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

In the paper:

Kac, Victor; Watanabe, Keiichi, Finite linear groups whose ring of invariants is a complete intersection. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221–223

it is said in Remark 2 on page 222 that for any finite subgroup $G$ of $GL_2(\mathbb{C})$ the ring of invariants $\mathbb{C}[X_1, X_2]^G$ is always a complete intersection ring without any reference.

Can anyone kindly tell me a reference for this result? Thank you.

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I am also considering finite subgroups of $GL_2\mathbb(C)$ which are not contained in $SL_2\mathbb(C)$ – Anjan Gupta Mar 11 '11 at 20:06
Here's a link to the paper: ams.org/journals/bull/1982-06-02/S0273-0979-1982-14989-8 . The remark is wrong, almost certainly a typo. – Graham Leuschke Mar 11 '11 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 '11 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 '11 at 1:14