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Nonsingular Hasse principle for rings

Nonsingular Hasse principle for rings J. reine angew. Math. 529 (2000), 75Ð100 Journal fur die reine und È angewandte Mathematik ( Walter de Gruyter Berlin Á New York 2000 Á By L. Darniere at Angers 1. Introduction There have been numerous model theoretic studies in the last twenty years on ®elds with families of topologies. Most of them involved various local-global principles (LGP) analogous to the geometric Hasse principle. In 1986, Rumely [11] proved that Hilbert's tenth problem had a positive answer for ~ the ring Z of all algebraic integers. His approach involved a new kind of LGP, adapted to the context of rings. This paved the way to new results on the model theory of rings, and indeed several important theorems concerning decidability, quanti®er elimination or model-completeness for various domains were obtained independently by van den Dries, Macintyre, Prestel and Schmid ([2], [3] and [9]) by using apparently di¨erent LGP's, related to Rumely's one. In this paper we introduce another LGP for rings we call LGPH. It turns out to contain all the preceding ones as particular cases, and to remain ®rst order (in the ordinary language of rings). On the other hand, the proof we give in this article of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal für die reine und angewandte Mathematik (Crelle's Journal) de Gruyter

Nonsingular Hasse principle for rings

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References (1)

Publisher
de Gruyter
Copyright
Copyright (c) 2000 by Walter de Gruyter GmbH & Co. KG
ISSN
0075-4102
eISSN
1435-5345
DOI
10.1515/crll.2000.098
Publisher site
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Abstract

J. reine angew. Math. 529 (2000), 75Ð100 Journal fur die reine und È angewandte Mathematik ( Walter de Gruyter Berlin Á New York 2000 Á By L. Darniere at Angers 1. Introduction There have been numerous model theoretic studies in the last twenty years on ®elds with families of topologies. Most of them involved various local-global principles (LGP) analogous to the geometric Hasse principle. In 1986, Rumely [11] proved that Hilbert's tenth problem had a positive answer for ~ the ring Z of all algebraic integers. His approach involved a new kind of LGP, adapted to the context of rings. This paved the way to new results on the model theory of rings, and indeed several important theorems concerning decidability, quanti®er elimination or model-completeness for various domains were obtained independently by van den Dries, Macintyre, Prestel and Schmid ([2], [3] and [9]) by using apparently di¨erent LGP's, related to Rumely's one. In this paper we introduce another LGP for rings we call LGPH. It turns out to contain all the preceding ones as particular cases, and to remain ®rst order (in the ordinary language of rings). On the other hand, the proof we give in this article of

Journal

Journal für die reine und angewandte Mathematik (Crelle's Journal) – de Gruyter

Published: Dec 4, 2000

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