Duality in algebra and topology

By W. Dwyer, John Greenlees, S. Iyengar. 2005

178 citations recorded by the source corpus.

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) Poincare duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincare duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b57a6a71d08fa5bd5ce006319853e878b78e0066

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Citations in the corpus

Homotopy completion and topological Quillen homology of structured ring spectra (2011, background): “The following Milnor type short exact sequences are well known in stable homotopy theory (for a recent reference, see [16]); they can be established as a consequence of [9, IX].”
On the Homology of Completion and Torsion (2010, result): “we consider in Theorem 11.3 above, there is a similar result in [Ef], proved under extra regularity assumptions. After writing the ﬁrst version of our paper, we learned a similar result was proved in [DGI], again under extra regularity assumptions. Appendix A. Derived Morita Theory Derived Morita th”
Grothendieck–Neeman duality and the Wirthmüller isomorphism (2015, background): “Dwyer, Greenlees and Iyengar [DGI06] have developed a rich framework which captures several dualities in the style of Pontryagin-Matlis duality, and we show how this connects with our theory in Example 7.4.”
On differential graded categories (2006, methodology): “ns via dg endomorphism rings of categories of complete, respectively torsion, modules. Their 10 BERNHARD KELLER results are applied in a unifying study of duality phenomena in algebra and topology in [41]. One of the original motivations for the theorem was D. Happel’s description [60] [61] of the b”
Morita theory in abelian, derived and stable model categories (2003, methodology): “ative degree from S to itself. A large part of the material in this section is taken from a joint paper with Shipley [SS03]. Two other papersdevoted to Morita theory in the context of ringspectra are [DGI] by Dwyer, Greenlees and Iyengar and [BL] by Baker and Lazarev. 4.1. Stable model categories. R”
Motivic cell structures (2003, background): “Another place these ideas have appeared is in the work of Dwyer, Greenlees, and Iyengar [ DGI ], who imported them into homological algebra.”
A universal characterization of higher algebraic K-theory (2010, background): “When 1 is a compact object in C, this functor induces an equivalence between the category of modules over EndC(1) and LocC(1) (see [14] for a nice discussion of this kind of “generalized Morita theory”).”
Finiteness in derived categories of local rings (2004, background): “opological context one has to express it in a homotopy invariant form. The idea of formulating statements about a ring in terms of its derived category is a familiar one, but because the framework of [13] was more general than that, we were forced to concentrate on the crudest and most robust proper”
Homology of perfect complexes (2006, methodology): “Techniques in [16] allow for a different approach to Theorem 7.”
Weight structures and simple dg modules for positive dg algebras (2010, background): “9 of [13], as kindly explained to us by Srikanth Iyengar [16].”
Nearby Lagrangians with vanishing Maslov class are homotopy equivalent (2010, methodology): “A discussion of the sort of results we shall use in the setting of spectra is given in [8] where they call such algebras connective.”
Derived Completions in Stable Homotopy theory (2007, methodology): “Suppose that B is ﬁnitely built from A in the sense of [8] , and that the A-module M is also ﬁnitely built from A.”