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.



Is completion of isolated singularity isolated?
I believe that in your situation, $B$ indeed has an isolated singularity at the maximal ideal $\mathfrak{n} \subseteq B$. Let me first give two possible definitions for “isolated singularity”; please let me know if there are standard definitions for these notions, and I will edit this answer accordingly!


**Definition.** Let $X$ be a locally noetherian scheme.


1. We say that $X$ has an *isolated singularity* at a closed point $x \in X$ if there is an open neighborhood $U \ni x$ such that the scheme $U \smallsetminus \{x\}$ is regular.
2. Suppose that $X$ is a scheme over a field $k$. We say that $X$ has a *geometric isolated singularity* at a closed point $x \in X$ if there is an open neighborhood $U \ni x$ such that the scheme $U \smallsetminus \{x\}$ is geometrically regular over $k$.


Adopting either definition, the ring $B$ in your situation has a isolated singularity (resp. geometric isolated singularity) at the maximal ideal $\mathfrak{n} \subseteq B$. We can in fact prove the following more general result:


**Proposition.** *Let $X$ be a locally noetherian scheme (resp. locally noetherian scheme over a field $k$) with an isolated singularity (resp. geometric isolated singularity) at a closed point $x \in X$, such that $\mathcal{O}\_{X,x}$ is a $G$-ring. Then, $\operatorname{Spec}\widehat{\mathcal{O}}\_{X,x}$ has an isolated singularity (resp. geometric isolated singularity) at the unique closed point $\widehat{x}$.*


Note that this proposition applies to your situation, since $A$ is of fintie type over a field, hence a $G$-ring in the sense of [Mat89, p. 256] by [Mat89, Cor. to Thm. 32.6] ($A$ is also excellent, but we only need the $G$-ring part of the definition of excellence).


*Proof.* Let $U \subseteq \operatorname{Spec} \mathcal{O}\_{X,x}$ be the intersection of $\operatorname{Spec} \mathcal{O}\_{X,x}$ and a neighborhood of $x$ in $X$ satisfying the condition in the definition above. Let $f\colon \operatorname{Spec} \widehat{\mathcal{O}}\_{X,x} \to \operatorname{Spec} \mathcal{O}\_{X,x}$ be the morphism induced by the completion homomorphism. The morphism
$$f^{-1}(U) \smallsetminus \{\widehat{x}\} \longrightarrow U \smallsetminus \{x\}\tag{1}\label{eq:completion}$$
is regular in the sense of [Mat89, p. 255] since the morphism $f$ is regular by the $G$-ring condition, and regular morphisms are preserved under base change [EGAIV$\_2$, Prop. 6.8.3(iii)].


For isolated singularities, we apply [Mat89, Thm. 23.7] to the (localizations of the) morphism \eqref{eq:completion} to conclude that $f^{-1}(U) \smallsetminus \{\widehat{x}\}$ is regular.


For geometric isolated singularities, the composition
$$f^{-1}(U) \smallsetminus \{\widehat{x}\} \longrightarrow U \smallsetminus \{x\} \longrightarrow \operatorname{Spec} k$$
is regular by [Mat89, Thm. 32.1], i.e., $f^{-1}(U) \smallsetminus \{\widehat{x}\}$ is geometrically regular over $k$. $\blacksquare$


### References


[EGAIV$\_2$] Alexander Grothendieck and Jean Dieudonné. “Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II.” *Inst. Hautes Études Sci. Publ. Math.* (1965), no. 24, 1–231. DOI: [10.1007/BF02684322](https://doi.org/10.1007/BF02684322). MR: [199181](https://mathscinet.ams.org/mathscinet-getitem?mr=199181).


[Mat89] Hideyuki Matsumura. *Commutative ring theory.* Second ed. Translated from the Japanese by M. Reid. Cambridge Stud. Adv. Math. 8. Cambridge Univ. Press, Cambridge, 1989. DOI: [10.1017/CBO9781139171762](https://doi.org/10.1017/CBO9781139171762). MR: [1011461](https://mathscinet.ams.org/mathscinet-getitem?mr=1011461).


