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**General Néron desingularization based on the idea of Popescu.**
*(English)*
Zbl 0821.13003

In Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008) and 104, 85-115 (1986; Zbl 0592.14014) the reviewer stated the following theorem:

“A morphism of noetherian (commutative) rings is regular if and only if it is a filtered inductive limit of standard smooth morphisms” (that is a special type of finite type smooth morphisms). – This is a positive answer to a question of M. Raynaud [Colloque d’algèbre commutative, Rennes 1972, Publ. Sem. Math. Univ. Rennes, No. 13 (1972; Zbl 0252.13010)] which has many consequences: The Bass-Quillen conjecture holds in the equicharacteristic case [cf. the reviewer, Nagoya Math. J. 113, 121-128 (1989; Zbl 0663.13006)], excellent henselian local rings have the property of Artin approximation – a positive answer to a conjecture of M. Artin [Actes Congr. Intern. Math. 1970, No. 1, 419-423 (1971; Zbl 0232.14003)]\(\dots\) Another form of this theorem stated by M. Cipu and the reviewer [Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 30, 63-76 (1984; Zbl 0581.14006)] says that if \(A \to A'\) is a regular morphism of noetherian rings, then every \(A\)-algebra of finite type \(B\), every \(A\)-morphism from \(B\) to \(A'\) can be factorized through a \(B\)-algebra of finite type \(B'\), which is standard smooth over \(A\) and which is “as smooth as possible” over \(B\) (that is except the singular locus of \(B\) over \(A)\). This new form relies completely on the quoted theorem and it is a positive answer to a conjecture of M. Artin [see Contemp. Math. 13, 223-227 (1982; Zbl 0528.13021)].

Unfortunately, lemma (9.5) of the reviewer’s first cited paper (in Nagoya Math. J. 100) does not hold in the case of condition \(\text{(iii}_ 2)\) as the author notices and repairs in the present paper. Mostly the paper rewrites in a nice way the original proof giving good names to concepts like containerizer, standardizer. The reviewer enjoys this new presentation.

Also the present paper introduces a new concept – the so-called “residual smoothing”. This is a difficult notion which hides many technical details and makes the reading hard. The reviewer has to confess that perhaps it was more difficult for him to understand this notion than for the author to understand and to repair the original proof. After reading a preliminary version of this paper, the reviewer gave a much easier version [Nagoya Math. J. 118, 45-53 (1990; Zbl 0685.14009)] of his original proof (loc. cit.). Meanwhile M. André [“Cinq exposés sur la desingularization” (Preprint 1991)] has also given an interesting version. Also a preprint of M. Spivakovsky should be mentioned [“Smoothing of ring homomorphisms, approximation theorems and the Bass- Quillen conjecture” (Preprint 1992)], following a different approach.

“A morphism of noetherian (commutative) rings is regular if and only if it is a filtered inductive limit of standard smooth morphisms” (that is a special type of finite type smooth morphisms). – This is a positive answer to a question of M. Raynaud [Colloque d’algèbre commutative, Rennes 1972, Publ. Sem. Math. Univ. Rennes, No. 13 (1972; Zbl 0252.13010)] which has many consequences: The Bass-Quillen conjecture holds in the equicharacteristic case [cf. the reviewer, Nagoya Math. J. 113, 121-128 (1989; Zbl 0663.13006)], excellent henselian local rings have the property of Artin approximation – a positive answer to a conjecture of M. Artin [Actes Congr. Intern. Math. 1970, No. 1, 419-423 (1971; Zbl 0232.14003)]\(\dots\) Another form of this theorem stated by M. Cipu and the reviewer [Ann. Univ. Ferrara, Nuova Ser., Sez. VII, 30, 63-76 (1984; Zbl 0581.14006)] says that if \(A \to A'\) is a regular morphism of noetherian rings, then every \(A\)-algebra of finite type \(B\), every \(A\)-morphism from \(B\) to \(A'\) can be factorized through a \(B\)-algebra of finite type \(B'\), which is standard smooth over \(A\) and which is “as smooth as possible” over \(B\) (that is except the singular locus of \(B\) over \(A)\). This new form relies completely on the quoted theorem and it is a positive answer to a conjecture of M. Artin [see Contemp. Math. 13, 223-227 (1982; Zbl 0528.13021)].

Unfortunately, lemma (9.5) of the reviewer’s first cited paper (in Nagoya Math. J. 100) does not hold in the case of condition \(\text{(iii}_ 2)\) as the author notices and repairs in the present paper. Mostly the paper rewrites in a nice way the original proof giving good names to concepts like containerizer, standardizer. The reviewer enjoys this new presentation.

Also the present paper introduces a new concept – the so-called “residual smoothing”. This is a difficult notion which hides many technical details and makes the reading hard. The reviewer has to confess that perhaps it was more difficult for him to understand this notion than for the author to understand and to repair the original proof. After reading a preliminary version of this paper, the reviewer gave a much easier version [Nagoya Math. J. 118, 45-53 (1990; Zbl 0685.14009)] of his original proof (loc. cit.). Meanwhile M. André [“Cinq exposés sur la desingularization” (Preprint 1991)] has also given an interesting version. Also a preprint of M. Spivakovsky should be mentioned [“Smoothing of ring homomorphisms, approximation theorems and the Bass- Quillen conjecture” (Preprint 1992)], following a different approach.

Reviewer: Dorin-Mihail Popescu (Bucureşti)

### MSC:

13B40 | Étale and flat extensions; Henselization; Artin approximation |

13H05 | Regular local rings |

14E15 | Global theory and resolution of singularities (algebro-geometric aspects) |

13F40 | Excellent rings |