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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 $Tor_{i}^{R_m}(M_m, R/m)$.

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By a well-known paper 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.

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  • $\begingroup$ Thank you, this is not well-known to me... I had a brief look at section 5 and 6 in particular prop 6.3. Matrix factorization of $x$ is powerful. Is there a general algorithm finding one 2x2 matrix factorization of $x$ with given cokernel $M=R/I$ where $I=(x_1,x_2)$ ? $\endgroup$ Feb 13, 2021 at 5:46
  • $\begingroup$ A typo, I mean with given cokernel $I$. $\endgroup$ Feb 13, 2021 at 5:58
  • $\begingroup$ If the hypersurface has degree $2$ and the ideal is two-generated, then the matrix would have size $2$ as well. One relation is Koszul, and the other one comes from the defining equation, similar to this example. $\endgroup$ Feb 13, 2021 at 6:02

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