1998/12/16 by Ezra Miller, Miller, Ezra
Mathematics · #13D02 #13P10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #math.AC #math.CO #msc:13D02 #msc:13P10
paper · pdf · doi:10.48550/arxiv.math/9812095
37 pages, LaTeX, 8 eps figures. Also available at http://math.berkeley.edu/~enmiller
arxiv created 1998/12/16 · arxiv updated 2009/11/30
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.