A colocalization spectral sequence
Abstract
Colocalization is a right adjoint to the inclusion of a subcategory. Given a ring-spectrum R, one would like a spectral sequence which connects a given colocalization in the derived category of R-modules 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' local-cohomology 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 non-commutative local-cohomology spectral sequence converging to the shifted cohomology of the loop space, a result dual to the local-cohomology theorem of Dwyer, Greenlees and Iyengar. An application to the abutment term of the Eilenberg-Moore spectral sequence is also presented.
Cite
@article{arxiv.0910.5251,
title = {A colocalization spectral sequence},
author = {Shoham Shamir},
journal= {arXiv preprint arXiv:0910.5251},
year = {2012}
}
Comments
The new version contains a uniqueness result, in Section 8, which shows that the current spectral sequence indeed generalizes the Greenlees spectral sequence. In addition, due to its intimidating nature the appendix has been removed from this version; its results will be placed elsewhere