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I have some confusion about the Hilbert Syzygy theorem, can anyone give me an detailed example about how to find an Hilbert's chain of syzygies? Also, for a non regular local ring $R$, give me an example showing that the chain of syzygies goes to infinity.

Another question is $\mathbb{C}[[t^{2},t^{3}]]=\mathbb{C}[[x,y]]/(y^{2}-x^{3})$, I want to know how to get this isomorphism.

The third one is that the countable direct product of $\mathbb{Z}$ is not free, how to prove this?

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    Ok, thanks for your advice:)2012-12-28

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It seems that you didn't follow the good advices to split this question into three. This can suggest that you are not very interested in your own question(s), so I'll give you only some hints and a link.

(1) The package Macaulay2 can help you more than anyone to find all the syzygies you want. For the second part take $R=\mathbb Z_4$ and the $R$-module $\hat 2\mathbb Z_4$.

(2) Prove that $\mathbb C[X,Y]/(Y^2-X^3)\simeq \mathbb C[t^2,t^3]$ and then take the completions.

(3) See here, Example 3.5.

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    Thanks so much, I am interested in this, but I am quite busy taking the quals , so I will do something after my exams, thanks again.2013-01-04