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
(2000),
pp. 180234 [MR2001k:13028], [pdf]
(last updated on 2018/04/25)
Abstract: Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nngraded 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 Zngraded 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 Nngraded 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 Zngraded 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=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 HiI(S) is equivalent to Hochster's for Hnim(S/I). © 2000 Academic Press.


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