Artinian level algebras of codimension 3

Abstract

In this paper, we continue the study of which $h$-vectors $\stackrel{˝}{=}(1,3,...,h_{d-1},h_d,h_{d+1})$ can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition ${\beta }_{2,d+2}(I^{\mathrm{lex}})={\beta }_{1,d+1}(I^{\mathrm{lex}})$, where $I^{\mathrm{lex}}$ 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 $I^{\mathrm{lex}}$ by canceling some adjacent terms of the same shift. We prove that when ${\beta }_{1,d+2}(I^{\mathrm{lex}})={\beta }_{2,d+2}(I^{\mathrm{lex}})$, $R/I$ 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 $\stackrel{˝}{=}(1,3,...,h_{d-1},h_d,h_{d+1})$, the Hilbert function of an Artinian algebra of codimension 3 with the condition $h_{d-1}=h_d<h_{d+1}$, (a) if $h_d\le 3d+2$, then $h$-vector $\H$ cannot be level, and (b) if $h_d\ge 3d+3$, then there is a level algebra with Hilbert function $\H$ for some value of $h_{d+1}$.

15 pages