Injective comodules and Landweber exact homology theories
Mark Hovey
Wesleyan University
Middletown, CT
mhovey@wesleyan.edu
We classify the indecomposable injective E(n)_{*}E(n)-comodules, where
$E(n)$ is the Johnson-Wilson homology theory. They are suspensions of
the J_{n,r}, where J_{n,r} is the E(n)-homology of the rth monochromatic
piece M_{r} E(r) of E(r) and $0\leq r\leq n$. The endomorphism ring of
J_{n,r} is the ring of operations in the completed E(r) theory; this
ring of operations is not really known so far as I know, though it is
closely related to the stabilizer group S_r. A byproduct of this study
is the folklore isomorphism below, apparently not written down before but
known to Hopkins, Greenlees, Sadofsky, and others:
E^{*}(X) = \Hom_{E(n)_{*}} (E(n)_{*}M_{n}X, K)
where E is completed E(n) theory and K is the n-fold desuspension of
E(n)_{*}/I_{n}^{\infty}).