Duality in algebra and topology

Abstract

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.

Complexes of injectivekG-modules

“…For a general finite group, the relationship between K(Inj kG) and D dg (C * (BG; k)) is much closer, and provides some sort of context for understanding what is going on in [Dwyer et al 2006]. Traces of arguments from that paper can be seen from time to time in this paper.…”

“…In the work of Dwyer, Greenlees and Iyengar [2006], a close relationship was established between D(Mod kG) and D dg (C * (BG; k)). For a general finite group, the relationship between K(Inj kG) and D dg (C * (BG; k)) is much closer, and provides some sort of context for understanding what is going on in [Dwyer et al 2006].…”

Commutative Algebra in the Cohomology of Groups

“…There is also an orientation issue. Namely, Dwyer, Greenlees and Iyengar [Dwyer et al 2002] give another proof for compact Lie groups and also a version for p-compact groups. For example, if H * (BG; k) is a polynomial ring on generators ζ 1 , .…”

Nullification and cellularization of classifying spaces of finite groups

“…Among the most recent works on it, we can quote [CDI02], where the authors generalize the notion of dimension of a CW-complex to the A-cellular framework, or [DGI02], where the concept of cellularization is extended to algebraic categories of R-modules.…”