Ideals with Maximal Local Cohomology Modules

This paper finds its motivation in the pursuit of ideals whose local cohomology modules have maximal Hilbert functions. In [8], [9] we proved that the lexicographic (resp. squarefree lexicographic) ideal of a family of graded (resp. squarefree) ideals with assigned Hilbert function provides sharp upper bounds for the local cohomology modules of any of the ideals of the family. Moreover these bounds are determined explicitely in terms of the Hilbert function, which is the specified starting data. In the present paper a characterization of the class of ideals with the desired property is accomplished. In order to be more precise we set some notation to be used henceforth. Let R . = K[X1, . . . , Xn] denote the polynomial ring in n variables over a field K of characteristic 0 with its standard grading, m . = (X1, . . . , Xn) the maximal homogeneous ideal of R, I ⊂ R a homogeneous ideal and I lex its lexicographic ideal. The canonical module of R will be denoted by ωR ≃ R(−n). If M stands for a graded R-module, then Hilb(M, t) will denote its Hilbert series in terms of t. The local cohomology modules H i m (M) of M will be considered with support on the maximal graded ideal m and with their natural grading. We write h(M)j for the dimension as a K-vector space of H i m (M)j . The dual of the local cohomology modules according to the Local Duality Theorem will be denoted with E(M) . = ExtR(M,ωR). We write I sat . = I : m for the saturation of an ideal I with respect to m.