The basic theory of real closed spaces. (English) Zbl 0634.14014
Regensburger Mathematische Schriften 15. Regensburg: Univ. Regensburg, Fakultät für Mathematik. xiv, 257 p. (1987).
This is the updated version of the author’s Habilitationsschrift [“Real closed spaces” (1984; Zbl 0586.14016)] including a more general definition of affine real closed spaces.
Real closed spaces provide, using the theory of real spectra, an abstract generalization, in the case of general rings, of semi-algebraic sets equipped with semi-algebraic (continuous) functions.
Chapter 1 is devoted to the introduction of a closure operator on the category of schemes. Chapter 2 and 3 introduce and discuss real closed sheaves. Chapter 4 makes the connection with semi-algebraic geometry. In chapter 5 a general theory of real closed spaces is started. Using this theory more precise results on the connection between semi-algebraic geometry and affine real closed spaces are obtained in chapter 6.
Real closed spaces provide, using the theory of real spectra, an abstract generalization, in the case of general rings, of semi-algebraic sets equipped with semi-algebraic (continuous) functions.
Chapter 1 is devoted to the introduction of a closure operator on the category of schemes. Chapter 2 and 3 introduce and discuss real closed sheaves. Chapter 4 makes the connection with semi-algebraic geometry. In chapter 5 a general theory of real closed spaces is started. Using this theory more precise results on the connection between semi-algebraic geometry and affine real closed spaces are obtained in chapter 6.
Reviewer: M.F.Roy
MSC:
14Pxx | Real algebraic and real-analytic geometry |