kurihara, On two types of complete discrete valuation fields. Compos. Math. 63 (1987), 237–257. | Numdam | MR 906373 | Zbl 0674.12007

[13] M. Kurihara, A note on p-adic etale cohomology. Proc. Japan Acad. Ser. A 63 (1987), 275–278. | MR 931263 | Zbl 0647.14006

[14] M. Kurihara, Abelian extensions of an absolutely unramified local field with general residue field. Invent. math. 93 (1988), 451–480. | MR 948109 | Zbl 0666.12012

[15] M. Kurihara, The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law. J. reine angew. Math. 498 (1998), 201–221. | MR 1629866 | Zbl 0909.19001

[16] J. Nakamura, On the structures of the Milnor K-groups of some complete discrete valuation fields. K-Theory 19 (2000), 269–309. | MR 1756261 | Zbl 1008.11068

[17] J. Nakamura, On the Milnor K-groups of complete discrete valuation fields. Doc. Math. 5 (2000), 151–200 (electronic). | MR 1756354 | Zbl 0948.19001

[18] A.N. Parshin, Class field theory and algebraic K-theory. Uspekhi Mat. Nauk. 30 no 1 (1975), 253–254, (English transl. in Russian Math. Surveys). | Zbl 0302.14005

[19] J.-P. Serre, Corps locaux (3 e édition), Hermann, Paris, (1968). | MR 354618 | Zbl 0137.02601

[20] T. Tsuji, Syntomic complexes and p-adic vanishing cycles. J. reine angew. Math. 472 (1996), 69–138. | MR 1384907 | Zbl 0838.14015

[21] T. Tsuji, p-adic étale cohomology and crystalline cohomology in the semi-stable reduction case. Invent. math. 137 (1999), 233–411. | MR 1705837 | Zbl 0945.14008

[22] W. Van der Kallen, The K 2 of rings with many units. Ann. Sci. Éc. Norm. Sup. 4 e série t. 10 (1977), 473–515. | Numdam | MR 506170 | Zbl 0393.18012

[23] S.V. Vostokov, Explicit form of the law of reciprocity. Izv. Acad. Nauk. SSSR 13 (1979), 557–588. | Zbl 0467.12018

[24] I. Zhukov, Milnor and topological K-groups of multidimensional complete fields. St. Petersburg Math. J. 9 (1998), 69–105. | MR 1458420 | Zbl 0899.11058