Right now I'm trying to find the Hilbert Function , and the corresponding Hilbert Polynomial for the ring $M=k[x,y,z,w]/(x,y) \cap (z,w)$. I just finished reading the first chapter of Eisenbud, so I don't have that much of an advanced toolbox for things like this yet, (I know almost nothing about Gröbner bases). My first attempt was to construct a free resolution, let $ R = k[x,y,z,w]$. We then have the free resolution: $0 \rightarrow R \rightarrow^{d_3} R^4 \rightarrow^{d_2} R^4 \rightarrow^{d_1} R \rightarrow^{d_0} M \rightarrow 0, $where $d_i$ sends the free generators onto the given generators of the next coming module. So, $R^4$ sends the generators onto the generators of the kernel of $d_0$ etc. My answer right now, which I've been getting from this, is that the Hilbert Function should be : $H_m(s) =\binom{3 + s}{3}- 4\binom{1 +s}{ 3} + 3\binom{2 + s}{ 3}.$
However, I have been testing this out with macaulay2, and it doesn't agree with my answer, so I suspect I'm not right. Any help would be nice.