Duality in algebra and topology

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.