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Completions of valuation rings. (English) Zbl 1216.13013

Kachi, Yasuyuki (ed.) et al., Recent progress in arithmetic and algebraic geometry. Proceedings of the 31st annual Barrett lecture series conference, Knoxville, TN, USA, April 25–27, 2002. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3401-0/pbk). Contemporary Mathematics 386, 13-34 (2005).
Summary: Let \(k\) be a field of characteristic zero, \(K\) an algebraic function field over \(k\), and \(V\) a \(k\)-valuation ring of \(K\). Zariski’s theorem of local uniformization shows that there exist algebraic regular local rings \(R_i\) with quotient field \(K\) which are dominated by \(V\), and such that the direct union of the \(R_i\)’s is \(V\). We investigate the ring \(T\), which is the direct union of the completions of the \(R_i\)’s. We give necessary and sufficient conditions for \(T\) to be a valuation ring. We then focus on the case in which the valuation has rank one.
For the entire collection see [Zbl 1078.14002].

MSC:

13F30 Valuation rings
13J10 Complete rings, completion
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