Local cohomology of BP_*BP-comodules

Mark Hovey and Neil Strickland
Wesleyan University      University of Sheffield
mhovey@wesleyan.edu      N.P. Strickland@sheffield.ac.uk

In the paper torsion-comod on this archive, we showed that the category
of E(n)_*E(n)-comodules is a localization of the category of
BP_*BP-comodules.  In this paper, we study the resulting localization
functor L_n on the category of BP_*BP-comodules.  It is an algebraic
analogue of the usual topological localization L_n.  It is left exact,
so has right derived functors L_n^i.  We show that these derived
functors are closely related to the local cohomology groups of
BP_*-modules studied by Greenlees and May; in fact, they coincide with
Cech cohomology with respect to I_{n+1}.  We also construct a spectral
sequence of comodules analogous to the Greenlees-May spectral sequence
(of modules) converging to BP_*(L_n X) whose E_2-term involves
L_n^i(BP_*X).  The proofs require getting a partial understanding of
injective objects in the category of BP_*BP-comodules.