Let $(R,\mathfrak{m})$ be a complete local ring, $a\_{\lambda}$ be a decreasing net of ideals in $R$, indexed by a directed set. Consider the completion under $a\_{\lambda}$-topology $A=\underleftarrow{\lim} R/\mathfrak{a}\_{\lambda}$. Is $A$ still complete under the $\mathfrak{m}$-topology?



On the Completion of a complete local ring
I assume that $(R, \frak{m} )$ is a complete Noetherian local ring. Set $\frak{a} = \bigcap\_\lambda \frak{a}\_\lambda$. By passing to $R/\frak{a}$ we may assume that the $\frak{a}\_\lambda$-topology is separated. Now, we use a Theorem of Chevalley (1946) which says that in the complete Noetherian local ring the $\frak{m}$-adic topology is weaker than every separated topology. Then every Cauchy sequence in the $\frak{a}\_\lambda$-topology is also a Cauchy sequence in the $\frak{m}$-adic topology. Hence $R = \underleftarrow{\lim} R/\frak{a}\_\lambda$ is complete. 


