The spherical spectrum of a graded ring
type: Item
instance of: scholarly article
title: The spherical spectrum of a graded ring
zbMATH Open document ID: 0821.14003
DOI: 10.1007/BF01263607
author: G. Stengle
author: Hasan Yousef
published in: Geometriae Dedicata
publication date: 1994-11-22T00:00:00Z
review text: At its most basic level, the real spectrum functor assigns to a commutative ring \(R\) a quasicompact topological space \(X_ R\) consisting of orderings on \(R\) with prime support. In the case of an affine algebra \(A\) over a real closed field \(K\), constructible sets in \(X_ A\) correspond to semialgebraic subsets of \(K^ n\), and in general the real spectrum plays a role in real algebraic (or semialgebraic) geometry paralleling that of the Zariski spectrum in algebraic geometry. The authors introduce the notion of the \textit{spherical spectrum} which assigns to a positively graded \(G = \bigoplus_{k \geq 0} G_ k\) a topological space \(S_ G\) of relevant homogeneous orderings, which are defined to be points \(\alpha\) of \(X_ G\) which do not contain all \(g\) of positive degree and with the property that, if \(g \in \alpha\), then so is the leading term of \(g\). The spherical spectrum is the real counterpart of the functor Proj and it is very useful to have a paper which sets out the basic definitions. The authors discuss the relation between \(S_ G\) and \(X_ G\). They prove the following theorem which highlights the role of odd elements: Let \(n\) be odd and let \(G(n) = \bigoplus_{k \geq 0} G_{nk}\). Then \(G\) and \(G(n)\) have homeomorphic spherical spectra. The authors go on to prove that for odd degree \(f \in G\), the spherically open set defined by \(f > 0\) is isomorphic to \(X_{G_{(f)}}\), giving a covering of \(S_ G\) by ``affine subsets''. Finally, the authors work out some of the details of the case where \(G\) is a graded affine \(K\)-algebra over a real closed field \(K\). Letting \(\widetilde S^ n\) denote the spherical spectrum of \(K[x_ 0, \dots, x_ n]\) for a real closed field \(K\), they find the anticipated correspondence between points of \(\widetilde S^ n\) with semialgebraic cones in \(K^{n + 1}\) as well as a ``Segre embedding'' of \(\widetilde S^ m \times \widetilde S^ n\) in \(\widetilde S^{m + n+1}\).
Mathematics Subject Classification ID: 14A05
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 13A02
zbMATH DE Number: 679542
zbMATH Keywords: Segre embedding
zbMATH Keywords: real spectrum
zbMATH Keywords: orderings
zbMATH Keywords: spherical spectrum
zbMATH Keywords: semialgebraic cones
reviewed by: Robert O. Robson
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 679542