Orbit spaces of unimodular rows over smooth real affine algebras
type: Item
instance of: scholarly article
title: Orbit spaces of unimodular rows over smooth real affine algebras
zbMATH Open document ID: 1388.13027
author: Mrinal Kanti Das
author: Soumi Tikader
author: Md. Ali Zinna
published in: Inventiones Mathematicae
publication date: 2018-04-17T00:00:00Z
review text: The main result is as follows: Let \(X=\text{Spec}(R)\) be a smooth real affine variety of even dimension \(d\geq2\), whose real points \(X(\mathbb R)\) constitute an orientable manifold. Then the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is a free abelian group of rank equal to the number of compact connected components of \(X(\mathbb R)\). In contrast, if \(d\geq3\) is odd, then the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is a \(\mathbb Z/2\mathbb Z\)-vector space (possibly trivial). That the set of isomorphism classes of (oriented) stably free modules of rank \(d\) is an abelian group is known from \textit{Wilberd van der Kallen} [J. Algebra 82, 363--397 (1983; Zbl 0518.20035)], but these groups \(Um_{d+1}(R)/\mathrm{SL}_{d+1}(R)\) have been hard to compute. Thus the authors have to combine many ingredients to arrive at the main result. They make a good effort to explain these ingedients to a general reader. One compares \(Um_{d+1}(R)/\mathrm{SL}_{d+1}(R)\) with a group \(Um_{d+1}(\mathbb R(X))/E_{d+1}(\mathbb R(X))\), where \(\mathbb R(X)\) is a convenient localization of \(R\). One also compares it with the Euler class groups \(E^d(R)\) and \(E^d(\mathbb R(X))\). In fact the authors employ homomorphisms in both directions between \(Um_{d+1}(R)/E_{d+1}(R)\) and \(E^d(R)\). The one from \(E^d(R)\) to \(Um_{d+1}(R)/E_{d+1}(R)\) is more recent. It deals with naive homotopy classes -- also known as homotopy orbits -- in a quadric over \(R\) with equation \(z^2+\sum_ix_iy_i=1\). This relies on \textit{A. Stavrova} [J. K-Theory 13, 199--248 (2014; Zbl 1314.19002)]. Higher Mennicke symbols and \(K\)-cohomology are also discussed.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 19A13
Mathematics Subject Classification ID: 19A15
Mathematics Subject Classification ID: 14C25
zbMATH DE Number: 6860750
zbMATH Keywords: unimodular row
zbMATH Keywords: stably free module
zbMATH Keywords: Euler class group
zbMATH Keywords: weak Mennicke symbol
zbMATH Keywords: naive homotopy
zbMATH Keywords: homotopy orbit
MaRDI profile type: Publication
full work available at URL: s00222-017-0764-y
OpenAlex ID: W2766810899
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DOI: 10.1007/S00222-017-0764-Y
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description: scientific article; zbMATH DE number 6860750