The Alexander duality functors and local duality with monomial support
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.
scientific article; zbMATH DE number 1525271
scientific article