I believed for a while that the reason you can't find a finite free resolution (with the free modules finitely generated) of $(x - 1, y - 1)$ as a module over $\mathbb{C}[x, y]/(y^2 - x^3)$ is that $y^2 = x^3$ isn't smooth.
I thought the singularity created "noise," so that you must localise at $(x - 1, y - 1),$ allowing you to find a finite projective resolution.
Then I tried to find a finite free (finitely generated) resolution of $(x + 1, y)$ as a module over $\mathbb{C}[x, y]/(y^2 - (x^3 + 1))$ and couldn't, despite this being the coordinate ring of a smooth elliptic curve.
So when can we find finite free (finitely generated) resolutions over smooth elliptic curves? Furthermore, if I'm right that $(x + 1, y)$ has no finite free (finitely generated) resolution, can I get any intuition as to why?