  We establish a form of the Gotzmann representation of the Hilbert polynomial
based on rank and generating degrees of a module, which allow for a
generalization of Gotzmann's Regularity Theorem. Under an additional assumption
on the generating degrees, the Gotzmann regularity bound becomes sharp. An
analoguous modification of the Macaulay representation is used along the way,
which generalizes the theorems of Macaulay and Green, and Gotzmann's
Persistence Theorem.
