Néron-Popescu desingularization. (English) Zbl 0954.13003

Kang, Ming-Chang (ed.), Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26-30, 1995. Cambridge, MA: International Press. 135-192 (1998).
A regular morphism of Noetherian rings is a filtered colimit of smooth morphisms. This result was proved by the reviewer [D. Popescu, Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008); 104, 85-115 (1986; Zbl 0592.14014) and 118, 45-53 (1990; Zbl 0685.14009)], by M. André [“Cinq exposés sur la desingularization”, Handwritten manuscript, École Polytechnique Fédérale de Lausanne (1971)], by T. Ogoma [J. Algebra 167, No. 1, 57-84 (1994; Zbl 0821.13003)] and by M. Spivakovsky [J. Am. Math. Soc. 12, No. 2, 381-444 (1999; Zbl 0919.13009)].
The exposition under review contains the easiest, self contained presentation of the proof which could be used also by beginners: Low dimensional Quillen cohomology groups are introduced directly using an argument of Faltings, local criterion of flatness is given only in a simple form used in the proof, Zariski’s main theorem is written after Peskine, etc. A strong motivation of the result is given by its applications to Artin approximation and the Bass-Quillen conjecture. The reviewer used this paper a lot in his exposition “Artin approximation” [in “Handbook of Algebra”, ed. M. Hazewinkel (Amsterdam 2000; Zbl 0949.00006), section 3A, 321-356 (2000)].
For the entire collection see [Zbl 0920.00026].


13B40 Étale and flat extensions; Henselization; Artin approximation
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
14B12 Local deformation theory, Artin approximation, etc.