Duality in Algebra and Topology
W. G. Dwyer, J. P. C. Greenlees, and S. Iyengar
We take some classical ideas from commutative algebra, mostly ideas
involving duality, and apply them in a topological setting. To
accomplish this we interpret properties of ordinary commutative rings
in such a way that they can be extended to differential graded
algebras or more generally to structured ring spectra. This framework
allows us to view all of the following dualities
o Poincare duality for manifolds
o Gorenstein duality for commutative rings
o Benson-Carlson duality for cohomology rings of finite groups
o Poincar duality for groups
o Gross-Hopkins duality in chromatic stable homotopy theory
as examples of a single phenomenon. We give a new formula for the
Brown-Comenetz dual of the sphere spectrum; this turns out to be one
instance of a general construction that in another setting gives the
dualizing module of a Gorenstein ring. We also prove the local
cohomology theorem for p-compact groups and reprove it for compact Lie
groups. The key observation is that the cochain algebra on BG has a
simple duality property which extends Poincare duality.
Department of Mathematics, University of Notre Dame, Notre Dame, IN
46556. USA, dwyer.1@nd.edu
Department of Pure Mathematics, Hick Building, Sheffield S3 7RH. UK,
j.greenlees@sheffield.ac.uk
202 Mathematical Sciences Building, University of Missouri, Columbia,
MO 65211. USA, iyengar@math.missouri.edu