Journal of the American Mathematical Society

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



A new proof of D. Popescu's theorem
on smoothing of ring homomorphisms

Author: Mark Spivakovsky
Journal: J. Amer. Math. Soc. 12 (1999), 381-444
MSC (1991): Primary 13B40, 13C10, 14B05, 14B12, 14E40
MathSciNet review: 1647069
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Abstract: 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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Additional Information

Mark Spivakovsky
Affiliation: Department of Mathematics, University of Toronto, Erindale College, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6

Keywords: Smooth homomorphism, N\'{e}ron desingularization, Artin approx\-i\-ma\-tion
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
Article copyright: © Copyright 1999 American Mathematical Society