Abstract Riemann surfaces of integral domains and spectral spaces
The authors consider a domain R and the abstract Riemann surface X(R) of the valuation overrings of R. They prove that X(R) is a spectral space in the sense of \textit{M. Hochster} [Trans. Am. Math. Soc. 142, 43-60 (1969; Zbl 0184.294)]. It is also proved that the canonical projection \(f_ R: X(R)\to Spec(R)\) is a homeomorphism if and only if R is a Prüfer domain. It is remarked that \(f_ R\) is always a closed map and there are given sufficient conditions for \(f_ R\) to be open. The map \(f_ R\) is spectral. It is deduced that X(R) is homeomorphic to an inverse limit of finite \(T_ 0\)-spaces and that \(f_ R\) is a homeomorphism if and only if it is an order-isomorphism. The authors remark that the functor Spec is not invertible on the category of abstract Riemann surfaces.
scientific article; zbMATH DE number 4053764