Hilbert functions of monomial ideals containing a regular sequence

TL;DR

This paper proves that certain monomial ideals containing a regular sequence satisfy the Eisenbud-Green-Harris conjecture and that Cohen-Macaulay properties are preserved, with implications for the structure of associated simplicial complexes.

Contribution

It establishes the conjecture for ideals containing a specific product of linear forms and regular sequences, extending known results to a broader class of monomial ideals.

Findings

Ideals containing a product of linear forms and a regular sequence satisfy the Eisenbud-Green-Harris conjecture.

Cohen-Macaulay property is preserved in these ideals.

The h-vector of Cohen-Macaulay simplicial complexes matches that of balanced complexes with specified parameters.

Abstract

Let $M$ be an ideal in $K[x_1,...,x_n]$ ($K$ is a field) generated by products of linear forms and containing a homogeneous regular sequence of some length. We prove that ideals containing $M$ satisfy the Eisenbud-Green-Harris conjecture and moreover prove that the Cohen-Macaulay property is preserved. We conclude that monomial ideals satisfy this conjecture. We obtain that $h$-vector of Cohen-Macaulay simplicial complex $\Delta$ is the $h$-vector of Cohen-Macaulay $(a_1-1,...,a_t-1)$-balanced simplicial complex where $t$ is the height of the Stanley-Reisner ideal of $\Delta$ and $(a_1,...,a_t)$ is the type of some regular sequence contained in this ideal.