## Symmetric ring spectra and topological Hochschild homology

Citations: | 3 - 1 self |

### BibTeX

@TECHREPORT{Shipley_symmetricring,

author = {Brooke Shipley},

title = {Symmetric ring spectra and topological Hochschild homology},

institution = {},

year = {}

}

### OpenURL

### Abstract

### Citations

553 |
Homological Algebra
- Cartan, Eilenberg
- 1956
(Show Context)
Citation Context ...he right and left R-module structure maps of R acting on M . Let R s be the smash product over k of s copies of R, i.e., R ^ k ^ k R. The following denition mimics the Hochschild complex as in [CE]. Denition 4.1.2. tHH k (R; M) is the simplicial k-module with s-simplices M ^ k R s . The simplicial face and degeneracy maps are given by d i = 8 > > : r ^ (id R ) s 1 if i = 0 (id M ) ^ (id R ... |

361 |
Homotopy limits completions and localizations
- Bous, Kan
- 1972
(Show Context)
Citation Context ... understood, but also to show that the homotopy colimit preserves level equivalences of symmetric spectra, see Proposition 2.3.2. We use the basic construction of the homotopy colimit for spaces from [BK]. Denition 2.3.1. Let B be a small category and F : B ! Sp a diagram of symmetric spectra. Let F l denote the diagram of spaces at level l. Then (hocolim B Sp F ) l = hocolim B S F l : This den... |

205 | Symmetric spectra
- Hovey, Shipley, et al.
(Show Context)
Citation Context ...ion The category of symmetric spectra introduced by Je Smith is a closed symmetric monoidal category whose associated homotopy category is equivalent to the traditional stable homotopy category, see [=-=HSS]-=-. In this paper, we study symmetric ring spectra, i.e., the monoids in the category of symmetric spectra. The category of symmetric ring spectra is closely related to the category of \functors with sm... |

164 | Homotopy theory of Γ-spaces, spectra, and bisimplicial sets - Bousfield, Friedlander - 1978 |

149 | Algebras and modules in monoidal model categories
- Schwede, Shipley
- 2000
(Show Context)
Citation Context ...ollary 4.2.10. See also Remark 2.2.2. Outline. In thesrst section we recall various denitions from [HSS], dene symmetric ring spectra, and discuss homotopy colimits. Section 2.2 uses the results of [S=-=S]-=- and [HSS] to establish model category structures for symmetric ring spectra, for R-modules over any symmetric ring spectrum R, and for R-algebras over any commutative symmetric ring spectrum R. In se... |

144 |
personal communication
- Smith, Bretscher, et al.
- 1980
(Show Context)
Citation Context ...lence. Hence, by Proposition 2.3.2, it induces a level equivalence on the homotopy colimits, F (0) ! hocolim I F . RING SPECTRA AND THH 13 Finally, we need the following proposition due to Je Smith, [=-=S]-=-. Let T be the category with objects n = f1; : : : ; ng and morphisms the standard inclusions. Homotopy colimits over T are weakly equivalent to telescopes. Let ! be the ordered set of natural numbers... |

135 |
Homotopy theories and model categories, in Handbook of Algebraic Topology
- Dwyer, Spalinski
- 1995
(Show Context)
Citation Context ...iated to the stable model category is equivalent to the stable homotopy category of spectra, see [HSS]. Hence, the stable model category is the model category which we refer to most often. See [Q] or =-=-=-[DS] for the basic denitions for model categories. Denition 2.1.6. Let f : X ! Y be a map in Sp . The map f is a level equivalencesif each fn : Xn ! Yn is a weak equivalence of spaces, ignoring the n... |

50 | On the K-theory of finite algebras over Witt vectors of perfect fields, Topology 36 - Hesselholt, Madsen - 1997 |

45 |
Farjoun, Cellular Spaces, Null spaces and homotopy Localization, Lect. Not
- Dror
- 1996
(Show Context)
Citation Context ...t level n, this functor is weakly equivalent to the (n + k)th Postnikov functor on spaces which is given by localization with respect to the set of maps f@[m+n+k+ 2] ! [m+n+k+ 2] : m 0g: See also [F2]. Then (C k T ) n RING SPECTRA AND THH 17 is n + k connected and i (C k T ) n ! i Tn is an isomorphism for i > n + k. Note that any levelsbrant spectrum is level equivalent to the homotopy colimit... |

19 |
Homotopical Algebra, volume 43
- Quillen
- 1967
(Show Context)
Citation Context ...y associated to the stable model category is equivalent to the stable homotopy category of spectra, see [HSS]. Hence, the stable model category is the model category which we refer to most often. See =-=-=-[Q] or [DS] for the basic denitions for model categories. Denition 2.1.6. Let f : X ! Y be a map in Sp . The map f is a level equivalencesif each fn : Xn ! Yn is a weak equivalence of spaces, ignorin... |

17 |
Topological Hochschild homology, preprint
- Bökstedt
- 1985
(Show Context)
Citation Context ... invariants of the stable homotopy type of X . There is also a spectral sequence for calculating the classical stable homotopy groups of DX , see Proposition 2.3.4. The category of FSPs was dened in [B] in order to dene the topological Hochschild homology for an associative ring spectrum R. In section 4, three dierent denitions of topological Hochschild homology for a symmetric ring spectrum are ... |

10 |
Farjoun, Homotopy and homology of diagrams of spaces, Algebraic topology
- Dror
- 1985
(Show Context)
Citation Context ...sion of the wedge summand corresponding to the identity map, m L 0 (S l ^ K) ! m L 0 (S l ^ hom I (m; m)+ ^ K): Because the homotopy colimit of a free diagram is weakly equivalent to the colimit, see =-=[F1-=-], the homotopy colimit of this map is the mapsl : m L 0 (S l ^ K) ! (DFmK) l mentioned in the lemma. We show that the map of diagrams is 2l m 1 connected at each spot. At each n 2 I , l (n), factors... |

6 | On the K-theory of algebras over Witt vectors of perfect Topology (to appear - Hesselholt, Madsen |

2 | Homotopy theory of-spaces, spectra, and bisimplicial sets - Bous, Friedlander - 1978 |

2 | Diagram spectra and generalized FSPs, in preparation - Mandell, May, et al. |