Is $R=K[|x^3,x^2y,xy^2,y^3|]$ a Gorenstein or a regular ring?

Question

Let $K$ be a field and $R=K[|x^3,x^2y,xy^2,y^3|]$ the ring of formal power series. Is $R$ a Gorenstein ring?

$R$ is Cohen-Macaulay of dimension 2.

So, I have to check if $\operatorname{Ext}^2_{K}(K,R)=K.$ I need some hints about that.

Answer 1

No, this is a quotient singularity where $K_X$ has index three. See the answer of Karl Schwede to the following question.

https://mathoverflow.net/questions/55526/example-of-a-variety-with-k-x-mathbb-q-cartier-but-not-cartier