# (Infinite) free resolution of $R/(x-z, y-w)$ for $R=\mathbb C[x,y,z,w]/(xy-zw)$

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

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.
• 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 $I=(x_1,x_2)$ ? Feb 13, 2021 at 5:46
• A typo, I mean with given cokernel $I$. Feb 13, 2021 at 5:58
• 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. Feb 13, 2021 at 6:02