Relative Hochster--Takayama formula and Cohen--Macaulay monomial ideal quotients

TL;DR

This paper extends Hochster and Takayama's formulas to quotients of monomial ideals, providing new criteria for Cohen-Macaulayness and applying these to symbolic powers and graph edge ideals.

Contribution

It develops a relative Hochster--Takayama formula for monomial ideal quotients and introduces a Cohen-Macaulayness criterion based on relative homology.

Findings

Characterization of Cohen-Macaulay monomial ideal quotients using relative homology.

Classification of Cohen-Macaulayness of symbolic power quotients for squarefree monomial ideals.

Analysis of Cohen-Macaulay properties of symbolic-ordinary discrepancy modules and edge ideals.

Abstract

Hochster's and Takayama's formulas describes the multigraded components of local cohomology modules of monomial ideals in terms of simplicial complexes. In this paper, we develop a relative version of these formulas for quotients $I/J$ of monomial ideals, expressing the multigraded pieces of local cohomology modules of $I/J$ as reduced relative (co)homology of pairs of degree complexes. As an application, we obtain a relative Reisner criterion characterizing Cohen-Macaulay monomial ideal quotients. We further apply this relative Hochster--Takayama framework to modules arising from symbolic power filtrations, including symbolic quotients $I^{(t)}/I^{(t+1)}$ and symbolic-ordinary discrepancy module $I^{(t)}/I^t$. In particular, for a squarefree monomial ideal $I$, we give a precise classification of when $I^{(t)}/I^{(t+1)}$ is Cohen-Macaulay for all or, equivalently, for some $t \ge 2$. When $I$ is the edge ideal of a graph, we characterize the Cohen-Macaulayness of $I^{(t)}/I^t$ for all or, equivalently, for some sufficiently large $t$, and analyze the behavior of its dimension function.