nLab
Brown-Comenetz duality

Contents

Contents

Idea

Brown-Comenetz 76 introduced the spectrum I /I_{\mathbb{Q}/\mathbb{Z}} which represents the cohomology theory that assigns to a spectrum XX the Pontrjagin duals Hom (π *X,/)Hom_{\mathbb{Z}}(\pi_{- \ast} X, \mathbb{Q}/\mathbb{Z}) of its homotopy groups.

If XX is a spectrum whose stable homotopy groups are finite groups, then the natural “double-duality” map XI /I /XX \to I_{\mathbb{Q}/\mathbb{Z}} I_{\mathbb{Q}/\mathbb{Z}} X (the unit of the corresponding continuation monad) is an equivalence (see at dualizing object in a closed category).

The Brown-Comenetz spectrum I /I_{\mathbb{Q}/\mathbb{Z}} has an “integral” refinement which is the Anderson spectrum I I_{\mathbb{Z}} inducing Anderson duality.

Properties

  • Eilenberg-Mac Lane spectra are Brown-Comenetz self-dual: I /H𝔽 pH𝔽 pI_{\mathbb{Q}/\mathbb{Z}} H \mathbb{F}_p \cong H \mathbb{F}_p.

References

  • E. H. Brown, Jr. and M. Comenetz, Pontrjagin duality for generalized homology and cohomology theories, Amer. J. Math. 98 (1976), no. 1, 1–27.

Last revised on April 6, 2017 at 11:31:47. See the history of this page for a list of all contributions to it.