Lefschetz properties in algebra and geometry

Abstract

The weak and strong Lefschetz properties on graded artinian algebras have been an object of study along the last few decades. Precisely, let be $A$ a graded artinian algebra. We say that $A$ has the Strong Lefschetz property (SLP) if the multiplication by a $d$th power of a general linear form have maximal rank (i.e. $\times L^{d} : A_{i} \rightarrow A_{i+d}$ is injective or surjective for every $i$). We say that $A$ has the Weak Lefschetz property (WLP) if occurs the same with $d = 1$. These properties have connections among different areas such as algebraic geometry, commutative algebra and combinatorics. Sometimes quite surprising, these connections give new approaches and relate problems, a priori, very distant.