Extending valuations to formal completions

1 Introduction This paper is an extended version of the talk given by Miguel Olalla at the International Conference on Valuation Theory in El Escorial in July 2011. Its purpose is to provide an introduction to our joint paper [7] without grinding through all of its technical details. We refer the reader to [7] for details and proofs; only a few proofs are given in the present paper. Notation. Throughout this paper, we will use the following notation: (R, m, k) a local noetherian domain K = QF (R) its field of fractions R ν a valuation ring dominating R ν |K : K * Γ the restriction of ν to K. ˆ R the m−adic completion of R. The ringˆR is local and its maximal ideal is m ˆ R. Our overall goal is to study extensionsˆν of ν tô R. 1 The ringˆR is not in general an integral domain, so we can only hope to extend ν to a pseudo-valuationˆν ofˆR. Let P be the support ofˆν, that is, the prime ideal which is mapped byˆν to ∞. That means precisely that we want to extend ν to a valuationˆν − of the quotientˆR