An exact sequence for \(K^M_*/2\) with applications to quadratic forms
type: Item
instance of: scholarly article
title: An exact sequence for \(K^M_*/2\) with applications to quadratic forms
zbMATH Open document ID: 1124.14017
author: Dmitri Orlov
author: A. S. Vishik
author: Vladimir Voevodsky
published in: Annals of Mathematics. Second Series
publication date: 2007-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0101023
review text: The paper under review is a natural extension of the third author's fundamental paper ``Motivic cohomology with \(\mathbb{Z}/2\)-coefficients'' [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)], in which he had presented a refined version of his spectacular proof of the famous Milnor conjecture in algebraic \(K\)-theory. Accordingly, the authors of the present paper freely refer to the methods and results developed in that foregoing work without reproducing them here. Actually, great deal of the research presented in the paper under review was conducted as early as in the spring of 1995, when all three authors were staying at Harvard University, and the final version, in its present form, was submitted a few years ago, when these three authors continued their collaboration at the IAS in Princeton. In this regard, the paper seems to have some remarkable history which probably is well-known to most leading experts in the field. As for the topics treated in the present paper, the set-up is as follows: Let \(k\) be a field of characteristic zero. For a sequence \(\underline a= (a_1,\dots,a_n)\) of invertible elements of \(k\) one has a natural homomorphism \(K^M_*(k)/2\to K^M_{*+n}(k)/2\) in Milnor's \(K\)-theory modulo elements divisible by 2, which is given by multiplication with the symbol corresponding to the sequence \(\underline a\). The main goal of the present paper is then to construct a four-term exact sequence involving the above natural homomorphism as middle term, from which some useful information about the kernel and the cokernel of it can be obtained. The proof of the existence of such an explicit exact sequence is technically rather involved and consists of two independent parts. At first, the authors define the so-called norm quadric \(Q_{\overline a}\) associated to the sequence \(\underline a\) and, using the techniques from V. Voevodsky's above-mentiond paper, derive then a four-term exact sequence relating the kernel and the…
reviewed by: Werner Kleinert
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14F42
Mathematics Subject Classification ID: 19D45
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E70
Mathematics Subject Classification ID: 13K05
zbMATH DE Number: 5157389
zbMATH Keywords: applications of methods of algebraic \(K\)-theory
zbMATH Keywords: Milnor \(K\)-theory
zbMATH Keywords: higher symbols
zbMATH Keywords: cohomology operations
zbMATH Keywords: quadratic forms
zbMATH Keywords: Witt vectors
MaRDI profile type: Publication
OpenAlex ID: W2171724054
Wikidata QID: Q30050405
arXiv ID: math/0101023
DOI: 10.4007/ANNALS.2007.165.1
description: scientific article