Artinian level algebras of codimension 3

AbstractIn this paper, we continue the study of which h-vectors H=(1,3,…,hd−1,hd,hd+1) can be the Hilbert function of a level algebra by investigating Artinian level algebras of codimension 3 with the condition β2,d+2(Ilex)=β1,d+1(Ilex), where Ilex is the lex-segment ideal associated with an ideal I. Our approach is to adopt a homological method called the Cancellation Principle: the minimal free resolution of I is obtained from that of Ilex by canceling some adjacent terms of the same shift.We prove that when β1,d+2(Ilex)=β2,d+2(Ilex), R/I can be an Artinian level k-algebra only if either hd−1<hd<hd+1 or hd−1=hd=hd+1=d+1 holds. We also show that for H=(1,3,…,hd−1,hd,hd+1), the Hilbert function of an Artinian algebra of codimension 3 with the condition hd−1=hd