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 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 Carmieli and Lior Yanovski