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Recall that Koszul Complex $K.(f,g)$ of polynomials $f,g \in k[x_1,\ldots,x_n]=:R$ is defined as:$$0 \to R \overset{\phi_1} \to R^2 \overset{\phi_2} \to R \to 0$$ where $\phi_1$ and $\phi_2$ are defined by the matrices $\phi_1=\begin{bmatrix}g\\-f\end{bmatrix}$ and $ \phi_2=\begin{bmatrix}f & g\end{bmatrix}$

Is $K.(f,g)$ always a minimal free resolution?

I know that for $K.(f,g)$ to be minimal one sufficient condition is that $f,g$ homogeneous and $\{f,g\}$ a regular sequence. So i think the answer to above question is no, but I'm unable in producing a counterexample.

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    $\begingroup$ What if they have a common factor? $\endgroup$ – Hoot Feb 16 '16 at 18:23
  • $\begingroup$ Or are both $0$? $\endgroup$ – Remy Feb 18 '16 at 6:27

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