Extensions of valuations to the completion of a local domain. (English) Zbl 0742.13012
Let \((R,m)\) be a normal local domain of dimension \(n>1\) and \(V\) a valuation domain birationally dominating \(R\). If the \(m\)-adic completion \(\hat R\) is a domain then there exists a valuation domain \(W\) extending \(V\) and birationally dominating \(\hat R\) [see S. Abhyankar, Ann. Math. (2) 63, 491–526 (1956; Zbl 0108.16803), lemma 13]. The aim of this paper is to show that if \(\hat R\) is a normal domain then there exists a unique valuation domain \(W\) as above. This statement generalizes H. Göhner’s result from J. Algebra 34, 403–429 (1975; Zbl 0308.13023) and M. Spivakovsky’s result from case \(n=2\) [see Am. J. Math. 112, No. 1, 107–156 (1990; Zbl 0716.13003)].
Reviewer: Dorin-Mihail Popescu (Vechta)
MSC:
| 13F30 | Valuation rings |
| 13J10 | Complete rings, completion |
| 13A18 | Valuations and their generalizations for commutative rings |
| 13B35 | Completion of commutative rings |
| 13B22 | Integral closure of commutative rings and ideals |
References:
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