. A C ] 1 6 D ec 1 99 8 Alexander Duality for Monomial Ideals and Their Resolutions

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. AMS Classification: 13D02; 13P10 Introduction Alexander duality in its most basic form is a relation between the homology of a simplicial complex Γ and the cohomology of another simplicial complex Γ∨, called the dual of Γ. Recently there has been much interest ([15], [6], [8], [2]) in the consequences of this relation when applied to the monomial ideals which are the Stanley-Reisner ideals IΓ and IΓ∨ for the given simplicial complex and its Alexander dual. This has the limitation that StanleyReisner ideals are always squarefree. The first aim of this paper is to define Alexander duality for arbitrary monomial ideals and then generalize some of the relations between IΓ and IΓ∨. A second goal is to demonstrate that Bass numbers are the proper vessels for the translation of Alexander duality into commutative algebra. The final goal is to reveal the connections between Alexander duality and the recent work on cellular resolutions. There are two “minimal” ways of describing an arbitrary monomial ideal: via the minimal generators or via the (unique) irredundant irreducible decomposition. Given a monomial ideal I, Definition 1.5 describes a method for producing another monomial ideal I∨ whose minimal generators correspond to the irredundant irreducible components of I. Miraculously, this is enough to guarantee that the minimal generators of I correspond to the irreducible components of I∨. It is particularly easy to verify that this reversal of roles takes place for the squarefree ideals I = IΓ and I ∨ = IΓ∨ above (Proposition 1.10). A connection with linkage and canonical modules is described in Theorem 2.1. One can also deal with Alexander duality as a combinatorial phenomenon, thinking of Γ as an order ideal in the lattice of subsets of {1, . . . , n}. The Alexander dual Γ∨ is then given by the complement of the order ideal, which gives an order ideal in the opposite lattice. For squarefree monomial ideals all is well since the only monomials we care about are represented precisely by the lattice of subsets of {1, . . . , n}. For general monomial ideals we instead consider the larger lattice Z, by which we mean the poset with its natural partial order . Then a monomial ideal I can be regarded as a dual order ideal in Z, and I∨ is constructed (roughly) from the complementary set of lattice points, which is an order ideal—see Definition 2.9. It is Theorem 2.13 which proves the equivalence of the two definitions. Bass numbers first assert themselves in Section 3. Their relations to Betti numbers for monomial modules (Corollary 3.6 and Theorem 3.12) are derived as consequences of graded