piF = directImageComplex F
We give an application of this function to create pure free resolutions. A much more general tool for doing this is in the script pureResolution.
A "pure free resolution of type (d_0,d_1,..,d_n)" is a resolution of a graded CohenMacaulay module M over a polynomial ring such that for each i = 1,..,n, the module of ith syzygies of M is generated by syzygies of degree d_i. Eisenbud and Schreyer constructed such free resolutions in all characteristics and for all degree sequences $d_0 < d_1 < \cdots < d_n$ by pushing forward appropriate twists of a Koszul complex. (The construction was known for the EagonNorthcott complex since work of Kempf).
If one of the differences $d_{i+1}  d_i$ is equal to 1, then it turns out that one of the maps in the pure resolution is part of the map of complexes directImageComplex k_j, where k_j is a map in this Koszul complex. Here is a simple example, where we produce one of the complexes in the family that included the EagonNorthcott complex (see for example "Commutative Algebra with a View toward Algebraic Geometry", by D. Eisenbud.)






The direct image complexes each have only one nonzero term,and so D has only one nonzero component. According to Eisenbud and Schreyer, this is the last map in a pure resolution. Since the dual of a pure resolution is again a resolution, we can simply take the dual of this map and resolve to see the dual of the resolution (or dualize again to see the resolution itself, which is the EagonNorthcott complex itself in this case.


