Bulletin of the Belgian Mathematical Society - Simon Stevin

Nonstandard Generic Points

Guy Wallet

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Abstract

Starting from the Zariski topology, a natural notion of nonstandard generic point is introduced in complex algebraic geometry. The existence of this kind of point is a strong form of the Nullstellensatz. This notion is connected with the classical concept of generic point in the spectrum $\text{Spec}({\cal A}_{n,\mathbb C})$ of the corresponding algebra ${\cal A}_{n,\mathbb C}$. The nonstandard affine space ${^*\mathbb C}^n$ appears as an affine unfolding of the geometric space $\text{Spec}({\cal A}_{n,\mathbb C})$. This affine space is the disjoint union of the sets whose elements are the nonstandard generic points of prime and proper ideals of ${\cal A}_{n,\mathbb C}$: this structure leads to the definition of algebraic points in ${^*\mathbb C}^n$. A natural extension to analytic points in ${^*\mathbb C}^n$ is given by Robinson's concept of generic point in local complex analytic geometry. The end of this paper is devoted to a generalization of this point of view to the real analytic case.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 13, Number 5 (2007), 1033-1057.

Dates
First available in Project Euclid: 1 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1170347824

Digital Object Identifier
doi:10.36045/bbms/1170347824

Mathematical Reviews number (MathSciNet)
MR2293226

Zentralblatt MATH identifier
1121.03088

Subjects
Primary: 03H05: Nonstandard models in mathematics [See also 26E35, 28E05, 30G06, 46S20, 47S20, 54J05] 14P05: Real algebraic sets [See also 12D15, 13J30] 14P15: Real analytic and semianalytic sets [See also 32B20, 32C05] 14-99 32B05: Analytic algebras and generalizations, preparation theorems 32B10: Germs of analytic sets, local parametrization

Keywords
Generic point nonstandard analysis Nullstellensatz Zariski topology

Citation

Wallet, Guy. Nonstandard Generic Points. Bull. Belg. Math. Soc. Simon Stevin 13 (2007), no. 5, 1033--1057. doi:10.36045/bbms/1170347824. https://projecteuclid.org/euclid.bbms/1170347824


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