Completions of local morphisms and valuations

Abstract

In this note we study injective local morphisms phi : (R, m) --> (S, n) of local excellent domains. In particular we are interested in the problem when the n-adic topology on S restricts to a topology on R that is linearly equivalent to the m-adic topology. Using a valuative criterion, we prove this in case R is analytically irreducible and S/R is essentially of finite type, and we recover and extend a weak version of Gabrielov's rank condition.