CYCLIC COMODULES, THE HOMOLOGY OF j, AND j-HOMOLOGY

This short paper arose as a warm-up to the more difficult computations at the primes 2 and 3 of connective versions of Behrens’ Q(2) spectrum, a spectrum conjecturally describing half of the K(2)-local sphere [2]. The natural starting case is to understand the K(1)-local story, giving rise to the present discussion of the homology of the connective image of J spectrum. The homology results are by no means new: a great many authors, including Davis, Knapp, and Angeltveit-Rognes have given descriptions of the results [3, 5, 1]. The description herein contains an easier description using very general properties of comodules over a coalgebra or over a Hopf algebra, and so can be readily adapted to other situations. We also present a computation of the homotopy Hopf algebra of HFp∧jHFp for odd primes p, a Hopf algebra suitable for computing the j-homology of a finite j-module using a modified Adams spectral sequence similar to that employed by the author in [4].