Rings, Modules, and Algebras in Stable Homotopy Theory

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RINGS, MODULES, AND ALGEBRAS IN STABLE HOMOTOPY THEORY

Abstract

Contents

Introduction

Prologue: the category of L-spectra

  1. Background on spectra and the stable homotopy category
  2. External smash products and twisted half-smash products
  3. The linear isometries operad and internal smash products
  4. The category of L-spectra
  5. The smash product of L-spectra
  6. The equivalence of the old and new smash products
  7. Function L-spectra
  8. Unital properties of the smash product of L-spectra

Structured ring and module spectra

  1. The category of S-modules
  2. The mirror image to the category of S-modules
  3. S-algebras and their modules
  4. Free A1 and E1 ring spectra and comparisons of definitions
  5. Free modules over A1 and E1 ring spectra
  6. Composites of monads and monadic tensor products
  7. Limits and colimits of S-algebras

The homotopy theory of R-modules

  1. The category of R-modules; free and cofree R-modules
  2. Cell and CW R-modules; the derived category of R-modules
  3. The smash product of R-modules
  4. Change of S-algebras; q-cofibrant S-algebras
  5. Symmetric and extended powers of R-modules
  6. Function R-modules
  7. Commutative S-algebras and duality theory

The algebraic theory of R-modules

  1. Tor and Ext; homology and cohomology; duality
  2. Eilenberg-Mac Lane spectra and derived categories
  3. The Atiyah-Hirzebruch spectral sequence
  4. Universal coefficient and Knneth spectral sequences
  5. The construction of the spectral sequences
  6. Eilenberg-Moore type spectral sequences
  7. The bar constructions B(M; R; N) and B(X; G; Y )

R-ring spectra and the specialization to MU

  1. Quotients by ideals and localizations
  2. Localizations and quotients of R-ring spectra
  3. The associativity and commutativity of R-ring spectra
  4. The specialization to MU-modules and algebras

Algebraic K-theory of S-algebras

  1. Waldhausen categories and algebraic K-theory
  2. Cylinders, homotopies, and approximation theorems
  3. Application to categories of R-modules
  4. Homotopy invariance and Quillen's algebraic K-theory of rings
  5. Morita equivalence
  6. Multiplicative structure in the commutative case
  7. The plus construction description of KR
  8. Comparison with Waldhausen's K-theoryofspaces

R-algebras and topological model categories

  1. R-algebras and their modules
  2. Tensored and cotensored categories of structured spectra
  3. Geometric realization and calculations of tensors
  4. Model categories of ring, module, and algebra spectra
  5. The proofs of the model structure theorems
  6. The underlying R-modules of q-cofibrant R-algebras

Bousfield localizations of R-modules and algebras

  1. Bousfield localizations of R-modules
  2. Bousfield localizations of R-algebras
  3. Categories of local modules
  4. Periodicity and K-theory

Topological Hochschild homology and cohomology

  1. Topological Hochschild homology: rst definition
  2. Topological Hochschild homology: second definition
  3. The isomorphism between thhR(A) and A S1

Some basic constructions on spectra

  1. The geometric realization of simplicial spectra
  2. Homotopical and homological properties of realization
  3. Homotopy colimits and limits
  4. Sigma-cofibrant, LEC, and CW prespectra
  5. The cylinder construction

Spaces of linear isometries and technical theorems

  1. Spaces of linear isometries
  2. Fine structure of the linear isometries operad
  3. The unit equivalence for the smash product of L-spectra
  4. Twisted half-smash products and shift desuspension
  5. Twisted half-smash products and cofibrations

The monadic bar construction

  1. The bar construction and two deferred proofs
  2. Cofibrations and the bar construction

Epilogue: The category of L-spectra under S

  1. The modified smash products CL, BL,and?L
  2. The modified smash products CR, BR,and?R

Bibliography