Geodesic
A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms
Spivakovsky, Mark1
1 Department of Mathematics, University of Toronto, Erindale College, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6
Journal of the American Mathematical Society, Tome 12 (1999) no. 2, pp. 381-444

Voir la notice de l'article provenant de la source American Mathematical Society

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.
DOI : 10.1090/S0894-0347-99-00294-5

Spivakovsky, Mark 1

1 Department of Mathematics, University of Toronto, Erindale College, 3359 Mississauga Road, Mississauga, Ontario, Canada L5L 1C6 
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Spivakovsky, Mark. A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms. Journal of the American Mathematical Society, Tome 12 (1999) no. 2, pp. 381-444. doi: 10.1090/S0894-0347-99-00294-5

[1] Andrã©, Michel Homologie des algèbres commutatives 1974

[2] Artin, M. Algebraic approximation of structures over complete local rings Inst. Hautes Études Sci. Publ. Math. 1969 23 58

[3] Artin, M. Algebraic structure of power series rings 1982 223 227

[4] Artin, M., Denef, J. Smoothing of a ring homomorphism along a section 1983 5 31

[5] Artin, Michael, Rotthaus, Christel A structure theorem for power series rings 1988 35 44

[6] Elkik, Renã©E Solutions d’équations à coefficients dans un anneau hensélien Ann. Sci. École Norm. Sup. (4) 1973

[7] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I Inst. Hautes Études Sci. Publ. Math. 1964 259

[8] Lang, Serge Algebra 1965

[9] Matsumura, Hideyuki Commutative algebra 1970

[10] Nagata, Masayoshi Local rings 1975

[11] Nica, Vasile, Popescu, Dorin A structure theorem on formally smooth morphisms in positive characteristic J. Algebra 1986 436 455

[12] Ogoma, Tetsushi General Néron desingularization based on the idea of Popescu J. Algebra 1994 57 84

[13] Popescu, Dorin General Néron desingularization Nagoya Math. J. 1985 97 126

[14] Popescu, Dorin Letter to the editor: “General Néron desingularization and approximation” [Nagoya Math. J. 104 (1986), 85–115 Nagoya Math. J. 1990 45 53

[15] Popescu, Dorin General Néron desingularization and approximation Nagoya Math. J. 1986 85 115

[16] Rotthaus, Christel On the approximation property of excellent rings Invent. Math. 1987 39 63

[17] Teissier, Bernard Résultats récents sur l’approximation des morphismes en algèbre commutative (d’après André, Artin, Popescu et Spivakovsky) Astérisque 1995

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