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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Isomorphism of complete local noetherian rings and strong approximation


Author: Lou van den Dries
Journal: Proc. Amer. Math. Soc. 136 (2008), 3435-3448
MSC (2000): Primary 13B40, 13J10; Secondary 13L05.
Published electronically: May 8, 2008
MathSciNet review: 2415027
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Abstract: About a year ago Angus Macintyre raised the following question. Let $ A$ and $ B$ be complete local noetherian rings with maximal ideals $ \mathfrak{m}$ and $ \mathfrak{n}$ such that $ A/\mathfrak{m}^n$ is isomorphic to $ B/\mathfrak{n}^n$ for every $ n$. Does it follow that $ A$ and $ B$ are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.


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Additional Information

Lou van den Dries
Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
Email: vddries@math.uiuc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09401-X
PII: S 0002-9939(08)09401-X
Keywords: Complete local noetherian ring, strong approximation
Received by editor(s): December 18, 2006
Received by editor(s) in revised form: September 4, 2007
Published electronically: May 8, 2008
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.