The spherical spectrum of a graded ring
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 \(\widetilde S^{m + n+1}\).
scientific article; zbMATH DE number 679542
scientific article