### Abstract

About a year ago Angus Macintyre raised the following question. Let A and B be complete local noetherian rings with maximal ideals m and n such that A/m^{n} is isomorphic to B/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.

Original language | English (US) |
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Pages (from-to) | 3435-3448 |

Number of pages | 14 |

Journal | Proceedings of the American Mathematical Society |

Volume | 136 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Isomorphism of complete local noetherian rings and strong approximation.** / Van Den Dries, Lou.

Research output: Contribution to journal › Article

*Proceedings of the American Mathematical Society*, vol. 136, no. 10, pp. 3435-3448. https://doi.org/10.1090/S0002-9939-08-09401-X

}

TY - JOUR

T1 - Isomorphism of complete local noetherian rings and strong approximation

AU - Van Den Dries, Lou

PY - 2008/10/1

Y1 - 2008/10/1

N2 - About a year ago Angus Macintyre raised the following question. Let A and B be complete local noetherian rings with maximal ideals m and n such that A/mn is isomorphic to B/nn 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.

AB - About a year ago Angus Macintyre raised the following question. Let A and B be complete local noetherian rings with maximal ideals m and n such that A/mn is isomorphic to B/nn 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.

UR - http://www.scopus.com/inward/record.url?scp=77950670677&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950670677&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-08-09401-X

DO - 10.1090/S0002-9939-08-09401-X

M3 - Article

AN - SCOPUS:77950670677

VL - 136

SP - 3435

EP - 3448

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 10

ER -