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?