# Need Counterexample to show Koszul complex is not minimal free resolution?

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.

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