##
**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.

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.

Reviewer: P.Landweber (New Brunswick)

### 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 |