Ein semialgebraischer Beweis der topologischen Form des Hauptsatzes von Zariski. (A semi-algebraic proof of the topological form of Zariski's main theorem)
Let R be a real closed field, \(C=R(\sqrt{-1})\) its algebraic closure. If X is a C-variety then X(C), the set of C-rational points of X, is a semi- algebraic space (using the identification \(C=R^{2n})\) in the sense of \textit{H. Delfs} and \textit{M. Knebusch} [``Locally semialgebraic spaces'', Lect. Notes Math. 1173 (1985; Zbl 0582.14006)]. Similar to the construction of the sheaf of Nash functions on the real spectrum of a ring [cf. \textit{M. F. Roy} in Géometrie algébrique réelle et formes quadratiques, Journ. S. M. F., Univ. rennes 1981, Lect. Notes Math. 959, 406-432 (1982; Zbl 0497.14009)] a sheaf of isoalgebraic functions is defined on X(C). The space X(C) together with this sheaf is an isoalgebraic space. Locally isoalgebraic spaces are obtained by gluing isoalgebraic spaces together. A locally isoalgebraic space X carries a natural structure of a locally semi-algebraic space which is denoted by \(| X|\). The main result is: If X is a connected normal isoalgebraic space and \(A\subset X\) is locally semi-algebraic, dim(A)\(\leq \dim | X| -2\), then \(X\setminus A\) is connected. From this the topological Zariski main theorem and the Riemann extension theorem can be deduced.
scientific article; zbMATH DE number 4128966