Ring of invariants of finite subgroup of $GL_2(\mathbb{C})$

In the paper 'FINITE LINEAR GROUPS WHOSE RING OF INVARIANTS IS A COMPLETE INTERSECTION' by VICTOR KAC AND KEI-ICHI WATANABE published in BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 6, Number 2, March 1982, it is said in remark 2 of page no 222 that for any finite subgroup $G$ of $GL_2(\mathbb{C})$ the ring of invariance $\mathbb{C}[X_1, X_2]^G$ is always a complete itersection ring without any reference. Can anyone kindly tell me a reference for this result? Thanking you.

edited Mar 11 2011 at 20:31

Jim Humphreys
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asked Mar 11 2011 at 19:12

Anjan Gupta
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en.wikipedia.org/wiki/Du_Val_singularity – S. Carnahan♦ Mar 11 2011 at 19:21

I am also considering finite subgroups of $GL_2\mathbb(C)$ which are not contained in $SL_2\mathbb(C)$ – Anjan Gupta Mar 11 2011 at 20:06

Here's a link to the paper: ams.org/journals/bull/1982-06-02/… . The remark is wrong, almost certainly a typo. – Graham Leuschke Mar 11 2011 at 20:24

For an explicit example, take $G$ to be the three-element group generated by $\mathrm{diag}(e^{2\pi i/3},e^{2\pi i/3})$. The invariant ring is $\mathbb{C}[x^3,x^2y,xy^2,y^3]$, which is not even Gorenstein. – Graham Leuschke Mar 11 2011 at 20:32

As far as I know, the article of Kac and Watanabe is the state of the art. Just ignore this typo. Or see this section of Neusel's book, and the references there: books.google.com/… – Graham Leuschke Mar 12 2011 at 1:14

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