Generalizations and applications of alexander self-duality in combinatorial commutative algebra

Abstract

We describe Alexander duality in three equivalent contexts: square-free monomial ideals, simple hypergraphs, and simplicial complexes. The first major result gives necessary and sufficient conditions for Alexander self-duality in each of these objects. As a consequence, we show a novel equivalence between Alexander self-duality and intersecting, 3-chromatic hypergraphs, about which Erdős and Lovász posed a number of still open questions in the 1970's. A recent topological study of Alexander self-duality gives an enumeration algorithm for all such hypergraphs, which is only realizable for small n. We describe improvements for a computational implementation of this enumeration algorithm that allow us to enumerate all intersecting hypergraphs and all non 2-colorable hypergraphs with six vertices or fewer. Further, we develop a technique called symmetric polarization to give generalized versions of these results for generalized Alexander self-duality as it is defined for monomial ideals.