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Abstract:
Let A and B be complete local noetherian rings with maximal ideals M
and N, and suppose that A/M^k is ring isomorphic to B/N^k for all
k. Does it follow that A and B are ring isomorphic? This question was
asked by Angus Macintyre more than a year ago. I will indicate a
proof using strong approximation that the answer is yes if the residue
field A/M is algebraic over its prime field, and a counter example by
Ofer Gabber where the residue field has transcendence degree 1 over
the field of rationals.
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