Let $A$ be an integral complete local ring over a field which is complete intersection.

Let $B$ be a normalization of $A$.

## Q. Is $B$ Gorenstein?

I guess that even the normalization of Gorenstein local ring should be Gorenstein.

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## Q. Is $B$ Gorenstein?

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Let $A$ be an integral complete local ring over a field which is complete intersection.

Let $B$ be a normalization of $A$.

I guess that even the normalization of Gorenstein local ring should be Gorenstein.

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Consider $A=k[[x^3,x^2y,y^3]]\subset k[[x^3, x^2y, xy^2, y^3]]=B$. $B$ is the integral closure of $A$, $A$ is a hypersurface, but $B$ is not Gorenstein.

normal(projective) variety! $\endgroup$