Title: Moduli spaces for Structured Ring Spectra
Authors: P.G. Goerss and M.J. Hopkins
Authors' email address: pgoerss@math.northwestern.edu
Abstract:
In this document we make good on all the assertions we made
in the previous paper ``Moduli spaces of commutative ring spectra''
wherein we laid out a theory a moduli spaces and problems for the existence
and uniqueness of commutative ring spectra. In particular, we
develop a theory of moduli spaces of algebra structures on spectra,
and give a decomposition of the moduli space as a tower of fibrations
wherein the successive fibers can be calculated using Andre'-Quillen
cohomology. By examining the obstructions to lifting a basepoint
up the tower, we then produce successively defined obstructions to
the realizing an algebra structure.
A point worth emphasizing is that the moduli problems here begin with
algebra: for example, we may have a homology theory E and a commutative
ring A in the category comodules associated to E and we wish to discuss
the homotopy type of the space of all commutative (in the strict sense)
ring spectra X so that the E-homology of X is A as a commutative ring.
We do not, a priori, assume that this moduli space is non-empty, or even
that there is a spectrum whose E-homology is A.
For a variety of applications we are not simply interested in this
absolute problem, but in a relative version as well. Fortunately,
Andre'-Quillen cohomology is inherently relative and the
theory adapts well to this case.
The main idea, which goes back to Dwyer, Kan, and Stover, is to try to
construct a simplicial ring spectrum, whose geometric realization
will realize A. Then we use the new simplicial direction and apply Postnikov
tower techniques to get the decomposition of the moduli space. Making this
work requires a certain amount of technical detail. In particular,
we need to be very careful with resolution model categories and their
localizations at a homology theory.