# the localization of R at P is a regular local ring then R is regular local ring

We know the fact that if $R$ is a regular local ring then $R_{P}$ the localization of $R$ at $P$, $P\in\mathrm{Spec}(R)$ is a regular local ring.

So, I wonder the converse is true or not? My counter-example is taking $R=\mathbb Z$.

-If $P=(0)$ then $R_{P}$ is a field so $R_{P}$ is a regular local ring

-If $P=(p)$, $p$ is a prime number then $\dim R_{p}=1=v(PR_{P})$ (generate by $\frac{p}{1}$) so it is also a regular local ring.

However, $\mathbb Z$ is not a regular local ring. Can you check that to me please?

• I know it is not true. I just wanna know my counter example is true or false. Jan 9, 2017 at 9:58
• Because I do not know if the localization of R at P is a regular local ring with every P in Spec(R) then is R a regular local ring? It is not true and I try to find the example of that. Jan 9, 2017 at 10:02
• Oh I see your meaning. You mean that $R$ is regular iff $R_{P}$ is regular local ring with every prime ideal $P \in Spec(R)$. But it is just quite similar to my question. The theorem (Serre 1955) is: Suppose that $(R;m)$ is a regular local ring. Then $R_{P}$ is again a regular local ring for every a prime ideal $P \in$R$. I wonder the converse of this theorem is true or not? so we just find some rings which is not local then the converse is not true. Jan 9, 2017 at 10:26 • No, it is if every$P\in Spec(R)$the localization of$R$at P is regular local ring then is R regular local ring? Jan 9, 2017 at 10:29 • But how about your meaning? Can you give me the example? @user26857 Jan 9, 2017 at 10:45 ## 1 Answer Let$I$be a prime ideal of$R_P$, then$I^c$is a prime ideal of$\mathbb{Z}$, so$I^c=q\mathbb{Z}$for some prime number$q$. We have$I=I^{ce}=(q\mathbb{Z})^e=(\frac{q}{1})$. However,$q\mathbb{Z}$is maximal so$I$is also maximal. So an arbitrary nonzero prime ideal of$R_P$is maximal, which means$\dim R_P=1=v(PR_P)\$

• Yes, I see that is what I mention. :D Jan 9, 2017 at 8:28
• O...k. So what do need to check?
– T C
Jan 9, 2017 at 8:28
• I just wanna know it is true or false :). Jan 9, 2017 at 8:33
• Well, it's true then
– T C
Jan 9, 2017 at 8:37