Artinian level algebras of codimension 3

TL;DR

This paper investigates the conditions under which certain Hilbert functions can be realized by Artinian level algebras of codimension 3, using homological methods and properties of lex-segment ideals.

Contribution

It characterizes specific Hilbert functions that can or cannot be realized by Artinian level algebras of codimension 3, introducing a homological approach based on the Cancellation Principle.

Findings

When $eta_{1,d+2}(I^{ m lex})=eta_{2,d+2}(I^{ m lex})$, level algebra existence is restricted to specific $h$-vector patterns.

For $h_{d-1}=h_d<h_{d+1}$ and $h_{d-1}=h_d=h_{d+1}=d+1$, level algebra realization is constrained.

If $h_d eq 3d+2$, the paper determines the existence of level algebras with given Hilbert functions.

Abstract

In this paper, we continue the study of which $h$-vectors $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition $\beta_{2,d+2}(I^{\rm lex})=\beta_{1,d+1}(I^{\rm lex})$, where $I^{\rm lex}$ is the lex-segment ideal associated with an ideal $I$. Our approach is to adopt an homological method called {\it Cancellation Principle}: the minimal free resolution of $I$ is obtained from that of $I^{\rm lex}$ by canceling some adjacent terms of the same shift. We prove that when $\beta_{1,d+2}(I^{\rm lex})=\beta_{2,d+2}(I^{\rm lex})$, $R/I$ can be an Artinian level $k$-algebra only if either $h_{d-1}<h_d<h_{d+1}$ or $h_{d-1}=h_d=h_{d+1}=d+1$ holds. We also apply our results to show that for $\H=(1,3,..., h_{d-1}, h_d, h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\leq 3d+2$, then $h$-vector $\H$ cannot be level, and (b) if $h_d\geq 3d+3$, then there is a level algebra with Hilbert function $\H$ for some value of $h_{d+1}$.