# Normalization of complete intersection

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.

• No. Every variety is the normalization of a hypersurface. mathoverflow.net/questions/68246/… May 26, 2016 at 2:15
• @KarlSchwede: Well, every normal (projective) variety! May 26, 2016 at 6:02
• :-) that is true May 26, 2016 at 6:39
• Cross-link to meta discussion about why this question was closed. May 26, 2016 at 17:06

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.