Alexander Duality for Monomial Ideals and Their Resolutions

TL;DR

This paper generalizes Alexander duality to all monomial ideals, linking their algebraic invariants and resolutions, and introduces a new canonical resolution method for these ideals.

Contribution

It extends Alexander duality beyond squarefree ideals to arbitrary monomial ideals and connects it with cellular resolutions and Bass numbers.

Findings

Established a natural expression of duality via Bass numbers.

Linked cellular resolutions of monomial ideals to those of their duals.

Constructed a new canonical resolution for monomial ideals.

Abstract

Alexander duality has, in the past, made its way into commutative algebra through Stanley-Reisner rings of simplicial complexes. This has the disadvantage that one is limited to squarefree monomial ideals. The notion of Alexander duality is generalized here to arbitrary monomial ideals. It is shown how this duality is naturally expressed by Bass numbers, in their relations to the Betti numbers of a monomial ideal and its Alexander dual. Relative cohomological constructions on cellular complexes are shown to relate cellular free resolutions of a monomial ideal to free resolutions of its Alexander dual ideal. As an application, a new canonical resolution for monomial ideals is constructed.