Model theory and spectra
The aim of the present paper is to consider some spectrum functors which have significance in algebraic geometry. The author sets up a spectrum functor \({\mathcal T}\)-spec from the category of commutative rings with unit into the category of topological spaces, where \({\mathcal T}\) is a theory of commutative rings with unit admitting elimination of quantifiers, and \({\mathcal T}\)-spec A consists of equivalence classes of homomorphisms. Closed formulae are defined as those equivalent to positive quantifier- free formulae and are used to define a topology on \({\mathcal T}\)-spec A making it a spectral space. Criteria for \({\mathcal T}\)-closed formulae and spectrally closed constructible subsets, respectively, are given. Some examples illustrate additional properties such as the existence of open projections and enough prime models for a theory \({\mathcal T}\). Using results of \textit{G. Cherlin} and \textit{M. Dickmann} [Ann. Pure Appl. Logic 25, 213-231 (1983; Zbl 0538.03028)], the author discusses the theory RCVR of real closed valuation rings. The constructed spectrum functor RCVR- spec is new, therefore more details are described, especially for real Prüfer rings. An affine space of dimension n over a fixed model R of a theory \({\mathcal T}\) with enough prime models is defined as a certain n-fold fibred product. Such affine spaces enjoy an ``Artin-Lang'' property. The projection from the affine space \({\mathcal A}^ m\) to the affine space \({\mathcal A}^ n\) is open if \({\mathcal T}\) has open projections (both spaces are endowed with the same type of topology). Sheaves of definable sections and sheaves of closed and bounded definable sections (denoted by \({\mathcal D}\) and \({\mathcal S}\), respectively), are considered, where a section over \(X\subset {\mathcal T}\)-spec A is a function s: \(X\to {\mathcal T}\)-spec A[T] such that \(pr(s(\alpha))=\alpha\). Closed and bounded sections are continuous and a closed section is bounded if and only if it can be extended in a certain way. The sheaves \({\mathcal R}\) (sheaf of rational functions), \({\mathcal S}\), and \({\mathcal D}\) are examples for definable sheaves of functions. Another example is the canonical Nash sheaf on the real spectrum. The final result deals with definable statements concerning the \({\mathcal L}\)- structure of constructible sets in \({\mathcal T}\)-spec A. The notions are defined in a model-theoretic way.
scientific article; zbMATH DE number 4143961
scientific article