## Homology and cohomology of E∞ ring spectra (2005)

Venue: | MATHEMATISCHE ZEITSCHRIFT |

Citations: | 17 - 0 self |

### BibTeX

@MISC{Basterra05homologyand,

author = {Maria Basterra and Michael A. Mandell},

title = { Homology and cohomology of E∞ ring spectra},

year = {2005}

}

### OpenURL

### Abstract

Every homology or cohomology theory on a category of E∞ ring spectra is Topological André–Quillen homology or cohomology with appropriate coefficients. Analogous results hold more generally for categories of algebras over operads.

### Citations

286 | The Geometry of Iterated Loop Spaces - May - 1972 |

165 | Modules, and Algebras in Stable Homotopy Theory. With an appendix by - Elmendorf, Kriz, et al. - 1997 |

157 |
Categories and cohomology theories, Topology 13
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(Show Context)
Citation Context ...akes the trivial based set ∗ to the trivial based set ∗. This constructs a Γ-space associated to X, and the previous lemma identifies the suspension BX as the classifying space of this Γ-space. Segal =-=[17]-=- proved that when a Γ-space is “special”, the loop space of the classifying space is a group completion. In this case, special means that the map from X ∐ · · · ∐X → X × · · · ×X is a weak equivalence... |

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131 | Model categories of diagram spectra
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- 2001
(Show Context)
Citation Context ... is a collection of maps An → A ′ n that commute with the structure maps. We define the homotopy groups of a spectrum A = {An} by πqA = Colim ˜πq+nAn, where ˜π∗A = Ker(π∗A → π∗B). Standard techniques =-=[6, 10]-=- allow us to prove in Section 7 that the category of CB/B-spectra forms a closed model category, with weak equivalences the maps that induce isomorphisms on homotopy groups. The resulting homotopy cat... |

128 |
Equivariant Stable Homotopy Theory
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(Show Context)
Citation Context ...has appeared in the literature, we outline a proof at the end of the section. The canonical maps ΣB n X → B n+1 X make {B n X} a spectrum in the category of spaces, or in the terminology of Lewis–May =-=[8]-=-, an “indexed prespectrum”. When X is cofibrant, this has the further property that the adjoint structure map B n X → ΩB n+1 X is a weak equivalence for n > 0 and is group completion for n = 0. Also w... |

126 |
Modules, and Algebras in Stable Homotopy Theory
- Rings
- 1996
(Show Context)
Citation Context ...ction 9 for details). The purpose of this paper is to study homology and cohomology theories on categories of E∞ ring spectra, or equivalently, on the modern categories of EKMM commutative S-algebras =-=[5]-=-, where the initial object is the sphere spectrum S and the coproduct is the modern symmetric monoidal smash product. Because the final object in the category of commutative S-algebras is the trivial ... |

83 | Stable model categories are categories of modules
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(Show Context)
Citation Context ...the equivalence of homotopy categories arises from a Quillen equivalence. Although the technical hypotheses do not quite apply, this theorem is closely related to the title theorem of Schwede–Shipley =-=[16]-=- that stable categories are categories of modules. Theorem 3 is in marked contrast to the corresponding situation for simplicial commutative algebras studied by Schwede [15], where the stable category... |

68 | Homotopy theories and model categories - Dwyer, Spaliński - 1995 |

59 | Spectra and symmetric spectra in general model categaries
- Hovey
(Show Context)
Citation Context ... is a collection of maps An → A ′ n that commute with the structure maps. We define the homotopy groups of a spectrum A = {An} by πqA = Colim ˜πq+nAn, where ˜π∗A = Ker(π∗A → π∗B). Standard techniques =-=[6, 10]-=- allow us to prove in Section 7 that the category of CB/B-spectra forms a closed model category, with weak equivalences the maps that induce isomorphisms on homotopy groups. The resulting homotopy cat... |

