Regularity and $h$-polynomials of monomial ideals

Abstract

Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $°x_i=1$ and $I\subset S$ a homogeneous ideal of $S$ with $\dim S/I=d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(\lambda )/(1-\lambda {)}^d$, where $h_{S/I}(\lambda )=h_0+h_1\lambda +h_2{\lambda }^2+\cdots +h_s{\lambda }^s$ with $h_s\ne 0$ is the $h$-polynomial of $S/I$. It is known that, when $S/I$ is Cohen--Macaulay, one has $\reg(S/I) = °h_{S/I}(λ)$, where $\reg(S/I)$ is the (Castelnuovo--Mumford) regularity of $S/I$. In the present paper, given arbitrary integers $r$ and $s$ with $r\ge 1$ and $s\ge 1$, a monomial ideal $I$ of $S=K[x_1,\dots ,x_n]$ with $n\gg 0$ for which $\reg(S/I) = r$ and $°h_{S/I}(\lambda )=s$ will be constructed. Furthermore, we give a class of edge ideals $I\subset S$ of Cameron--Walker graphs with $\reg(S/I) = °h_{S/I}(λ)$ for which $S/I$ is not Cohen--Macaulay.

arXiv comment11 pages