Ambidexterity in the T(n)-Local Stable Homotopy Theory

The monochromatic layers of the chromatic filtration on spectra, that is the K(n)-local (stable 00-)categories Sp_{K(n)} enjoy many remarkable properties. One example is the vanishing of the Tate construction due to Hovey-Greenlees-Sadofsky. The vanishing of the Tate construction can be considered as a natural equivalence between the colimits and limits in Sp_{K(n)} parametrized by finite groupoids. Hopkins and Lurie proved a generalization of this result where finite groupoids are replaced by arbitrary \pi-finite 00-groupoids.

There is another possible sequence of (stable 00-)categories who can be considered as “monochromatic layers”, those are the T(n)-local 00-categories Sp_{T(n)}. For the Sp_{T(n)} the vanishing of the Tate construction was proved by Kuhn. We shall prove that the analog of Hopkins and Lurie’s result in for Sp_{T(n)}. Our proof will also give an alternative proof for the K(n)-local case.

This is a joint work with Shachar Carmeli and Lior Yanovski