Complete cohomology theories and the homology of their omega spectra
Let \(E^*(-)\) be a multiplicative cohomology theory which is represented by an \(\Omega\)-spectrum \(\{\mathbf{E}_r\}\). The author studies in this work the effect upon the homology \(H_*(\mathbf{E}_r; {\mathbb F}_p)\) when the theory \(E^*(-)\) is completed with respect to an ideal of the coefficient ring. This provides further evidence, in the form of several useful theorems, that the homology of \(\Omega\)-spectra is best studied in the context of coalgebraic algebra (that is, with systematic use of comodules). The paper is thus a natural complement for the author's previous papers [\textit{M. J. Hopkins} and \textit{J. R. Hunton}, Topology 34, 29--36 (1995; Zbl 0862.55005) and \textit{J. R. Hunton} and \textit{P. R. Turner}, Topology 38, 621--634 (1999; Zbl 0929.55007)]. It is for example shown that under various mild hypotheses (including a flatness condition and a Mittag-Leffler condition) that the homology of the completed \(\Omega\)-spectrum is obtained from the homology of the original \(\Omega\)-spectrum by a coalgebraic module tensor product, and is thus a (formal or actual) completion of this coalgebraic module. The results apply to Morava \(E\)-theory and to Baker's completion of elliptic cohomology, among many others. One corollary is that the \({\mathbb F}_p\)-homology of the \(r\)th space representing \(n\)th completed Morava \(E\)-theory is a polynomial algebra if \(r\) is even, and exterior if \(r\) is odd. This has an application to the construction of a \(v_n\)-periodic unstable Adams spectral sequence [\textit{M. Bendersky} and \textit{J. R. Hunton}, to appear].
scientific article; zbMATH DE number 1801884