Sequential topologies and quotients of Milnor \(K\)-groups of multidimensional local fields (Q2783045)

Let \(F=k_n\) be an \(n\)-dimensional local field, i.e. a complete local discretely valued field such that its residue field \(k_{n-1}\) is \((n-1)\)-dimensional and where \(k_0\) is perfect of characteristic \(p>0\). In the present paper, the author studies the structure of the quotient \(K_m^{\text{top}}(F)=K_m(F)/\bigcap_{l\geq 1}lK_m(F)\), where \(K_m(F)\) denotes the \(m\)th Milnor \(K\)-group. The motivation is provided by the fact that the properties of the Milnor \(K\)-groups and these quotients can be effectively applied to class field theory of (multidimensional) local fields as has been done, e.g., by K.~Kato, A. N.~Parshin, S. V.~Vostokov, and the author. One of the main tools in the present paper is the study of several topologies on \(K_m(F)\). It is shown that if \(k_0\) is finite, then the intersections of all subgroups open with respect to these various topologies coincide and are equal to \(\bigcap_{l\geq 1}lK_m(F)\) (and thus equal to the kernel of the reciprocity map \(K_m(F)\to \text{Gal}(F^{ab}/F)\) if \(m=n\)). Furthermore, it is shown that then one also has that \(\bigcap_{l\geq 1}lK_m(F)\) is divisible and that the \(r\)-torsion of \(K_m^{\text{top}}(F)\) is generated by the \(r\)-torsion of \(F^*\), provided that \(F\) contains a primitive \(r\)-th root of unity. NEWLINENEWLINENEWLINEIn an appendix to this paper [St. Petersbg. Math. J. 13, No. 3, 503-507 (2002); translation from Algebra Anal. 13, No. 3, 222-228 (2002; Zbl 1003.11054)], \textit{O. Izhboldin} constructed an example of a field \(F\) containing a primitive \(p\)-th root of unity such that the \(p\)-torsion of \(K_m(F)/\bigcap_{l\geq 1}lK_m(F)\) is not generated by the \(p\)-torsion of \(F\). The author points out that such examples for irregular prime numbers \(p\) can also be constructed in a quite different manner based on results by \textit{G.~Banaszak} [Compos. Math. 86, 281-305 (1993; Zbl 0778.11066)].