I am trying to find out the kernel (syzygy) in a free resolution.
Here $R = K[x,y,z]$ where $K$ is a field. I am trying to resolve the ideal $M=(x,y,z)$. I have the following resolution.
$ \phi_0 : R \rightarrow R/M$ by natural map. I get kernel as $(x,y,z)$. Now I need to map $R^3$ to $R$.
$\phi_1 : R^3 \rightarrow R$ by mapping the 3 generators of $R^3$ to $x, y$ and $z$ respectively. Now I get that
$\begin{bmatrix} 0 &z &-y \\-z &0 &x \\y &-x &0 \end{bmatrix}$ will be in the kernel. But I am not being able to show that the kernel is itself generated by this.