Sequentially Cohen-macaulay Modules and Local Cohomology

Let I ⊂ R be a graded ideal in the polynomial ring R = K[x1, . . . , xn] where K is a field, and fix a term order <. It has been shown in [17] that the Hilbert functions of the local cohomology modules of R/I are bounded by those of R/ in(I), where in(I) denotes the initial ideal of I with respect to <. In this note we study the question when the local cohomology modules of R/I and R/ in(I) have the same Hilbert function. A complete answer to this question can be given for the generic initial ideal Gin(I) of I, where Gin(I) is taken with respect to the reverse lexicographical order and where we assume that char(K) = 0. In this case our main result (Theorem 3.1) says that the local cohomology modules of R/I and R/Gin(I) have the same Hilbert functions if and only if R/I is sequentially Cohen-Macaulay. In Section 1 we give the definition of sequentially CM-modules which is due to Stanley [18], and in Theorem 1.4 we present Peskine’s characterization of sequentially CM-modules in terms of Ext-groups. This characterization is used to derive a few basic properties of sequentially CM-modules which are needed for the proof of the main result. In the following Section 2 we recall some well-known facts about generic initial modules, and also prove thatR/Gin(I) is sequentially CM, see Theorem 2.2. Section 3 is devoted to the proof of the main theorem, and in the final Section 4 we state and prove a squarefree version (Theorem 4.1) of the main theorem. Its proof is completely different from that of the main theorem in the graded case. It is based upon a result on componentwise linear ideals shown in [2] and the fact (see [11]) that the Alexander dual of a squarefree componentwise linear ideal defines a sequentially CM simplicial complex.