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General Néron desingularization and approximation. (English) Zbl 0592.14014
Let \(u:\quad A\to A'\) be a morphism of noetherian rings. Then \((i)\quad u\quad is\) regular iff it is a filtered inductive limit of finite type smooth morphisms. Under some separability conditions this result was the subject of our previous paper [Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008)] on which we rely now.
Let \({\mathfrak a}\subset A\) be a proper ideal and \(\hat A\) the completion of A in the \({\mathfrak a}\)-adic topology. Using (i) it follows: \((ii)\quad if\) (A,\({\mathfrak a})\) is henselian and \(A\to \hat A\) is regular then every finite system of polynomials over A has its set of solutions in A dense with respect to the \({\mathfrak a}\)-adic topology in the set of its solutions in \(\hat A.\) - When A is local and \({\mathfrak a}\) is maximal then this result is a positive answer to one of M. Artin’s conjectures [M. Artin, Actes Congr. internat. Math. 1970, Vol. 1, 419-423 (1971; Zbl 0232.14003)].
As a consequence of (ii) it follows: \((iii)\quad the\) completion of an excellent henselian factorial local ring is factorial, too. - Also from (ii) we get a complete solution of so called approximation on nested subrings, i.e.:
\((iv)\quad Let\) k be a field, \(k<X>\) the algebraic power series ring in \(X=(X_ 1,...,X_ r)\) over k, f a finite system of polynomials over \(k<X>\) and \(\hat y=(\hat y_ 1,...,\hat y_ n)\in k[[X]]^ n\) a formal solution such that \(\hat y_ i\in k[[X_ 1,...,X_{s_ i}]]\), \(1\leq i\leq n\) for some positive integers \(s_ i\leq r\). Then there exists a solution \(y=(y_ 1,...,y_ n)\) of f in \(k<X>\) such that \(y_ i\in k<X_ 1,...,X_{s_ i}>\), \(1\leq i\leq n\).

MSC:
14B25 Local structure of morphisms in algebraic geometry: étale, flat, etc.
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
13E05 Commutative Noetherian rings and modules
13J10 Complete rings, completion
13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13J15 Henselian rings
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