Monomial ideals, almost complete intersections and the Weak Lefschetz property

Abstract

Abstract. Many algebras are expected to have the Weak Lefschetz property, although this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field and on arithmetic properties of the exponent vectors of the monomials.

On the shape of a pure 𝑂-sequence

“…Note that if k is finite, the notion of a "general" linear form does not make sense, so we might as well assume that k is infinite. In this case, we will conclude (thanks to [54], Proposition 2.2) that R/I has the WLP if and only if the characteristic is not 2, 5 or 7.…”

“…We close this chapter with a short discussion of the characteristic. We show via a simple example, using the methods of [54], that the same monomial ideal can exhibit different behavior with respect to the WLP if the characteristic changes. Furthermore, the characteristics for which the WLP fails arise as the prime factors of the determinant of a certain matrix of integers, and the list of these primes can have "gaps."…”

“…Note first that thanks to [54], Proposition 2.1 (c), it is equivalent to show that the multiplication by L from degree 8 to degree 9 is surjective. Our approach is exactly the same as the analogous arguments in [54].…”

“…In three variables it was shown by Brenner and Kaid [12] that the WLP can fail when the type is 3, even for monomial almost complete intersections. This important class of algebras was, or is being, further studied in [11,21,54] with respect to the failure of the WLP.…”

Smooth monomial Togliatti systems of cubics

“…By semicontinuity, it follows that a general artinian complete intersection ideal I ⊂ R has the WLP but it is open whether every artinian complete intersection of height ≥ 4 over a field of characteristic zero has the WLP. It is worthwhile to point out that in positive characteristic, there are examples of artinian complete intersection ideals I ⊂ k[x, y, z] failing the WLP (see, e.g., Remark 7.10 in [8]). …”