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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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