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Rings with approximation property. (English) Zbl 0702.13007

Let A be a noetherian semilocal ring which satisfies the Artin approximation property. It is well known that A is henselian, universally catenary and that the formal fibres of A are geometrically normal. The present paper proves that the formal fibres of A are geometrically regular; and hence, that A is an excellent ring. [The converse, that every henselian excellent semilocal ring which contains the field of rational numbers has the Artin approximation property, has already been proved by the author in Invent. Math. 88, 39-63 (1987; Zbl 0614.13014).] The idea of the proof is to find “enough” equations for describing the singularity of the localization \(\hat A_ P\) where \(\hat A\) is the completion of A and P is a prime ideal in the singular locus of A. The main tool in the proof is the theorem (due to the author and Artin) that if R is an excellent discrete valuation ring, then the formal power series ring \(R[[X_ 1,...,X_ n]]\) is a direct limit of smooth R- algebras.
Reviewer: A.Kustin

MSC:

13B40 Étale and flat extensions; Henselization; Artin approximation
14B12 Local deformation theory, Artin approximation, etc.
13J15 Henselian rings

Citations:

Zbl 0614.13014
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References:

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