Title: Algebraic infinite delooping and derived destabilization,
Abstract: Working over the prime field of characteristic two, consequences of the
Koszul duality between the Steenrod algebra and the big Dyer-Lashof algebra are
studied, with an emphasis on the interplay between instability for the Steenrod
algebra action and that for the Dyer-Lashof operations. The central algebraic
framework is the category of length-graded modules over the Steenrod algebra
equipped with an unstable action of the Dyer-Lashof algebra, with compatibility
via the Nishida relations.
A first ingredient is a functor defined on modules over the Steenrod algebra
that arose in the work of Kuhn and McCarty on the homology of infinite loop
spaces. This functor is given in terms of derived functors of destabilization
from the category of modules over the Steenrod algebra to unstable modules,
enriched by taking into account the action of Dyer-Lashof operations.
A second ingredient is the derived functors of the Dyer-Lashof
indecomposables functor to length-graded modules over the Steenrod algebra.
These are related to functors used by Miller in his study of a spectral
sequence to calculate the homology of an infinite delooping. An important fact
is that these functors can be calculated as the homology of an explicit Koszul
complex with terms expressed as certain Steinberg functors. The latter are
quadratic dual to the more familiar Singer functors.
By exploiting the explicit complex built from the Singer functors which
calculates the derived functors of destabilization, Koszul duality leads to an
algebraic infinite delooping spectral sequence. This is conceptually similar to
Miller's spectral sequence, but there seems to be no direct relationship.
The spectral sequence sheds light on the relationship between unstable
modules over the Steenrod algebra and all modules.