Math @ Duke

Publications [#243914] of Ezra Miller
Papers Published
 Miller, E, The Alexander duality functors and local duality with monomial support,
Journal of Algebra, vol. 231 no. 1
(September, 2000),
pp. 180234, Elsevier BV [MR2001k:13028], [pdf], [doi]
(last updated on 2021/05/08)
Abstract: 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 Z graded localization. Results are obtained on the interaction of duality for resolutions with cellular resolutions and lcmlattices. Using injective resolutions, theorems of Eagon, Reiner, and Terai are generalized to all N graded modules: the projective dimension of M equals the supportregularity of its Alexander dual, and M is CohenMacaulay if and only if its Alexander dual has a supportlinear free resolution. Alexander duality is applied in the context of the Z graded local cohomology functors H () for squarefree monomial ideals I in the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I=m is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, that Terai's formula for the Hilbert series of H (S) is equivalent to Hochster's for H (S/I). © 2000 Academic Press. n n n n i i ni I I m


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