Duality in algebra and topology

TL;DR

This paper develops a unified conceptual framework linking various dualities in algebra and topology, extending classical ideas from commutative algebra to homotopy theory and related structures.

Contribution

It introduces a new framework that unifies multiple dualities across algebra and topology, and provides new proofs and results in these areas.

Findings

Unified view of dualities such as Poincare and Gorenstein duality

New proofs of classical duality results

Extension of algebraic duality concepts to homotopy-theoretic rings

Abstract

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can be extended to the more general rings that come up in homotopy theory. Amongst the rings we work with are the differential graded ring of cochains on a space, the differential graded ring of chains on the loop space, and various ring spectra, e.g., the Spanier-Whitehead duals of finite spectra or chromatic localizations of the sphere spectrum. Maybe the most important contribution of this paper is the conceptual framework, which allows us to view all of the following dualities: Poincare duality for manifolds, Gorenstein duality for commutative rings, Benson-Carlson duality for cohomology rings of finite groups, Poincare duality for groups, Gross-Hopkins duality in chromatic stable homotopy theory, as examples of a single phenomenon. Beyond setting up this framework, though, we prove some new results, both in algebra and topology, and give new proofs of a number of old results.