55 | André-Quillen cohomology of commutative S-algebras
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(Show Context)
Citation Context ...ological André–Quillen cohomology with various coefficients provides examples of cohomology theories on CR/B. For a cofibrant commutative R-algebra A and a cofibrant commutative A-algebra X, Basterra =-=[1]-=- constructs the cotangent complex LΩAX as the derived commutative X-algebra indecomposables (of the derived augmentation ideal) of X ∧A X. The cotangent complex LΩAX is an X-module, and restricting to... |

35 | H.: Abstract homotopy theory - Brown - 1965 |

22 |
Homotopy theory of A∞ ring spectra and applications to MU - modules, K-theory 24
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(Show Context)
Citation Context ...the category of A-bimodules. When A is cofibrant, one typically writes A e for A ∧R A op . More generally, A e denotes A ′ ∧R A ′op , for some fixed choice of cofibrant approximation A ′ → A. Lazarev =-=[7]-=- identifies Topological Quillen Cohomology in terms of Topological Hochschild Cohomology, and identifies the module of infinitesimal deformations of an associative algebra A as the homotopy fiber of t... |

20 |
Homotopical algebra, volume 43
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(Show Context)
Citation Context ...ot the cofibrations, this completes the proof of Theorem 9.1. Remark 9.5. It is sometimes useful to consider model categories that do not have all small colimits but (as in the original definition in =-=[14]-=-) are only assumed to have finite colimits. For these categories, it appears unlikely that the version of the Product Axiom above is equivalent to the one in the introduction, and it depends on the ap... |

17 | Ring spectra which are Thom complexes
- Mahowald
- 1979
(Show Context)
Citation Context ...where Z is the spectrum associated to X. According to Lewis [8, §IX], the Thom spectrum M obtained from a map of E∞ spaces X → BF naturally has the structure of an E∞ ring spectrum (see also Mahowald =-=[9]-=-), and the diagonal map M −→ M ∧ X+ is a map of E∞ ring spectra. The derived extension of scalars to E∞ M-algebras, M ∧ M −→ M ∧ X+ ∼ = M ∧ Σ ∞ X+ induces the Thom isomorphism and is a weak equivalenc... |

16 | Failure of Brown Representability in Derived Categories,'' preprint - Christensen, Keller, et al. |

15 | The uniqueness of infinite loop space machines
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- 1978
(Show Context)
Citation Context ...e cotangent complex of E∞ ring spectra. Up to equivalence, these do not depend on the operad, and we can understand these in terms of an equivalent commutative S-algebra. The work of May and Thomason =-=[12]-=- shows that up to isomorphism in the stable category, there is a canonical spectrum associated to X whose zeroth space is the group completion of X; it is any spectrum output by an “infinite loop spac... |

15 | Stable homotopy of algebraic theories
- Schwede
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(Show Context)
Citation Context ...theorem of Schwede–Shipley [16] that stable categories are categories of modules. Theorem 3 is in marked contrast to the corresponding situation for simplicial commutative algebras studied by Schwede =-=[15]-=-, where the stable category is equivalent to the homotopy category of modules over a ring spectrum that is generally very different from the ground ring; see also Theorem 2.8 below. For an object A in... |

9 | Equivariant Stable Homotopy Theory, volume 1213 - Lewis, May, et al. |

8 |
Amnon Neeman, Failure of Brown representability in derived categories, Topology 40
- Christensen, Keller
- 2001
(Show Context)
Citation Context ... is an equivalence of categories. A characterization of the category of homology theories on CR/B is slightly trickier because of the failure of Brown’s Representability Theorem for homology theories =-=[3]-=-. Given a homology theory h∗ on the category MB of B-modules, we obtain a homology theory hD ∗ on the category CR/B by setting h D ∗ (X, A) = h∗(LAb B AX, ∗). This describes a functor from the categor... |

2 | E∞ ring spaces and E∞ ring spectra, volume 577 - May - 1977 |