Constructive Néron Desingularization of algebras with big smooth locus

Abstract

An algorithmic proof of the General Néron Desingularization theorem and its uniform version is given for morphisms with big smooth locus. This generalizes the results for the one-dimensional case (cf.[10], [7]).

Simple General Neron Desingularization in local Q-algebras

“…Using [15, Lemma 2.4] we may replace B i by an A-algebra C of finite type with H C/A = C, that is C is a smooth algebra over A. We may choose C to be standard smooth over A using [15,Lemma 3.4] (see also [8,Lemma 4]).…”

“…Other authors gave similar proofs (see [23], [21], [9]) but most of them were not constructive. In [14], [12], [13], [5], [6], [7], [8], there exist some constructive proofs with algorithms in some special cases. We mention that all these proofs are simpler when we deal with Q-algebras because in this case all residue field extensions are separable and it is easier to handle the regularity of the morphisms.…”

“…, f r ), r ≤ n is a system of polynomials from I then we can define the ideal ∆ f generated by all r × r-minors of the Jacobian matrix (∂f i /∂Y j ). After Elkik [3] In [8], it is given the following constructive theorem.…”

“…The proof was based on the following proposition (see [8,Proposition 3], where the conditions "A ′ is Henselian, and A, A ′ have dimensions m are superfluous" and we remove them here). Proposition 3.…”