CI-sequences and almost complete intersections

In this paper, we study the Hilbert function and the graded Betti numbers of almost complete intersection artinian algebras. We show that that every Hilbert function of a complete intersection artinian algebra is the Hilbert function of an almost complete intersection algebra. In codimension 3 we focus on almost complete intersection artinian algebras whose Hilbert function coincides with that of a complete intersection defined by 3 forms of the same degree. We classify all the possible graded Betti numbers of such algebras and we specify what cancellations are allowed in a minimal graded free resolution.