The Alexander Duality Functors and Local Duality with Monomial Support
✍ Scribed by Ezra Miller
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 375 KB
- Volume
- 231
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated ގ 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 ޚ 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 ގ 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 i Ž . ޚ -graded local cohomology functors H y for squarefree monomial ideals I in I the polynomial ring S, proving a duality directly generalizing local duality, which is the case when I s ᒊ is maximal. In the process, a new flat complex for calculating local cohomology at monomial ideals is introduced, showing, as a consequence, i Ž .