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Let ( R ,  m ,  k ) be a local noetherian domain with field of fractions  K  and  R  ν  a valuation ring, dominating  R  (not necessarily birationally). Let  ν |  K  : K * ↠Γ be the restriction of  ν  to  K  by definition,  ν |  K   is centred at  R . Let  R^ denote the  m ‐adic completion of  R . In the applications of valuation theory to commutative algebra and the study of singularities, one is often induced to replace  R  by its  m ‐adic completion  R^  and  ν  by a suitable extension  ν^  −  to  R^ / P  for a suitably chosen prime ideal  P , such that  P ∩ R =(0). The purpose of this paper is to give, assuming that  R  is excellent, a systematic description of all such extensions  ν^  −  and to identify certain classes of extensions which are of particular interest for applications.

Extending a valuation centred in a local domain to the formal completion