Let $k$ be an algebraically closed field of characteristic different from $2,3$ and $5$, and let $R=k[[x,y,x_2,\dots,x_d]]/(f)$, where $f\in(x,y,x_2,\dots,x_d)^2$, $f\neq0$. By results of Buchweitz-Greuel-Schreyer and Knörrer, $R$ has finite Cohen-Macaulay representation type if and only if $R\cong k[[x,y,x_2,\dots,x_d]]/(g+x_2^2+\cdots+x_d^2)$, where $g\in k[x,y]$ is one of the following polynomials (one-dimensional ADE singularities):

- ($A_n$) $x^2+y^{n+1}$, $n\geq1$
- ($D_n$) $x^2y+y^{n-1}$, $n\geq4$
- ($E_6$) $x^3+y^4$
- ($E_7$) $x^3+xy^3$
- ($E_8$) $x^3+y^5$

I am interested in the class group $\mathrm{Cl(R)}$ of these rings, for any $d\geq1$. Has anyone already computed it?

For $d=2$, i.e., two-dimensional ADE singularities, the class group is known and can be computed using the Auslander-Reiten quiver of $R$ (see e.g. [Yos90, $\S13$]). However, I could not find similar (even partial) computations for $d>2$ in the literature. Could you point me to some references in this direction?

Thank you in advance!

**References**:

[Yos90] Yoshino, Yuji, *Cohen-Macaulay modules over Cohen-Macaulay rings*. London Mathematical Society Lecture Note Series, 146. Cambridge University Press, Cambridge, 1990.