Completions of local morphisms and valuations

Abstract.

In this note we study injective local morphisms $\varphi: (R,{\mathfrak m})\rightarrow (S,{\mathfrak n})$ of local excellent domains. In particular we are interested in the problem when the ${\mathfrak n}$–adic topology onS restricts to a topology on R that is linearly equivalent to the ${\mathfrak m}$–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.