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


1 Answer 1


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.