I have made the mistake multiple times of assuming that $(x,y)$ is a projective $\mathbb{C}[x,y]$ module by thinking that $(x,y) \oplus \mathbb{C} \cong\mathbb{C}[x,y]$ as $\mathbb{C}[x,y]$ modules, but I have been told this is false.
What then is a projective resolution of $(x,y)$ as a $\mathbb{C}[x,y]$ module? I know that a free resolution exists, but how do I compute it?
Let $A= \mathbb C[X,Y]$, $I$ your ideal. There is a surjection $\varepsilon : A^2 \to I$ that assigns $(p,q) \longmapsto Xp+Yq$. What is its kernel? Try to continue this way. You cannot get past something of length two, so you haven't much work ahead.