Book contents
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
Cohomology theories for highly structured ring spectra
Published online by Cambridge University Press: 23 October 2009
- Frontmatter
- Contents
- Preface
- The development of structured ring spectra
- Compromises forced by Lewis's theorem
- Permutative categories as a model of connective stable homotopy
- Morita theory in abelian, derived and stable model categories
- Higher coherences for equivariant K-theory
- (Co-)Homology theories for commutative (S-)algebras
- Classical obstructions and S-algebras
- Moduli spaces of commutative ring spectra
- Cohomology theories for highly structured ring spectra
- Index
Summary
Abstract. This is a survey paper on cohomology theories for A∞ and E∞ ring spectra. Different constructions and main properties of topological André-Quillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of A∞ and E∞ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures.
INTRODUCTION
In recent years algebraic topology witnessed renewed interest to highly structured ring spectra first introduced in [23]. To a large extent it was caused by the discovery of a strictly associative and symmetric smash product in the category of spectra in [8], [15]. This allowed one to replace the former highly technical notions of A∞ and E∞ ring spectra by equivalent but conceptually much simpler notions of S-algebras and commutative S-algebras respectively.
An S-algebra is just a monoid in the category of spectra with strictly symmetric and associative smash product (hereafter referred to as the category of S-modules). Likewise, a commutative S-algebra is a commutative monoid in the category of S-modules.
The most important formal property of categories of S-algebras and commutative S-algebras is that both are topological model categories in the sense of Quillen, [27] as elaborated in [8].
- Type
- Chapter
- Information
- Structured Ring Spectra , pp. 201 - 232Publisher: Cambridge University PressPrint publication year: 2004
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