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}).