Completions of local morphisms and valuations

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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.