Local cohomology of BP_*BP-comodules

Abstract

In a previous paper, the authors 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.