Sequentially Cohen-Macaulay modules and local cohomology

TL;DR

This paper establishes a characterization of sequentially Cohen-Macaulay modules via the equality of Hilbert functions of local cohomology modules for a graded ideal and its generic initial ideal in polynomial rings over characteristic zero fields.

Contribution

It provides a new criterion linking the sequentially Cohen-Macaulay property to local cohomology Hilbert functions, enhancing understanding of module structure in algebraic geometry.

Findings

Hilbert functions of local cohomology modules coincide for R/I and R/Gin(I) if and only if R/I is sequentially Cohen-Macaulay.

The result applies specifically to polynomial rings over fields of characteristic zero.

This characterization offers a new tool for identifying sequentially Cohen-Macaulay modules.

Abstract

The main result of the paper states that for a graded ideal I in a polynomial ring R over a field of characteristic 0, the Hilbert functions of the local cohomology modules of R/I and of R/Gin(I) coincide if and only if R/I is sequentially Cohen-Macaulay.