## Abstract

We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra A we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms ℓ give maximal rank for all the multiplication maps ×ℓ:[A]_{i}⟶[A]_{i+1}. This is a Zariski open set and its complement is the non-Lefschetz locus. For monomial complete intersections, we completely describe the non-Lefschetz locus. For general complete intersections of codimension three and four we prove that the non-Lefschetz locus has the expected codimension, which in particular means that it is empty in a large family of examples. For general Gorenstein algebras of codimension three with a given Hilbert function, we prove that the non-Lefschetz locus has the expected codimension if the first difference of the Hilbert function is of decreasing type. For completeness we also give a full description of the non-Lefschetz locus for artinian algebras of codimension two.

Original language | English |
---|---|

Pages (from-to) | 288-320 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 505 |

DOIs | |

State | Published - Jul 1 2018 |

### Bibliographical note

Funding Information:The first author was partially supported by the Grant VR2013-4545, the second author by the National Security Agency under Grant H98230-12-1-0204 and by the Simons Foundation under Grant #309556, the third author by the Grant MTM2016-78623-P and the fourth author by the National Security Agency under Grant H98230-12-1-0247 and by the Simons Foundation under Grant #317096.

Publisher Copyright:

© 2018 Elsevier Inc.

## Keywords

- Artinian algebra
- Complete intersection
- Gorenstein algebra
- Weak Lefschetz property

## ASJC Scopus subject areas

- Algebra and Number Theory