I have a ring $R=k[S]$ which defines an affine toric variety $X_\sigma$, where $S=M\cap \sigma^\vee$ is the semigroup from a rational polyhedral cone $\sigma$. Let $I$ be the ideal for the toric boundary divisor of $X_\sigma$, in other words, $I$ is obtained from the toric ideal of $S$ that is $>0$ on the closure of $\sigma$. I want to know if the $I$-adic completion $\hat{R}$ of $R$ is still an integral domain?
A simple example is $R=k[x,y]$, and $I$ is the principal ideal $(xy)$. Is the completion an integral domain?
Best Answer
The completion of $R=k[x,y]$ w.r.t. to $(xy)$ (or any principal ideal) is indeed a domain. It is isomorphic to $R[[t]]/(t-xy)$, so we must prove that the ideal $(t-xy)$ is prime. To do that we can localize along $\mathfrak{m}:=(x,y)$; then the local ring $R_{\mathfrak{m}}[[t]]$ is regular, hence factorial, so we must show that $t-xy$ is irreducible in $R_{\mathfrak{m}}[[t]]$; this is clear since it is irreducible in $k[[x,y,t]]$.