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The Alexander duality functors and local duality with monomial support

Publication ,  Journal Article
Miller, E
Published in: Journal of Algebra
September 1, 2000

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated Nn-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 Zn-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 Nn-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 Zn-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=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 Hn-im(S/I). © 2000 Academic Press.

Duke Scholars

Published In

Journal of Algebra

DOI

ISSN

0021-8693

Publication Date

September 1, 2000

Volume

231

Issue

1

Start / End Page

180 / 234

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4902 Mathematical physics
  • 0101 Pure Mathematics
 

Citation

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Chicago
ICMJE
MLA
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Miller, E. (2000). The Alexander duality functors and local duality with monomial support. Journal of Algebra, 231(1), 180–234. https://doi.org/10.1006/jabr.2000.8359
Miller, E. “The Alexander duality functors and local duality with monomial support.” Journal of Algebra 231, no. 1 (September 1, 2000): 180–234. https://doi.org/10.1006/jabr.2000.8359.
Miller E. The Alexander duality functors and local duality with monomial support. Journal of Algebra. 2000 Sep 1;231(1):180–234.
Miller, E. “The Alexander duality functors and local duality with monomial support.” Journal of Algebra, vol. 231, no. 1, Sept. 2000, pp. 180–234. Scopus, doi:10.1006/jabr.2000.8359.
Miller E. The Alexander duality functors and local duality with monomial support. Journal of Algebra. 2000 Sep 1;231(1):180–234.
Journal cover image

Published In

Journal of Algebra

DOI

ISSN

0021-8693

Publication Date

September 1, 2000

Volume

231

Issue

1

Start / End Page

180 / 234

Related Subject Headings

  • General Mathematics
  • 4904 Pure mathematics
  • 4902 Mathematical physics
  • 0101 Pure Mathematics