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Global approximation in dimension two. (English) Zbl 0707.13003

The paper under review is concerned with Artin approximation and related results for normed rings. First result:
Theorem 1.1. Let A be an excellent normal domain of dimension at most two, and I an ideal of A. Let \(\bar A\) be the I-adic completion of A and \(\tilde A\) be the henselization of A at \(\bar A.\) Then \(\tilde A\subset \bar A\) has the approximation property, with respect to the I-adic metric. - The proof uses some of Elkik’s theorems [R. Elkik, Ann. Sci. Éc. Norm. Supér., IV. Sér. 6(1973), 553-603 (1974; Zbl 0327.14001), theorems 2 and 2 bis].
Main result: Theorem 4.4: Let \(A\subset \bar A\) be a flat extension of Noetherian normal domains of dimension \(\leq 2\), such that \(\bar A\) is a quasi-completion of A relative to a norm \(\| \cdot \|\), and such that the map of local rings \(A_{\bar p\cap A}\to \bar A_{\bar p}\) is regular for every prime \(\bar p\subset \bar A\) of height at most 1. Let \(\tilde A\) be the henselization of A at \(\bar A.\) Assume that \((\bar A,\| \cdot \|)\) satisfies: (i) For every \(\epsilon >0\) there exists \(\delta >0\) such that for all \(x\in \bar A\), if \(\| x^ 2\| <\delta\) then \(\| x\| <\epsilon\); and
(ii) every height 2 prime of \(\bar A\) is the extension of a height 2 prime of A having the same residue field.
Then \(\tilde A\subset \bar A\) has the approximation property with respect to \(\| \cdot \|.\)
The proof of theorem 4.4 is parallel to the proof of theorem 1.1, using some normed version of Elkik’s results, proved in section 3. In order to prove the results from section 3, a normed version of Tougeron’s lemma is needed and proved in section 2 [see J. C. Tougeron, Thesis (Rennes 1967)]. Some interesting applications of theorem 4.4. are given in section 4.
Reviewer: O.Pasarescu

MSC:

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

Citations:

Zbl 0327.14001
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References:

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