## Journal of Commutative Algebra

### Survey Article: A tour of the weak and strong Lefschetz properties

#### Abstract

An artinian graded algebra, $A$, is said to have the weak Lefschetz property (WLP) if multiplication by a general linear form has maximal rank in every degree. A vast quantity of work has been done studying and applying this property, touching on numerous and diverse areas of algebraic geometry, commutative algebra and combinatorics. Amazingly, though, much of this work has a common ancestor" in a theorem originally due to Stanley, although subsequently reproved by others. In this paper we describe the different directions in which research has moved starting with this theorem, and we discuss some of the open questions that continue to motivate current research.

#### Article information

Source
J. Commut. Algebra, Volume 5, Number 3 (2013), 329-358.

Dates
First available in Project Euclid: 13 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jca/1389621375

Digital Object Identifier
doi:10.1216/JCA-2013-5-3-329

Mathematical Reviews number (MathSciNet)
MR3161738

Zentralblatt MATH identifier
1285.13002

#### Citation

Migliore, Juan; Nagel, Uwe. Survey Article: A tour of the weak and strong Lefschetz properties. J. Commut. Algebra 5 (2013), no. 3, 329--358. doi:10.1216/JCA-2013-5-3-329. https://projecteuclid.org/euclid.jca/1389621375