First we consider the holomorphic Koszul complex on $\mathbb{C}^2$: $$ 0\to \mathcal{O}(\mathbb{C}^2)\overset{\begin{pmatrix}-z_2\\z_1\end{pmatrix}}{\to} \mathcal{O}(\mathbb{C}^2)^{\oplus 2}\overset{(z_1,z_2)}{\to} \mathcal{O}(\mathbb{C}^2)\to \mathbb{C}\to 0. $$ It is well-known that the above complex is exact.

Now we consider the $C^{\infty}$-version of the Koszul complex $$ 0\to C^{\infty}(\mathbb{C}^2)\overset{\begin{pmatrix}-z_2\\z_1\end{pmatrix}}{\to} C^{\infty}(\mathbb{C}^2)^{\oplus 2}\overset{(z_1,z_2)}{\to} C^{\infty}(\mathbb{C}^2)\to \mathbb{C}\to 0. $$ It is clear that the above complex is not exact. For example $\bar{z_1}$ and $\bar{z_2}$ both vanish at the origin but are not in the image of multiplying $z_1$ and $z_2$.

My question is: What is the cohomology of the above complex?