The Alexander duality functors and local duality with monomial support
type: Item
instance of: scholarly article
title: The Alexander duality functors and local duality with monomial support
zbMATH Open document ID: 0968.13009
author: Ezra Miller
published in: Journal of Algebra
publication date: 2001-09-12T00:00:00Z
review text: The classical notion of Alexander duality arose in algebraic topology. Given the close links between square free monomial ideals in polynomial rings and simplicial complexes, it is not surprising that a form of Alexander duality was found in combinatorial commutative algebra. Other forms of duality are well known, for example Matlis duality for \(\mathbb{Z}^n\)-graded modules. The question was raised by \textit{J. Herzog, V. Reiner} and \textit{V. Welker} [Mich. Math. J. 46, No. 2, 211-223 (1999; Zbl 0967.13018)] whether Alexander duality makes sense for larger classes of modules than square free monomial ideals. The author in a preprint [\textit{E. Miller}, ``Alexander duality for monomial ideals and their resolution''; \url{http://www-math.mit.edu/~combin/abstracts/sep98/miller.html}] extended it to arbitrary monomial ideals and their resolutions. In the paper under review Alexander duality is made into a functor on the category of finitely generated \(\mathbb{N}^n\)-graded modules by comparing Matlis and Alexander duality. The paper then sets out `to demonstrate the utility and concreteness of injective resolutions for \(\mathbb{Z}^n\)-graded modules by exploring their structure and relations with free resolutions via Alexander and local duality'. It is shown that Alexander duality gives a duality between free and injective resolutions and this is used to relate various homological properties of a module with properties of its dual.
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 1525271
zbMATH Keywords: Alexander duality
zbMATH Keywords: local duality
zbMATH Keywords: free monomial ideals
zbMATH Keywords: Matlis duality
zbMATH Keywords: injective resolutions
zbMATH Keywords: free resolutions
reviewed by: Timothy Porter
describes a project that uses: CoCoA
MaRDI profile type: Publication
full work available at URL: jabr.2000.8359
OpenAlex ID: W2055975229
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DOI: 10.1006/JABR.2000.8359
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description: scientific article; zbMATH DE number 1525271