C ASTELNUOVO ‐Regularität und H ILBERT reihen

Let A be an equidimensional locally Cohen‐Macaulay graded k ‐algebra. Following [16] we have upper bounds for Castelnuovo's regularity of A (see [16], main theorem und theorem 1.). The aim of this paper is to characterize the equalities of these bounds and Castelnuovo's regularity by using the theory of Hilbert series. For this reason the inequalities of [16] are new proved by applying our methods. Moreover, we improve and extend main results and investigations of [11], [7], [18], [14], [16]. For monomial ideals the conjecture of D. B AYER and M. S TILLMAN [4] is proved.