If $(R,m)$ is a complete local ring (with respect to the $m-$adic topology) and $I$ a prime ideal in $R$, is $R/I$ complete (with respect to the $m/I-$adic topology)? It seems too strong, but I am unable to give a counterexample.
Is the quotient of a complete ring, complete?
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abstract-algebra
commutative-algebra
ring-theory
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0complete for which topology? – 2012-03-07
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0@seporhau: Clarified. I thought, it was convention to assume "with respect to the maximal ideal" unless otherwise specified. – 2012-03-07
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1What you'd like to be able to do is take a Cauchy sequence in $R/I$, then lift each element so that you have a Cauchy sequence in $R$, then take the limit and push it back down to the quotient. So the question becomes: under what conditions can you lift your Cauchy sequence mod $I$ to a Cauchy sequence in $R$?. – 2012-03-07
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2The completion functor from the category of $R$-modules to the catogory of $\hat R$-modules is exact. So the quotient by an ideal $I \subset R=\hat{R}$ is complete if and only if $I$ is complete as an $\hat R$-module. I tend to think that this is always the case at least for Noetherian $R$. – 2012-03-07
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0@DimaSustretov: Thank you. – 2012-03-08