Continuity of the norm map on Milnor \(K\)-theory
type: Item
instance of: scholarly article
zbMATH Open document ID: 1263.11106
author: Matthew Morrow
publication date: 2012-08-27T00:00:00Z
Mathematics Subject Classification ID: 11S70
Mathematics Subject Classification ID: 11D45
Mathematics Subject Classification ID: 19F05
Mathematics Subject Classification ID: 11S31
zbMATH DE Number: 6072573
zbMATH Keywords: norm map
zbMATH Keywords: unit filtration
zbMATH Keywords: reciprocity
MaRDI profile type: Publication
full work available at URL: is011011005jkt166
OpenAlex ID: W2103713736
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cites work: On the torsion in \(K_ 2\) of local fields
cites work: The exponential homomorphisms for the Milnor K-groups and an explicit reciprocity law
cites work: Explicit reciprocity laws for \(p\)-divisible groups over higher dimensional local fields
DOI: 10.1017/IS011011005JKT166
published in: Journal of \(K\)-Theory
title: Continuity of the norm map on Milnor \(K\)-theory
review text: For a finite extension \(L/F\) of complete, discrete valuation fields, the norm map on the Milnor \(K\)-groups \(K_m(L)\) is continuous w.r.t. the unit group filtrations, more precisely, for all \(i \geq 1,\) \(N_{L/F} (U^{ie} K_m(L)) \subseteq U^i K_m (F),\) where \(e\) is the ramification degree of \(L/F.\) This allows to define a norm map from the completion \(\widehat K_m (L) := \displaystyle\mathop{\lim_{\leftarrow\atop j}}\;K_m (L)/U^j \;K_m (L)\) to the analogous completion \(\widehat K(F),\) which is at the heart of Kato's higher local class field theory. Kato's original proof appealed to two-dimensional local rings and Weierstrass preparation. In the present article, the author gives an ``elementary'' proof relying on explicit calculations of symbols of polynomial expressions, using in particular the so called ``reciprocity property'' of the norm : the map \(\displaystyle\sum_v N_{F(v)/F} \partial_v\;:\;K_m (F(X)) \to K_{m-1} (F)\) is zero, where \(v\) ranges over all discrete valuations on \(F(X)\) which are trivial on \(F\) and \(F(v)\) denotes the residue field of \(F(X)\) at \(v\).
reviewed by: Thong Nguyen Quang Do
description: scientific article; zbMATH DE number 6072573