Growth conditions on Hilbert functions of modules
Gotzmann's Regularity Theorem uses a binomial representation of the Hilbert polynomial of a standard graded algebra to establish a bound on Castelnuovo-Mumford regularity. Using this and his Persistence Theorem, Gotzmann provided an explicit construction of the Hilbert scheme. This author will show that Gotzmann's Regularity Theorem cannot be extended to arbitrary modules. However, under an additional assumption on the generating degrees of a module, Gotzmann's Regularity Theorem will be proven. The modules satisfying the additional assumption will correspond to globally generated coherent sheaves. This will be used to provide an explicit construction of the Quot scheme.
The Gotzmann Regularity bound is known to be strict for standard graded algebras, but not for globally generated coherent sheaves. In order to address this, new representations for the Hilbert function and Hilbert polynomial are given that account for the rank and generating degrees of a module. Generalizations of the theorems of Macaulay, Green, and Gotzmann will be proven using these representations. The generalized Gotzmann number will give a strict upper bound for the regularity of modules generated in degree zero. Additionally, these representations will be used to prove a sharp inequality on the first and second Chern classes of a globally generated coherent sheaf.