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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms
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by Mark Spivakovsky
J. Amer. Math. Soc. 12 (1999), 381-444


We give a new proof of D. Popescu’s theorem which says that if $\sigma :A\rightarrow B$ is a regular homomorphism of noetherian rings, then $B$ is a filtered inductive limit of smooth finite type $A$-algebras. We strengthen Popescu’s theorem in two ways. First, we show that a finite type $A$-algebra $C$, mapping to $B$, has a desingularization $C\rightarrow D$ which is smooth wherever possible (roughly speaking, above the smooth locus of $C$). Secondly, we give sufficient conditions for $B$ to be a filtered inductive limit of its smooth finite type $A$-subalgebras. We also give counterexamples to the latter statement in cases when our sufficient conditions do not hold.
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Bibliographic Information
  • Mark Spivakovsky
  • Affiliation: Department of Mathematics, University of Toronto, Erindale College, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6
  • Email:
  • Received by editor(s): May 8, 1992
  • Received by editor(s) in revised form: July 24, 1998
  • Additional Notes: Research supported by the Harvard Society of Fellows, NSF, NSERC and the Connaught Fund

  • Dedicated: Dedicated to Professor H. Hironaka on the occasion of his sixtieth birthday
  • © Copyright 1999 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 12 (1999), 381-444
  • MSC (1991): Primary 13B40, 13C10, 14B05, 14B12, 14E40
  • DOI:
  • MathSciNet review: 1647069