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/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.
| Original language | English (US) |
|---|---|
| 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 |
Fingerprint
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics
Cite this
Isomorphism of complete local noetherian rings and strong approximation. / Van Den Dries, Lou.
In: Proceedings of the American Mathematical Society, Vol. 136, No. 10, 01.10.2008, p. 3435-3448.Research output: Contribution to journal › Article
}
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 -