
Mathematics 2009
A colocalization spectral sequenceAbstract: Colocalization is a right adjoint to the inclusion of a subcategory. Given a ringspectrum R, one would like a spectral sequence which connects a given colocalization in the derived category of Rmodules and an appropriate colocalization in the derived category of graded modules over the graded ring of homotopy groups of R. We show that, under suitable conditions, such a spectral sequence exists. This generalizes Greenlees' localcohomology spectral sequence. The colocalization spectral sequence introduced here is associated with a localization spectral sequence, which is shown to be universal in an appropriate sense. We apply the colocalization spectral sequence to the cochains of certain loop spaces, yielding a noncommutative localcohomology spectral sequence converging to the shifted cohomology of the loop space, a result dual to the localcohomology theorem of Dwyer, Greenlees and Iyengar. An application to the abutment term of the EilenbergMoore spectral sequence is also presented.
