On the structure of Milnor $K$-groups of certain complete discrete valuation fields

Kurihara, Masato

Journal de théorie des nombres de Bordeaux, Volume 16 (2004) no. 2, pp. 377-401.

For a typical example of a complete discrete valuation field $K$ of type II in the sense of [12], we determine the graded quotients ${gr}^i{K}_2\left(K\right)$ 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.

Pour un exemple typique de corps de valuation discrète complet $K$ de type II au sens de [12], nous déterminons les quotients gradués ${gr}^i{K}_2\left(K\right)$ 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.

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[11] M. Kolster, $K_2$ of non-commutative local rings. J. Algebra 95 (1985), 173–200. | Zbl

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

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

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

[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 | Zbl

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

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

[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

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

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

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

[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 | Zbl

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

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