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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.

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1 Answer 1

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When $d\geq 3$, these are isolated hypersurface singularities of dimension at least $4$, so are UFD by the Grothendieck's local Lefschetz Theorem.

When $d=2$ and the field has characteristic $0$, the class group is $\mathbb Z^{r-1}$ where $r$ is the number of branches of $g$. See 2.2.6 of Kollár's paper "Flip, flops, minimal models, etc". That result was for the Picard group of the punctured spectrum of the affine hypersurface, but it should agree with the class group of the completed local rings (you might need to use old results by Danilov here).

I do not know if anyone has worked out the case of positive char. But at least we know that the class group is torsion-free, see the references in this MO question and answer.

A good reference for local Picard group (which for a normal local point of dimension $2$ or higher is just the class group at that point) is Kollár's paper "Maps between local Picard groups", where you can also find reference to the Grothendieck-Lefchetz theorem mentioned above.

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    $\begingroup$ @uno: Then translate everything by $1$. $\endgroup$ Commented Apr 4, 2021 at 14:56
  • $\begingroup$ My bad here. There was a typo in the notation for $d$ in my original question. Then I changed it, but I didn't realize that meanwhile Long had already posted! By the way, thank you @HailongDao for your answer, very interesting that the only non-trivial cases are in dimensions $2$ and $3$ $\endgroup$
    – Alessio
    Commented Apr 4, 2021 at 17:46
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    $\begingroup$ @Alessio: now it makes sense. Looks like a few people have downvoted my answer because of that. (-: Ah well, I don't really want to bump my answer up by trivial editing, so let's leave it like that. $\endgroup$ Commented Apr 4, 2021 at 17:56
  • $\begingroup$ Sorry for that! :-( I don't know much how the point system works here on MO, but if I can do anything to bump your question up, I'll do. It's a good one! $\endgroup$
    – Alessio
    Commented Apr 5, 2021 at 8:24
  • $\begingroup$ @Alessio: it's ok, I don't really care, as long as it was useful. $\endgroup$ Commented Apr 5, 2021 at 16:50

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