For a Noetherian local ring $R$, Koszul complex is a useful tool to construct finite free resolution of any quotient of $R$ by regular sequences $R/(x_1,\dots, x_r)$.

Let $R=\mathbb C[x,y,z,w]/(xy-zw)$, and $M=R/(x-z,y-w) \cong \mathbb C[x,y]$. $x-z, y-w$ don't form a regular sequence of $R$, which can be checked directly or using that $R$ is Cohen-Macaulay and $\dim R- \dim M =1$.

The question is, how to construct a natural (locally) free resolution of $M$ and describe the boundary maps explicitly? It seems to be infinite.

Maybe a more local question is, how to construct a minimal free resolution after localization at $\mathcal{m}=(x,y,z,w)$? The motivation is to compute dimension of $Tor_{i}^{R_m}(M_m, R/m)$.