Completions of valuation rings

الأصل · EN

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 Rᵢ with quotient field K which are dominated by V, and such that the direct union of the Rᵢ's is V. We investigate the ring T, which is the direct union of the completions of the Rᵢ'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.