A colocalization spectral sequence

TL;DR

This paper introduces a spectral sequence connecting colocalizations in derived categories of ring spectra and graded modules, generalizing previous local-cohomology spectral sequences and applying to loop space cohomology.

Contribution

It constructs a new colocalization spectral sequence for ring spectra, extending Greenlees' local-cohomology spectral sequence and demonstrating its universality and applications.

Findings

Established the existence of the colocalization spectral sequence under certain conditions.

Applied the spectral sequence to cochains of loop spaces, deriving a non-commutative local-cohomology spectral sequence.

Provided insights into the abutment of the Eilenberg-Moore 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.