Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|cite|improve this question
Since inverse limits commute, I believe this is true. –  the L Nov 30 '11 at 13:53
@Liran: This is a bit too easy - you forgot about the quotient rings. –  Martin Brandenburg Nov 30 '11 at 20:11
Moreover (following Liran Shaul's comment), if we define $I=\cap_\lambda a_\lambda$, then isn't $A$ equal to $R/I$, so $\frak m$-complete? We can assume $I=0$, and then any sequence $(r_n)$ in $R$ that is Cauchy in the $a_\lambda$-topology is also Cauchy in the $\frak m$-topology, so has a limit in $R$. –  inkspot Nov 30 '11 at 20:21

1 Answer 1

up vote 5 down vote accepted

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.

share|cite|improve this answer
I'm not sure why there was some difficulty getting the latex to work out; it looks like it was choking on the \frak commands. I did the best I could. –  Todd Trimble Dec 1 '11 at 14:43
Thanks a lot, Todd Trimble. –  Pham Hung Quy Dec 2 '11 at 14:35
Can you please provide a reference for this Theorem of Chevalley? –  the L Jan 8 '12 at 13:09
Lemma 7 - Chevalley: On theory of local ring, Annals of Math. 1946, 690-708. Theorem 2.1 - P. Schenzel, On use of local cohomology in algebra and geometry, in: six lecture of commutative algebre, 1998 (you can seacher it on internet) –  Pham Hung Quy Jan 9 '12 at 12:16

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.