Shapes of free resolutions over a local ring
  We classify the possible shapes of minimal free resolutions over a regular local ring. This illustrates the existence of free resolutions whose Betti numbers behave in surprisingly pathological ways. We also give an asymptotic characterization of the possible shapes of minimal free resolutions over hypersurface rings. Our key new technique uses asymptotic arguments to study formal Q-Betti sequences. 
Christine Berkesch, Daniel Erman, Manoj Kummini, Steven V Sam
math.AC