Morava $E$-theory of filtered colimits

Abstract

Morava E-theory E V 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 E 2 -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.