Revisiting Zariski Main Theorem from a constructive point of view

This paper deals with the Peskine version of Zariski Main Theorem published in 1965 and discusses some applications. It is written in the style of Bishop's constructive mathematics. Being constructive, each proof in this paper can be interpreted as an algorithm for constructing explicitly the c...

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Bibliographic Details
Published inJournal of algebra Vol. 406; pp. 46 - 68
Main Authors Alonso, M.E., Coquand, T., Lombardi, H.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.05.2014
Subjects
Constructive mathematics
Matematik
Mathematics
Multivariate Hensel Lemma
Quasi-finite algebras
RINGS
Zariski Main Theorem
Multivariate Hensel Lemma
Constructive mathematics
Quasi-finite algebras
Zariski Main Theorem
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Summary:This paper deals with the Peskine version of Zariski Main Theorem published in 1965 and discusses some applications. It is written in the style of Bishop's constructive mathematics. Being constructive, each proof in this paper can be interpreted as an algorithm for constructing explicitly the conclusion from the hypothesis. The main non-constructive argument in the proof of Peskine is the use of minimal prime ideals. Essentially we substitute this point by two dynamical arguments; one about gcd's, using subresultants, and another using our notion of strong transcendence. In particular we obtain algorithmic versions for the Multivariate Hensel Lemma and the structure theorem of quasi-finite algebras.
ISSN:0021-8693
1090-266X
1090-266X
DOI:10.1016/j.jalgebra.2014.02.003