Title: Rings, modules, and algebras in stable homotopy theory
Authors: A. D. Elmendorf, I. Kriz, M. A. Mandell, and J. P. May
address: Purdue University Calumet, Hammond IN 46323
address: The University of Michigan, Ann Arbor, MI 48109-1003
address: The University of Chicago, Chicago, IL 60637
address: The University of Chicago, Chicago, IL 60637
Let $S$ be the sphere spectrum. We construct an associative,
commutative, and unital smash product in a complete and cocomplete
category $\sM_S$ of ``$S$-modules'' whose derived category $\sD_S$ is
equivalent to the classical stable homotopy category. This allows a
simple and algebraically manageable definition of ``$S$-algebras'' and
``commutative $S$-algebras'' in terms of associative, or associative
and commutative, products $R\sma_S R \darrow R$. These notions are
essentially equivalent to the earlier notions of $A_\infty$ and
$E_\infty$ ring spectra, and the older notions feed naturally into the
new framework to provide plentiful examples. There is an equally
simple definition of $R$-modules in terms of maps $R\sma_S M\darrow M$.
When $R$ is commutative, the category $\sM_R$ of $R$-modules also has
an associative, commutative, and unital smash product, and its derived
category $\sD_R$ has properties just like the stable homotopy
category.
Working in the derived category $\sD_R$, we construct spectral
sequences that specialize to give generalized universal coefficient
and K\"{u}nneth spectral sequences. Classical torsion products and Ext
groups are obtained by specializing our constructions to
Eilenberg-Mac~Lane spectra and passing to homotopy groups, and the
derived category of a discrete ring $R$ is equivalent to the derived
category of its associated Eilenberg-Mac~Lane $S$-algebra.
We also develop a homotopical theory of $R$-ring spectra in $\sD_R$,
analogous to the classical theory of ring spectra in the stable
homotopy category, and we use it to give new constructions as
$MU$-ring spectra of a host of fundamentally important spectra whose
earlier constructions were both more difficult and less precise.
Working in the module category $\sM_R$, we show that the category of
finite cell modules over an $S$-algebra $R$ gives rise to an
associated algebraic $K$-theory spectrum $KR$. Specialized to the
Eilenberg-Mac~Lane spectra of discrete rings, this recovers Quillen's
algebraic $K$-theory of rings. Specialized to suspension spectra
$\Sigma^{\infty}(\Omega X)_+$ of loop spaces, it recovers Waldhausen's
algebraic $K$-theory of spaces.
Replacing our ground ring $S$ by a commutative $S$-algebra $R$, we
define $R$-algebras and commutative $R$-algebras in terms of maps
$A\sma_R A\darrow A$, and we show that the categories of $R$-modules,
$R$-algebras, and commutative $R$-algebras are all topological model
categories. We use the model structures to study Bousfield
localizations of $R$-modules and $R$-algebras. In particular, we prove
that $KO$ and $KU$ are commutative $ko$ and $ku$-algebras and
therefore commutative $S$-algebras.
We define the topological Hochschild homology $R$-module $THH^R(A;M)$
of $A$ with coefficients in an $(A,A)$-bimodule $M$ and give spectral
sequences for the calculation of its homotopy and homology groups.
Again, classical Hochschild homology and cohomology groups are
obtained by specializing the constructions to Eilenberg-Mac~Lane
spectra and passing to homotopy groups.