The Alexander Duality Functors and Local Duality with Monomial Support

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated n-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution of its Alexander dual and contains all of the maps in the minimal free resolution of M over every n-graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcm-lattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all n-graded modules: the projective dimension of M equals the support-regularity of its Alexander dual, and M is Cohen–Macaulay if and only if its Alexander dual has a support-linear free resolution. Alexander duality is applied in the context of the n-graded local cohomology functors HiI(−) for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I = is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Teraiʹs formula for the Hilbert series of HiI(S) is equivalent to Hochsterʹs for Hn − i (S/I).