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 $Tor\_{i}^{R\_m}(M\_m, R/m)$.



(Infinite) free resolution of $R/(x-z, y-w)$ for $R=\mathbb C[x,y,z,w]/(xy-zw)$
By a well-known [paper](https://www.ams.org/journals/tran/1980-260-01/S0002-9947-1980-0570778-7/S0002-9947-1980-0570778-7.pdf) by Eisenbud, over hypersurfaces the resolution of any module becomes periodic of period at most $2$ once after $depth(R)-depth(M)$ steps.


In your case the first map is just embedding of $I=(x-z,y-w)$ into $R$, and since the depth difference is $1$, the maps are $\begin{bmatrix} &y &w \\ &z &x\end{bmatrix}$ and $\begin{bmatrix} &x &-w \\ &-z &y\end{bmatrix}$, and repeating afterwards.


