Lex-segment Ideals, Liaison and the Weak Lefschetz Property

The CoCoA VII Conference

Juan Migliore July, 2001

Abstract

All of this talk is based on joint work with Uwe Nagel, and part of it is joint also with Tadahito Harima and Junzo Watanabe.

Lex-segment ideals play a pivotal role in Computer Algebra and Commutative Algebra. They are the easiest way to construct a scheme with given Hilbert function, and they have the wonderful property discovered independently by Bigatti and by Hulett that among all ideals with given Hilbert function, they have the maximal graded Betti numbers. We will talk about some other unexpected roles that these ideals (and more generally Borel-fixed ideals) have played in commutative algebra.

We begin with a discussion of the construction of ideals with “any” given Hilbert function by using lex-segment ideals. For Hilbert functions that are eventually zero we get Artinian lex-segment ideals. More generally, from Artinian lex-segment ideals (or indeed any Artinian monomial ideal) one can obtain arithmetically Cohen-Macaulay, reduced unions of linear varieties of any dimension by extending the lifting mechanism of Hartshorne and Geramita-Gregory-Roberts.

Liaison is an equivalence relation among saturated ideals in a polynomial ring (or equivalently, among closed subschemes of projective space). Traditionally liaison was restricted to the study of links that were performed via complete intersections, and a wealth of results have been obtained over the years. However, recently a flurry of activity has centered around the idea of linking by Gorenstein ideals. An appealing conjecture here is that any two arithmetically Cohen-Macaulay ideals of the same height (= codimension) are in the same Gorenstein liaison class. (This is false for complete interesection liaison, and the simplest counterexample happens to be a lex-segment ideal.) Recent work of Hartshorne casts some doubt on this conjecture. However, we have the result that the conjecture is at least true “up to Hilbert function” and “up to flat deformation” by showing that it is true for any lifting of an Artinian monomial ideal and true for any Borel-fixed ideal.

A standard graded Artinian K-algebra A = ⨁_{i≥0} A_{i} is said to have the Weak Lefschetz property (WLP) if, for a general linear form ℓ, the induced map ×ℓ : A_{i} → A_{i+1} has maximal rank for all i. A is said to have the Strong Lefschetz property (SLP) if ×ℓ^{d} : A_{i} → A_{i+d} has maximal rank for all i and all d. Note that SLP implies WLP. All Artinian quotients of K[x, y] have the SLP (the proof of this uses Borel-fixed ideals), but in higher dimensional polynomial rings they are not equivalent.

If I is an Artinian lex-segment ideal in the polynomial ring R then R/I almost never has the Weak Lefschetz property. Nevertheless, lex-segment ideals played a key role in recent work on the subject. First, a characterization of possible Hilbert functions for algebras with the WLP was obtained. This included a construction for any such Hilbert function. Then it was shown that for any allowable Hilbert function there is a maximal set of graded Betti numbers among algebras with the WLP. The extremal Betti numbers are obtained, not surprisingly, by using lex-segment ideals in a suitable way in the above construction. Interestingly, if we replace WLP by SLP then exactly the same characterization and bounds hold, as is shown again by a slightly broader use of lex-segment ideals in the construction.

Finally, we consider the important special case of complete intersections. It was shown independently by Stanley and by Watanabe that Artinian monomial complete intersections possess the SLP, and by Watanabe that “most” Gorenstein ideals possess the SLP. Not all Gorenstein ideals possess it, but it was not known if all complete intersections do or not. We proved that all height three complete intersections in K[x_{1}, x_{2}, x_{3}] have the WLP. So far our methods do not yield the SLP. It is also not known if all Artinian Gorenstein ideals in the particular ring K[x_{1}, x_{2}, x_{3}] have the WLP – the only known counterexamples occur in higher codimension.