Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Do prime ideals expand to prime ideals in the completion?

I believe this is the case since I think $R/P\equiv \hat{R}/P\hat{R}$, although Atiyah-Macdonald explicitly mentions the preservation of quotients only with respect to powers of maximal ideal (I am assuming completion with respect to the maximal ideal here).

EDIT: I guess there is more going on here. If I take $S=k[x]$ with $k$ a field, then $(x+1)S$ is prime in $S$ but $x+1$ is a unit $k[[x]]$. So the modified question is:

When do prime ideals expand to prime ideals in the completion?

share|cite|improve this question
For the original question, the answer is no. Consider the ideal generated by $y^2-x^2(x+1)$ in $\mathbb C[x,y]_m$ where $m$ is the maximal ideal $(x,y)$ of $\mathbb C[x,y]$. In your edit, $S$ is no longer a local ring. The modified question is too general. – user18119 Mar 14 '12 at 21:09
@QiL: Thanks for pointing out that mistake and for the counterexample. – Jake Voigt Mar 14 '12 at 21:21
Dear @QiL, believe it or not, I hadn't seen your comment when I posted my answer. I suppose I had started to write and then came back to my answer after some interruption. Anyway you were first but the similarity of our examples is amusing. – Georges Elencwajg Mar 15 '12 at 1:21
@GeorgesElencwajg, don't worry about that. I understood as such. – user18119 Mar 15 '12 at 12:21
up vote 3 down vote accepted

No, this is not true: prime ideals do not extend to prime ideals after completion.

Consider $S=\mathbb C[X,Y]$ and its localization $R=S_M$at $M=(X,Y)\subset S$, so that $R=\mathbb C[X,Y]_{(X,Y)}$ is a local ring with maximal ideal $\mathfrak m=(X,Y)R$.
The completion of $R$ along $\mathfrak m$ is the ring of formal power series $\hat R=\mathbb C[[X,Y]]$.
The principal prime ideal $\mathfrak p=(Y^2-X^2-X^3)\subset R$ has as extension the principal ideal $\hat {\mathfrak p}=\mathfrak p\hat R=(Y^2-X^2-X^3)\hat R$ which is no longer prime:
Indeed $Y^2-X^2-X^3=(Y+X\sqrt {1+X})(Y-X\sqrt {1+X})\in \hat {\mathfrak p}\;$ although $Y+ X\sqrt {1+X},Y-X\sqrt {1+X}\notin \hat{\mathfrak p}$

share|cite|improve this answer

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.