Geodesic
Formal relations between analynic functions
Izvestiya. Mathematics , Tome 7 (1973) no. 5, pp. 1056-1088.

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In this paper we give conditions under which the completion of the kernel of a homomorphism of analytic rings $\varphi\colon A\to B$ coincides with the kernel of the corresponding homomorphism of the completions $\widehat\varphi\colon\widehat A\to\widehat B$. 
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A. M. Gabrièlov. Formal relations between analynic functions. Izvestiya. Mathematics , Tome 7 (1973) no. 5, pp. 1056-1088. http://geodesic.mathdoc.fr/item/IM2_1973_7_5_a4/

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