Stable homotopy over the Steenrod algebra
John H. Palmieri
In this paper I apply the results of "Axiomatic stable homotopy
theory" (Hovey-Palmieri-Strickland) to the study of the Steenrod
algebra A and its cohomology Ext_A. To do this, I work in the
category Stable(A), in which the objects are cochain complexes of
injective comodules over the dual of A.
In this category, one can set up basic homotopy theoretic tools (like
Postnikov towers); using these, I generalize some well-known results
about A-modules to the category Stable(A) (like the vanishing line
theorems of Anderson-Davis and Miller-Wilkerson. I then use the
methods and philosophy of modern stable homotopy theory to study
Stable(A). This gives generalizations of deeper results about
A-modules (such as the nilpotence theorems of the author), new proofs
of old results (such as "chromatic convergence" in Stable(A)), and
new results (such as a disproof of a version of the telescope
conjecture in the category Stable(A)).