Algebras with the Weak Lefschetz Property

Abstract

This is a survey on some works in which the Weak Lefschetz Property (WLP) for Artinian standard graded algebras is investigated, see for instance (Ragusa and Zappala, arXiv:1112.1498. To appear in Rend Circ Mat Palermo; Colloq Math 64:73–83, 2013; Favacchio et al, J Pure Appl Algebra 217:1955–1966, 2013). In particular, it is shown that the Hilbert function of an almost complete intersection Artinian standard graded algebra of codimension 3 is a Weak Lefschetz sequence, i.e. it is the Hilbert function of some Artinian algebra with WLP or equivalently it is unimodal and the positive part of their first differences is a O-sequence. Moreover we give both some numerical condition on the Hilbert function and other conditions on the graded Betti numbers in order to force Artinian Gorenstein standard graded algebras of codimension 3 to enjoy the WLP. For Artinian standard graded algebras with the WLP we study the behavior of their linear quotients both with respect to the Hilbert function and to the graded Betti numbers. From this we produce a new property denominated Betti Weak Lefschetz Property (β-WLP) which permits a good behavior of the grade Betti numbers for the linear quotients of Artinian standard graded algebras with the WLP. We find conditions on the generators’ degrees of a complete intersection Artinian graded algebra with the WLP which force the algebra to have the β-WLP.