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Background: There is a well-known theorem that if $R$ is Noetherian, $I$ is an ideal in $R$, $0\rightarrow M_1\rightarrow M_2\rightarrow M_3\rightarrow 0$ an exact sequence of finitely generated $R$-modules, and for each $i$, $\widehat{M}_i$ is the $I$-adic completion of $M_i$, then there is an induced natural exact sequence $0\rightarrow \widehat{M}_1\rightarrow \widehat{M}_2\rightarrow \widehat{M}_3\rightarrow 0$. As a consequence, $\widehat{M}_3$ is isomorphic to $\widehat{M}_2/\widehat{M}_1$.

Question: Is it also true that $\widehat{M}_3$ is isomorphic to $\widehat{M}_2/\widehat{M}_1$ with respect to their topologies? If it's true, how can I prove it?

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I've put your equations into LaTeX. –  Zev Chonoles Jul 12 '11 at 23:41
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The same theorem (namely the Artin--Rees lemma) that proves that $\hat{M}_1/\hat{M}_2 \buildrel \sim \over \longrightarrow \hat{M}_3$ also proves directly that this isomorphism is a homeomorphism when $\hat{M}_1/\hat{M}_2$ is equipped with the quotient topology induced by the $I$-adic topology on $\hat{M}_2$ and $\hat{M}_3$ is equipped with its $I$-adic topology. (It furthermore proves that the $I$-adic topology on $\hat{M}_2$ induces the $I$-adic topology on $\hat{M}_1$.)

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