Artin Level Modules

We introduce level modules and show that these form a natural class of modules over a polynomial ring. We prove that there exist compressed level modules, i.e., level modules with the expected maximal Hilbert function, given socle type and the number of generators. We also show how to use the theory of level modules to compute minimal free resolutions of the coordinate ring of points from the back, which reveals new examples where random sets of points fail to satisfy the minimal resolution conjecture.