Isomorphism of complete local noetherian rings and strong approximation
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- by Lou van den Dries PDF
- Proc. Amer. Math. Soc. 136 (2008), 3435-3448 Request permission
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.References
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Additional Information
- Lou van den Dries
- Affiliation: Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, Illinois 61801
- MR Author ID: 59845
- Email: vddries@math.uiuc.edu
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3435-3448
- MSC (2000): Primary 13B40, 13J10; Secondary 13L05
- DOI: https://doi.org/10.1090/S0002-9939-08-09401-X
- MathSciNet review: 2415027