Dwyer, W.G., Greenlees, J.P.C. and Iyengar, S. (2006) Duality in algebra and topology. Advances in Mathematics, 200 (2). pp. 357-402. ISSN 0001-8708Full text available as:
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.
|Copyright, Publisher and Additional Information:||Imported from arXiv. This is an author produced version of a paper subsequently published in 'Advances in Mathematics'. Uploaded in accordance with the publisher's self-archiving policy.|
|Institution:||The University of Sheffield|
|Academic Units:||The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)|
|Depositing User:||Beccy Shipman|
|Date Deposited:||10 Mar 2009 09:52|
|Last Modified:||06 Jul 2014 03:42|