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Residue homomorphisms in Milnor K-theory. (English) Zbl 0586.12011

Galois groups and their representations, Proc. Symp., Nagoya/Jap. 1981, Adv. Stud. Pure Math. 2, 153-172 (1983).
[For the entire collection see Zbl 0516.00012.]
The author studies a ”residue homomorphism” res: \(\hat K_{q+1}(\hat M)\to \hat K_ q(k)\), where \(\hat K_*\) denotes a certain completion of Milnor’s higher K-groups, k is a complete discrete valuation field, and \(\hat M\) is the field of series \(\sum_{n\in {\mathbb{Z}}}a_ n X^ n\) over k with \(ord_ k(a_ n)\) bounded below, and \(\lim_{n\to -\infty}a_ n=0.\)
The paper extends work of J.-L. Brylinski [Ann. Inst. Fourier 33, No.3, 23-38 (1983; Zbl 0524.12008)], and provides a simple interpretation for the p-primary part of the reciprocity map \(K_ n(K)\to Gal(K^{ab}/K)\) used in the higher-dimensional class field theory of the author. Here K, a ”local field of dimension n”, is assumed to be of characteristic p. Some of the proofs make use of similar results for the Quillen K-groups; the residue map comes from a boundary map in the exact sequence for a localization.

MSC:

11S70 \(K\)-theory of local fields
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
11S31 Class field theory; \(p\)-adic formal groups
14E20 Coverings in algebraic geometry