Class formations and higher dimensional local class field theory (Q676214)

Let \(K\) be an \(n\)-local field, \(K^M_n(K)\) the \(n\)-th Milnor \(K\)-group of \(K\). Furthermore, let \(G_K^{ab}\) be the Galois group of the maximal abelian extension of \(K\). The reciprocity map is a canonical homomorphism from \(K_n^M(K)\) in \(G_K^{ab}\) which for every finite abelian extension \(L/K\) induces an isomorphism \(K_n^M(K)/ N_{L/K}K_n^M(L)\cong \text{Gal}(L/K)\). The reciprocity map was constructed in characteristic \(p>0\) by \textit{A. Parshin} [Usp. Mat. Nauk 30, 253-254 (1975; Zbl 0302.14005)] and in general by \textit{K. Kato} [J. Fac. Sci., Univ. Tokyo, Sect. I A 26, 303-376 (1979; Zbl 0428.12013) and ibid. 27, 603-683 (1980; Zbl 0463.12006)]. The author uses the notion of generalized class formation as defined by \textit{Y. Koya} [Invent. Math. 101, 705-715 (1990; Zbl 0751.11055)] for a short and elegant construction of the reciprocity map. A main tool in this construction for \(n\geq 3\) is the decomposable part of motivic cohomology \(\check\mathbb{Z}(n)\) studied by \textit{B. Kahn} [Contemp. Math. 126, 79-87 (1992; Zbl 0759.19005)]. The case \(n=2\) was settled by Koya (e.c.).