Sean Grate: Betti tables forcing failure of the weak Lefschetz property

Abstract: For most rings, a lot of the data of the ring can be captured via its (minimal) free resolution. This can then be summarized with a Betti table which, in some sense, describes the complexity of the ring. If such a ring is also Artinian, the ring is said to have the weak Lefschetz property (WLP) if multiplication by some linear form is always full rank. Although Lefschetz properties are of interest to algebraists, many combinatorialists like to leverage constructions of Artinian algebras with the WLP to prove results about, for instance, log-concavity of sequences. Joint with Hal Schenck, we show that if the Betti table of an Artinian algebra has a certain substructure resembling a Koszul complex, then the Artinian algebra cannot have the WLP.