A rigidity property of local cohomology modules

TL;DR

This paper explores a new rigidity property of local cohomology modules, extending known results about Betti numbers and introducing the concept of partially sequentially Cohen-Macaulay modules.

Contribution

It establishes a local cohomology analogue of a rigidity theorem and introduces partially sequentially Cohen-Macaulay modules, broadening the understanding of homological invariants.

Findings

Proves a rigidity property for local cohomology modules.

Introduces the concept of partially sequentially Cohen-Macaulay modules.

Connects local cohomology properties with homological invariants.

Abstract

The relationships between the invariants and the homological properties of $I$, ${\rm Gin}(I)$ and $I^{\rm lex}$ 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.