Let $K$ be an algebraically closed field and let $A=K[x_1,\dots,x_n]/I$ be a $K$-algebra of finite type which has only an isolated singularity at the origin. Let $\mathfrak{m}=(x_1,\dots,x_n)$ and consider the localization $A_{\mathfrak{m}}$ and its $\mathfrak{m}$-adic completion $B$.

Is $B$ also an isolated singularity? The localization at $\mathfrak{m}$ should not be a problem, but completion might. In general, the completion of an isolated singularity need not to be isolated. However, I was wondering whether in this situation the answer might be positive. I am willing to assume some stronger hypothesis. Can anybody point out any results in this direction?

Thank you in advance.