On the algebraic \(K\)-theory of \(R[X,Y,Z]/(X^2+Y^2+Z^2-1)\)
type: Item
instance of: scholarly article
title: On the algebraic \(K\)-theory of \(R[X,Y,Z]/(X^2+Y^2+Z^2-1)\)
zbMATH Open document ID: 1252.14040
author: Marek Golasiński
author: Francisco Gomez Ruiz
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2012-01-10T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.bbms/1323787172
review text: The authors calculate the Brouwer degree of a certain polynomial map. It has been already proven by \textit{R. G. Swan} [Ann. Math. Stud. 113, 432--522 (1987; Zbl 0708.55014)] that \(\tilde{K_0}\) of the coordinate ring \(A=R[X,Y,Z]/(X^2+Y^2+Z^2-1)\) of the algebraic 2-sphere \(S^2(R)\) over a field \(R\) is isomorphic to the integers \({\mathbb Z}\), whenever \(R\) is of characteristic not two and contains the squared root of \(-1\). The authors use explicit idempotent matrices \(\gamma^p\) of order 2, that correspond to the integer \(p\) in the aforementioned isomorphism, and represent classes in \(\tilde{K_0}(A)\) to calculate the Brouwer degree of a polynomial map \(S^2(R)\rightarrow S^2(R)\), where \(R\) is a field of characteristic zero. They then proceed to apply the algebraic de Rham cohomology of Kähler differentials to define Brouwer degree for polynomial homomorphisms of \(A\) to itself, and relate the problem of finding hermitian representatives for \(R = K(\sqrt{-1})\), where \(K\) does not contain \(\sqrt{-1}\), to some problems of representing Brouwer degrees by polynomial maps. In particular they show that if \({\mathbb Q}[i] \subset R\), then there is a polynomial map from \(S^2(R)\) to itself of Brouwer degree \(p\) for every \(p\) in \({\mathbb Z}\). On the other hand, if \(R\) contains all algebraic real numbers, then there is a polynomial map from \(S^2(R)\) to itself of Brouwer degree \(p\) for any odd \(p\).
reviewed by: Paweł Gładki
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 19A49
Mathematics Subject Classification ID: 55R50
zbMATH DE Number: 5996087
zbMATH Keywords: algebraic 2-sphere
zbMATH Keywords: idempotent matrix
zbMATH Keywords: \(\tilde K_0\)
zbMATH Keywords: polynomial map
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5996087