Morava E-theory of Filtered Colimits

Morava E-theory E∨ n∗(−) is a much-studied theory in algebraic topology, but it is not a homology theory in the usual sense, because it fails to preserve coproducts (resp. filtered homotopy colimits). The object of this paper is to construct a spectral sequence to compute the Morava E-theory of a coproduct (resp. filtered homotopy colimit). The E2 term of this spectral sequence involves the derived functors of direct sum (resp. filtered colimit) in an appropriate abelian category. We show that there are at most n − 1 (resp. n) of these derived functors. When n = 1, we recover the known result that homotopy commutes with an appropriate version of direct sum in the K(1)-local stable homotopy category.