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I am trying to understand this theorem, and as an example I try this case: Let $R=\mathbb{C}[z_1^{\pm},\ldots,z_n^{\pm}]^{S_n}$ be the algebra of Laurent polynomials. Now $R$ should be somehow finitely generated over some polynomial algebra, which I would presume to be $A=\mathbb{C}[s_1,\ldots,s_n]\equiv \mathbb{C}[z_1,\ldots,z_n]^{S_n}$ where $s_i$ are symmetric polynomials. On the other hand, due to the terms $1/z_i$, I think $R$ would be an infinite rank module over $A$.

Probably something is terribly wrong in my reasoning. Which is the polynomial algebra $A$ given by the theorem? Are there any other nice and simple examples of this theorem?

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    $\begingroup$ If $S_n$ means you take Laurent polynomials invariant under permutations of variables then how is it a module over $\mathbb C[z_1,...,z_n]$? $\endgroup$
    – მამუკა ჯიბლაძე
    Feb 7, 2018 at 17:47
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    $\begingroup$ Ohh god you are right... $\endgroup$
    – Ioannis Zolas
    Feb 7, 2018 at 17:51
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    $\begingroup$ I just have no idea how to find the actual algebra and I am making stupid mistakes $\endgroup$
    – Ioannis Zolas
    Feb 7, 2018 at 17:52

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