The author considers the problem of calculating E∗ ( ∏ i Xi) from the E∗(Xi). Because of Hopkins’s chromatic splitting conjecture, he is particularly interested in the case when E = E(n) is the Johnson-Wilson spectrum, but the basic requirement is that E is a ring spectrum up to homotopy which is topologically flat in the sense that it is the minimal weak colimit of a filtered diagram of finite spectra Fα such that E∗(Fα) is a finitely projective E∗-module. He shows that one may realize an injective resolution and construct a spectral sequence for E∗ ( ∏ i Xi) with E2-term given by the derived functors of the E∗E-comodule product of the E∗(Xi). When E = E(n) and the Xi are E-local, the spectral sequence has a horizontal vanishing line at some stage and converges strongly. Furthermore, the derived functors can be calculated by relative injectives, and hence if the Xi are E-modules, the spectral sequence collapses, which shows that E∗ ( ∏ i Xi) is the E∗E-comodule product of the E∗(Xi)
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