General Néron desingularization. (English) Zbl 0561.14008

A morphism \(A\to B\) of noetherian commutative rings is regular iff it is a filtered inductive limit of smooth morphisms of finite type. This theorem allow to reduce the solvability in B of some polynomial equations over A to the solvability of some polynomial equations for which it is possible to apply the Implicit Function Theorem. The sufficiency is easy but the necessity is difficult even under some conditions of separability as it is given in this paper.
The technique of our ”desingularization” is mainly contained in the present paper the complete proof being given in our paper ”General Néron desingularization and approximation” (submitted to Nagoya Math. J.) which contains also some applications of the above theorem. A stronger result (based on our theorem) is given by M. Cipu and the author in Ann. Univ. Ferrara, Nuova Ser., Sez. VII 30, 63-76 (1984), which extends some theorems of M. Artin and J. Denef from Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, vol. II: Geometry, Prog. Math. 36, 5-31 (1983; Zbl 0555.14002).


14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13E05 Commutative Noetherian rings and modules


Zbl 0555.14002
Full Text: DOI