Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals

TL;DR

This paper provides criteria for Cohen-Macaulayness of monomial ideals and their symbolic powers, linking algebraic properties to graph theory and linear programming, with characterizations involving the structure of simplicial complexes.

Contribution

It introduces new criteria for Cohen-Macaulayness based on primary decomposition and characterizes symbolic powers of Stanley-Reisner ideals using simplicial complexes and matroid theory.

Findings

Cohen-Macaulayness of symbolic powers relates to the structure of the simplicial complex.

All symbolic powers are Cohen-Macaulay if and only if the complex is a matroid complex.

Cohen-Macaulayness can transfer between different symbolic powers.

Abstract

We present criteria for the Cohen-Macaulayness of a monomial ideal in terms of its primary decomposition. These criteria allow us to use tools of graph theory and of linear programming to study the Cohen-Macaulayness of monomial ideals which are intersections of prime ideal powers. We can characterize the Cohen-Macaulayness of the second symbolic power or of all symbolic powers of a Stanley-Reisner ideal in terms of the simplicial complex. These characterizations show that the simplicial complex must be very compact if some symbolic power is Cohen-Macaulay. In particular, all symbolic powers are Cohen-Macaulay if and only if the simplicial complex is a matroid complex. We also prove that the Cohen-Macaulayness can pass from a symbolic power to another symbolic powers in different ways.