Isomorphism of complete local noetherian rings and strong approximation

van den Dries, Lou

doi:10.1090/S0002-9939-08-09401-X

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

Steven Dale Cutkosky and Hema Srinivasan, An intrinsic criterion for isomorphism of singularities, Amer. J. Math. 115 (1993), no. 4, 789–821. MR 1231147, DOI 10.2307/2375013
J. Denef and L. Lipshitz, Ultraproducts and approximation in local rings. II, Math. Ann. 253 (1980), no. 1, 1–28. MR 594530, DOI 10.1007/BF01457817
Hideyuki Matsumura, Commutative algebra, W. A. Benjamin, Inc., New York, 1970. MR 0266911
Gerhard Pfister and Dorin Popescu, Die strenge Approximationseigenschaft lokaler Ringe, Invent. Math. 30 (1975), no. 2, 145–174 (German). MR 379490, DOI 10.1007/BF01425506
Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618