Rings, Modules, and Algebras in Stable Homotopy Theory

Abstract

ii iii Abstract. Let S be the sphere spectrum. We construct an associative, commutative, and unital smash product in a complete and cocomplete category M S of "S-modules" whose derived category 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 ∧ S R −→ R. These notions are essentially equivalent to the earlier notions of A ∞ and E ∞ 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 ∧ S M −→ M .When R is commutative, the category M R of R-modules also has an associative, commutative, and unital smash product, and its derived category D R has properties just like the stable homotopy category.Working in the derived category 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 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 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 Σ ∞ (ΩX) + 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 Ralgebras and commutative R-algebras in terms of maps A ∧ R A −→ 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 T HH 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. iv

Monoidal model categories

“…This is usually the case, but S is not cofibrant in the category of S-modules of [EKMM97]. Without the second condition, the homotopy category of a monoidal model category would not be a monoidal category, because there would not be a unit.…”

“…Examples of symmetric monoidal model categories include the categories of simplicial sets, compactly generated topological spaces, S-modules [EKMM97], symmetric spectra [HSS98], and topological symmetric spectra.…”

“…Monoidal model categories abound in nature: examples include simplicial sets, compactly generated topological spaces, and chain complexes of modules over a commutative ring. The thirty-year long search for a monoidal model category of spectra met success with the category of S-modules of [EKMM97] and the symmetric spectra of [HSS98].…”

Differential graded categories

“…This is based on the category of symmetric spectra as constructed in [67] (cf. [46] for a different construction of a symmetric monoidal model category for the category of spectra). We refer to [136] for an excellent exposition of these far-reaching results and their surprising applications in homotopy theory.…”

Controlled algebra for simplicial rings and algebraic K-theory

“…Ring spectra provide a natural generalization of rings, and simplicial rings are an intermediate step between rings and ring spectra. Algebraic K-theory can be defined for ring spectra, see in [EKMM97]. The statement of the Farrell-Jones Conjecture makes sense with connective ring spectra as coefficients.…”