Castelnuovo-Mumford Regularity and finiteness of Hilbert Functions
  The notion of regularity has been used by S. Kleiman in the construction of bounded families of ideals or sheaves with given Hilbert polynomial, a crucial point in the construction of Hilbert or Picard scheme. In a related direction, Kleiman proved that if I is an equidimensional reduced ideal in a polynomial ring S over an algebraically closed field, then the coefficients of the Hilbert polynomial of R = S/I can be bounded by the dimension and the multiplicity of R. Srinivas and Trivedi proved that the corresponding result does not hold for a local domain. However, they proved that there exist a finite number of Hilbert functions for a local Cohen-Macaulay ring of given multiplicity and dimension. The proofs of the above results are very difficult and involve deep results from Algebraic Geometry.   The aim of this paper is to introduce a unified approach which gives more general results and easier proofs of the above mentioned results. This approach is based on the fact that a class C of standard graded algebras has a finite number of Hilbert functions if and only if there are upper bounds for the regularity and the embedding dimension of the members of C. 
Maria Evelina Rossi, Ngo Viet Trung and Giuseppe Valla