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I have the following question.

Suppose $(R, \mathfrak{m})$ is a local Noetherian ring, $\mathfrak{a}$ is $\mathfrak{p}$-primary ideal. Is it true that $\hat{\mathfrak{a}}$ is a primary ideal in $\hat{R}$, where $\hat{}$ mean $\mathfrak{m}$-adic completion?

Is above question true if $R$ is excellent ring and $(0)$ is a primary ideal with $\mathfrak{n}=\sqrt{(0)}$, such that $R/\mathfrak{n}$ is normal?

In the second case we know that $R/\mathfrak{n}$ is analytically normal, hence $\hat{R}/\mathfrak{n}\hat{R}$ is a domain. Thus $\mathfrak{n} \hat{R}$ is a prime ideal, but I don’t know if $(0)$ is a primary ideal.

Any help would be appreciated!

abcd1234
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  • Let $A=\mathbb Q[[x,y]]/(xy)$, which is a complete local ring containing $\mathbb Q$ and of depth $1$. As explained in https://stacks.math.columbia.edu/tag/0AL7 , $A$ is the completion of a Noetherian local domain $R$. Since $R$ is an integral domain, hence $(0)$ in $R$ is prime, so primary. The completion of $(0)$, which is again $(0)$ in $A$ is not primary since $A$ has two minimal primes – uno Nov 15 '24 at 10:05

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