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Duality in algebra and topology. (English) Zbl 1155.55302
Summary: We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson-Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross-Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.
This paper is a natural continuation of the previous papers by the authors [Comment. Math. Helv. 81, No. 2, 383-432 (2006; Zbl 1096.13027), W. G. Dwyer and J. P. C. Greenlees, Am. J. Math. 124, No. 1, 199-220 (2002; Zbl 1017.18008)]

##### MSC:
 55P43 Spectra with additional structure ($$E_\infty$$, $$A_\infty$$, ring spectra, etc.) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 16D90 Module categories in associative algebras 18E30 Derived categories, triangulated categories (MSC2010) 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 57N65 Algebraic topology of manifolds
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