Cohomological Degrees and Hilbert Functions of Graded Modules

Making use of the recent construction of cohomological degrees functions, we give several estimates on the relationship between number of generators and degrees of ideals and modules with applications to Hilbert functions. They extend results heretofore known from generalized Cohen-Macaulay local rings to nearly arbitrary local rings. The rules of computation these functions satisfy enables comparison with Castelnuovo-Mumford's regularity in the graded case. As application, we derive sharp improvements on predicting the outcome of effecting Noether normalizations in tangent cones.