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

Zbl 0614.13014
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References:

 [1] Artin, M.: Algebraic approximation of structures over complete local rings. Publ. Math. IHES36, 23-58 (1969) · Zbl 0181.48802 [2] Artin, M.: Algebraic structure of power series rings. Contemp. Math.13, 223-227 (1982) · Zbl 0528.13021 [3] Artin, M., Denef, J.: Smoothing of a ring homomorphism along a section. Arithmetic and Geometry Vol. II. Boston: Birkh?user 1983 · Zbl 0555.14002 [4] Artin, M.: Rotthaus, C.: A structure theorem for power series rings. In: Algebraic geometry and commutative algebra, in Honor of Masayoshi Nagata, Vol. I, pp. 35-44: Tokyo Kinokuniya 1988 · Zbl 0709.13009 [5] Grothendieck, A., Dieudonne, J.: ?lements de g?om?trie alg?brique. Publ. Math. Inst. Hautes Etud. Sci.20 (1964) [6] Grothendieck, A., Dieudonne, J.: ?lements de g?om?trie alg?brique. Publ. Math. Inst. Hautes Etud. Sci.24 (1965) [7] Grothendieck, A., Dieudonne, J.: ?lements de g?om?trie alg?brique. Publ. Math. Inst. Hautes Etud Sci.32 (1967) [8] Kurke, H., Mostowski, T., Pfister, G., Popescu, D., Roczen, M.: Die Approximationseigenschaft lokaler Ringe. (Lecture Notes Math. vol. 634). Berlin Heidelberg New York: Springer 1968 · Zbl 0401.13013 [9] Matsumura, H.: Commutative algebra. New York: Benjamin 1968 · Zbl 0211.06501 [10] Matsumura, H.: Commutative ring theory. Cambridge: Cambridge University Press 1986 · Zbl 0603.13001 [11] Pfister, G., Popescu, D.: On three dimensional local rings with approximation property. Rev. Roum. Math. Pure Appl.26, 301-307 (1981) · Zbl 0486.13009 [12] Ploski, A.: Note on a theorem by M. Artin. Bull. Acad. Pol. Sci. Ser. Math.22, 1107-1109 (1974) · Zbl 0302.32002 [13] Popescu, D.: General N?ron desingularization. Nagoya Math. J.100, 97-126 (1985) · Zbl 0561.14008 [14] Rotthaus, C.: Potenzreihenerweiterung und formale Fasern in lokalen Ringen mit Approximationseigenschaft. Manuscr. Math.42, 53-65 (1983) · Zbl 0514.13009 [15] Rotthaus, C.: On the approximation property of excellent rings. Invent. Math.88, 39-63 (1987) · Zbl 0614.13014
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