The weak Lefschetz property and mixed multiplicities of monomial ideals (Q6604509)

Let \(k\) be a field. An artinian graded \(k\)-algebra \(A\) is said to have the Weak Lefschetz Property (WLP) if multiplication by some (hence by a general) linear form \(L\) induces a homomorphism of maximal between any two consecutive components of \(A\). It has the Strong Lefschetz Property (SLP) if the analogous statement holds for \(\times L^d\), for all \(d > 0\). A particular class of algebras which have been studied in the last decade from the point of view of the WLP are those of the form \(A(\Delta) = k[x_1,\dots,x_n]/((x_1^2,\dots,x_n^2) + I_\Delta)\), where \(I_\Delta\) is a square-free monomial ideal, the Stanley-Reisner ideal of some simplicial complex \(\Delta\). In this paper the author gives several nice results about algebras of this kind. The first main theorem gives a criterion for WLP in characteristic zero in terms of the analytic spread of what he calls the \(i\)-th incidence ideal of \(\Delta\). He gives an analogous result for SLP in the case that \(A(\Delta)\) is also level. As a corollary he gives a result deducing WLP in characteristic \(p\) in certain degrees (for certain \(p\)) assuming that it holds in characteristic zero. The last main result assumes only that \(A\) is an artinian monomial algebra with \(\dim_k(A_1) \leq \dim_k (A_2)\). He shows that then \(A\) has the WLP in degree 1 in characteristic zero if and only if it has the WLP in degree 1 in every odd characteristic. He gives a further combinatorial characterization of this latter result.