Remarks on the computation of minimal finite free resolutions

Several improvements to the computation of the minimal free resolution of finite modules have been made also recently and some of them concerning ideals of polynomials are Hilbert driven or depend on the knowledge of the regularity of the ideal. Here I show that some of the calculations usually made to determine the minimal free resolution of a homogeneous polynomial ideal $I$ are still redundant, provided that we know a set of minimal homogeneous generators, the regularity, and the Hilbert function of $I$. More precisely, I show that the construction of some syzygies can be avoided. As result, the known methods for the computation of a minimal free resolution are improved. The underlying idea of this work can be also used to predict if certain points in generic position on general rational curves satisfy the minimal resolution conjecture.