## Rings, modules, and algebras in stable homotopy theory. With an appendix by M. Cole.(English)Zbl 0894.55001

Mathematical Surveys and Monographs. 47. Providence, RI: American Mathematical Society (AMS). xi, 249 p. (1997).
The authors’ abstract: “Let $$S$$ be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category $${\mathcal M}_S$$ of “$$S$$-modules” whose derived category $${\mathcal D}_S$$ 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 $$R\wedge_S R\to R$$. 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 $$R\wedge_S M\to M$$. When $$R$$ is commutative, the category $${\mathcal M}_R$$ of $$R$$-modules also has an associative, commutative, and unital smash product, and its derived category $${\mathcal D}_R$$ has properties just like the stable homotopy category.
Working in the derived category $${\mathcal D}_R$$, we construct spectral sequences that specialize to give generalized universal coefficient and Künneth 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 $${\mathcal D}_R$$, 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 $${\mathcal M}_R$$, 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 $$A\wedge_R A\to A$$, 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.”
The basic construction underlying the work presented in this book is the twisted half-smash product of a suitable space and spectrum. The appendix by Michael Cole is devoted to new definitions of twisted half-smash products and function spectra.

### MathOverflow Questions:

Künneth formulas/theorem for bordism groups and cobordisms?

### MSC:

 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 19D99 Higher algebraic $$K$$-theory 55N20 Generalized (extraordinary) homology and cohomology theories in algebraic topology 55P42 Stable homotopy theory, spectra 19L99 Topological $$K$$-theory 55T25 Generalized cohomology and spectral sequences in algebraic topology 55N22 Bordism and cobordism theories and formal group laws in algebraic topology