# Is Gorenstein singularity a locally analytic property

Let $X$ be a variety over $\mathbb{C}$, and $X^{an}$ be the analytic space associated to $X$. $X$ has Gorenstein singularity at $x \in X$ iff the local ring $\mathcal{O}_{X,x}$ is a Gorenstein ring. Is this result true:

$\mathcal{O}_{X,x}$ is a Gorenstein $\iff$ $\mathcal{O}_{X^{an},x}$ is a Gorenstein.

I want to show the following thing: Suppose $X,Y$ are varieties over $\mathbb{C}$, and $x \in X^{an}, y\in Y^{an}$. Suppose $U \subset X^{an}, V\subset Y^{an}$ are open sets in the Eucliden topology, and there exists an analytic isomorphism $f : U \to V$ sending $x$ to $y$. Then, if $X$ has Gorenstein singularity at $x$, it also has Gorenstein singularity at $y$. Certainly, if the aforementioned result holds, this result follows from it.

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The key point is that if $X$ is a locally finite type $\mathbf{C}$-scheme and $x \in X(\mathbf{C})$ then $O_{X,x}$ and $O_{X^{\rm{an}},x}$ are both local noetherian rings with the same completion (induced by the evident canonical map from the algebraic local ring to the analytic one). So for any property of local noetherian rings which is equivalent to check on the completion (such as the Gorenstein or CM properties), one gets immediately the equivalence on algebraic and analytic sides by comparison through common complete local rings.