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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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    well, if I only give one question, it is too short that I can not submit, so I just get them together, I am a new user here, does this mean anything different? Thanks, Merry Christmas2012-12-25
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    Thanks so much for your advice, I will change my way of posting questions, and also I wonder if the question is too short, what should I do to get it submitted?2012-12-25
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    If the question itself is too short, you can add something about why you are interested in it, what you already know so far, etc., which is a good idea to do anyway.2012-12-25
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    Ok, thanks for your advice:)2012-12-28

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