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

Tai Huy Ha, Nguyen Cong Minh

TL;DR

This work develops a relative Hochster–Takayama theory for quotients of monomial ideals $I/J$, expressing multigraded local cohomology $H_{ rak m}^{i}(I/J)_{oldsymbol a}$ via reduced relative degree complex cohomology and establishing a Relative Reisner Criterion for Cohen–Macaulayness. It extends Takayama’s framework to arbitrary monomial quotients, and specializes to squarefree cases to connect with relative Stanley–Reisner theory. The authors apply the theory to symbolic power filtrations, proving that $I^{(t)}/I^{(t+1)}$ is CM for all (resp. some) $t$ iff the underlying simplicial complex is a matroid, and they classify Cohen–Macaulayness of symbolic–ordinary discrepancy modules $I^{(t)}/I^t$ for edge ideals, including detailed results for unicyclic and perfect graphs. They also develop dimension-stability results for these discrepancy modules and relate them to the Ratliff condition, offering a coherent topological perspective on CM and generalized CM properties via degree complexes. Overall, the work provides a unified combinatorial-topological framework for understanding CM-type properties of monomial-quotient modules and their symbolic filtrations, with explicit graph-theoretic classifications and dimension-stability phenomena.

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.

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

TL;DR

This work develops a relative Hochster–Takayama theory for quotients of monomial ideals , expressing multigraded local cohomology via reduced relative degree complex cohomology and establishing a Relative Reisner Criterion for Cohen–Macaulayness. It extends Takayama’s framework to arbitrary monomial quotients, and specializes to squarefree cases to connect with relative Stanley–Reisner theory. The authors apply the theory to symbolic power filtrations, proving that is CM for all (resp. some) iff the underlying simplicial complex is a matroid, and they classify Cohen–Macaulayness of symbolic–ordinary discrepancy modules for edge ideals, including detailed results for unicyclic and perfect graphs. They also develop dimension-stability results for these discrepancy modules and relate them to the Ratliff condition, offering a coherent topological perspective on CM and generalized CM properties via degree complexes. Overall, the work provides a unified combinatorial-topological framework for understanding CM-type properties of monomial-quotient modules and their symbolic filtrations, with explicit graph-theoretic classifications and dimension-stability phenomena.

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 of monomial ideals, expressing the multigraded pieces of local cohomology modules of 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 and symbolic-ordinary discrepancy module . In particular, for a squarefree monomial ideal , we give a precise classification of when is Cohen-Macaulay for all or, equivalently, for some . When is the edge ideal of a graph, we characterize the Cohen-Macaulayness of for all or, equivalently, for some sufficiently large , and analyze the behavior of its dimension function.
Paper Structure (11 sections, 28 theorems, 111 equations)

This paper contains 11 sections, 28 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complexes and Homology
  4. Degree Complexes and Symbolic Powers of Monomial Ideals
  5. Edge Ideals of Graphs and Other Conventions
  6. Relative Degree Complexes and Hochster--Takayama Formula
  7. Relative Complexes for Squarefree Monomial Ideals
  8. Cohen-Macaulay and Generalized Cohen-Macaulay Properties
  9. Symbolic Quotients
  10. Dimension Stability and Symbolic-Ordinary Discrepancy Modules
  11. Cohen-Macaulay Symbolic-Ordinary Discrepancy Modules

Key Result

Theorem 2.1

Let $I$ be a monomial ideal in $S$. For each $i\in\mathbb{Z}$ and ${\bold a} \in \mathbb{Z}^{n}$, there is an isomorphism of ${\Bbbk}$-vector spaces and $H_{\mathfrak{m}}^{i}(S/I)_{{\bold a}}=(0)$ if $G_{{\bold a}}\notin\Delta(\sqrt{I})$.

Theorems & Definitions (59)

  • Theorem 2.1: Takayama
  • Theorem 2.2: Reisner's Criterion
  • Lemma 3.1
  • proof
  • Theorem 3.2: Relative Hochster--Takayama
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4: Rigidity
  • proof
  • ...and 49 more