The Lefschetz properties of monomial complete intersections in positive characteristic

TL;DR

This paper extends the understanding of Lefschetz properties of monomial complete intersections to positive characteristic fields, providing new classifications and bounds for when these properties hold.

Contribution

It offers the first positive characteristic classification of monomial complete intersections with Lefschetz properties, including a bound on the characteristic for these properties.

Findings

Lefschetz properties hold for certain monomial complete intersections in positive characteristic.

Established lower bounds on characteristics guaranteeing Lefschetz properties.

Classified cases in characteristic two and for monomials of equal degree.

Abstract

Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary artinian monomial complete intersections in characteristic two. To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.