A rigidity property of local cohomology modules

Abstract

The relationships between the invariants and the homological properties of I, Gin(I) and I lex have been studied extensively over the past decades. A result of A. Conca, J. Herzog and T. Hibi points out some rigid behaviours of their Betti numbers. In this work we establish a local cohomology counterpart of their theorem. To this end, we make use of properties of sequentially Cohen-Macaulay modules and we study a generalization of such concept by introducing what we call partially sequentially Cohen-Macaulay modules, which might be of interest by themselves.

On the notion of sequentially Cohen–Macaulay modules

“…The next theorem provides a characterization of partially sequentially Cohen-Macaulay modules. It was proved for the first time in [35,Theorem 3.5] in the ideal case. Here, we generalize the result to finitely generated modules and fix the gap in the original proof thanks to Proposition 8.…”

“…Notice that this is a stronger condition, though, since h i (R/I) h i (R/ Gin(I)) h i (R/I lex ), coefficientwise, see [34,Theorems 2.4 and 5.4]. Actually, in [35,Theorem 4.4], the following result is proved.…”

“…We are going to study next the notion of partially sequentially Cohen-Macaulay module, as introduced in [35], which naturally generalizes that of sequentially Cohen-Macaulay modules. Thanks to Schenzel's Theorem 2, the definition can be given in terms of the dimension filtration of the module.…”

“…In [35], it is claimed that if x ∈ R is M-regular, it is possible to prove that M is i-sCM if and only if M/xM is (i − 1)-sCM following the same lines of the proof [36, Theorem 4.7], which is not utterly correct, as we already pointed out in Example 3. The claim is indeed false: if x is M-regular and M/xM is (i − 1)-sCM, then M is not necessarily i-sCM, as the following example shows.…”

“…3, we recollect the definition of partial sequential Cohen-Macaulayness introduced by the third and the fourth author in [35], by requiring that only the queue of the dimension filtration is a Cohen-Macaulay filtration, see Definition 5 and some basic properties in the graded case in Lemmata 2 and 3. The next two results, Lemma 4 and Proposition 8, are dedicated to filling a gap in the proof of a fundamental lemma in [35], and we present the first of our generalizations of Theorem 1 in Theorem 4. We start Sect.…”