working in $\mathbb{R}[x,y,z]$ I would like to know free resolutions of ideals generated by 4 elements.
If $I$ is generated by 3 elements and these do not have a common zero then they are a regular sequence and thus the only syzygies are the Koszul syzygies.
What can one say if $I$ is generated by 4 elements (with no common zero)? I think there always exist 3 elements in $I$ that form a regular sequence but I dont really have any more ideas.
This might just be well known but I have no references if it is.
Thx in advance