Welcome to MathOverflow

Question

1

2

Let $\hat{R}$ is $m$-adic completion of a local ring $(R,m)$.What is the relation between $Min R$ and $Min \hat{R}$. we know that $\hat{R}$ is faithfully flat $R$-module. $Min R$=set of all minimal prime divisors of zero. I think

$p\in Min(R)$ iff $\hat{p}=p\hat{R}\in Min\hat{R}$.

edited May 12 2012 at 23:19

asked May 12 2012 at 6:34

anna
266●1●8

4

If $p$ is a minimal prime of $R$, then $p \hat{R}$ need not be prime. See mathoverflow.net/questions/18496/… – Dustin Cartwright May 12 2012 at 8:53

Mahdi Majidi-Zolbanin · Accepted Answer · 2012-05-13 02:23:24Z UTC

1

What you are sayinng is not true in general. For example, the completion of a domain may not be a domain. An example is given in exercises of Bourbaki's Commutative Algebra.

answered May 13 2012 at 2:23

Mahdi Majidi-Zolbanin
1,792●1●2●11

To be more precise, the example in Bourbaki exercises is the following: $K$ is an algebraically closed field of char $0$, $R=K[X,Y]$, $p=X(X^2+Y^2)+(X^2-Y^2)$. Then one can show the ideal $pR$ is prime. So $R/pR$ is a domain. If $\mathfrak{m}$ is the maximal ideal of $R/pR$ generated by $X$ and $Y$, then the exercise asks to show that the $\mathfrak{m}$-adic completion of $R/pR$ is not a domain. The given hint is to show that $p$ decomposes into a product of two formal power series in $K[[X,Y]]$. – Mahdi Majidi-Zolbanin May 15 2012 at 3:12

relation between Min(R) and Min(R^)

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.