Valuations and diameters of Milnor \(K\)-rings
type: Item
instance of: scholarly article
title: Valuations and diameters of Milnor \(K\)-rings
zbMATH Open document ID: 1197.19001
author: Ido Efrat
published in: Israel Journal of Mathematics
publication date: 2009-10-02T00:00:00Z
review text: It is convenient to begin the review by quoting the main theorem of the paper, after which we will define the necessary objects: ``Let \(F\) be a (commutative) field and let \(S\) be a subgroup of \(F^{\times}\) containing \(-1\). Suppose that the graph of \(K_*^M(F)/S\) is finite and of diameter \(\geq 4\). Then there exists a nontrivial valuation \(\nu\) on \(F\) such that \(S\) is open in the \(\nu\)-topology.'' In this way the paper provides a method for obtaining, from graph- and \(K\)-theoretic data, a suitable valuation on the field. The construction of valuations from Milnor \(K\)-theory goes back to work by \textit{B. Jacob} [J. Algebra 68, 247--267 (1981; Zbl 0457.10007)] on Pythagorean fields, while graphs appear in the following similar, non-commutative result by \textit{A. S. Rapinchuk} and \textit{Y. Segev} [Invent. Math. 144, No.~3, 571--607 (2001; Zbl 0999.16016)]: Let \(D\) be a finite-dimensional division algebra over a finitely-generated field and \(S\) a finite index, normal subgroup of \(D^{\times}\). If the diameter of the commuting graph of \(D^{\times}/S\) is \(\geq4\), then there exists a non-trivial valuation on \(D\) for which \(S\) is open (the commutating graph of a group has the non-trivial elements as vertices, with an edge between two elements if and only if they commute). Here \(K_*^M(F)/S=\bigoplus_{r=0}^\infty K_r^M(F)/S\) is the author's relative Milnor \(K\)-ring [\textit{I. Efrat}, Pac. J. Math. 226, No.~2, 259--275 (2006; Zbl 1161.19002)] which, in degree \(r\), is defined to be \((F^{\times}/S)^{\otimes r}\) modulo the relations \(a_1S\otimes\dots\otimes a_rS\) whenever \(1\in a_iS+a_jS\) for some \(i\neq j\). The associated graph is defined to have vertices for each non-trivial coset of \(F^{\times}/S\), with two vertices \(aS, bS\) connected by an edge if and only if \(\{a,b\}_S\) (\(=\) the image of \(aS\otimes bS\) in \(K_2^M(F)/S\)) is zero. Finally, the diameter of a graph is the greatest integer \(d\) (possibly \(\infty…
reviewed by: Matthew Morrow
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 19F99
Mathematics Subject Classification ID: 19C99
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 5610598
zbMATH Keywords: generalized Milnor K-theory
zbMATH Keywords: valuations
zbMATH Keywords: field arithmetic
zbMATH Keywords: k-structures
MaRDI profile type: Publication
full work available at URL: s11856-009-0064-3
OpenAlex ID: W1989348712
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DOI: 10.1007/S11856-009-0064-3
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description: scientific article; zbMATH DE number 5610598