## topological AndréQuillen homology and stabilization

Venue: | Topology Appl. 121 (2002) No.3 |

Citations: | 13 - 1 self |

### BibTeX

@ARTICLE{Basterra_topologicalandréquillen,

author = {Maria Basterra and Randy Mccarthy},

title = {topological AndréQuillen homology and stabilization},

journal = {Topology Appl. 121 (2002) No.3},

year = {},

pages = {556}

}

### OpenURL

### Abstract

The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: André-Quillen homology, E∞homology).

### Citations

375 |
Homotopical algebra
- Quillen
- 1967
(Show Context)
Citation Context ...onstructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: André-Quillen homology, E∞homology). Introduction In Homotopical Algebra =-=[11]-=-, Daniel Quillen introduced the concept of model category structure giving an axiomatization of the necessary conditions that a category must satisfy in order to support a homotopy theory. Within this... |

69 |
On the (co)-homology of commutative rings
- Quillen
- 1970
(Show Context)
Citation Context ...ruction to calculate H Γ ∗ (A/k; M). For the rest of this section we consider the special case of augmented algebras: Let A be a ring and let (B•, η, ɛ) be an augmented cofibrant simplicial A algebra =-=[12]-=-, i.e. a simplicial algebra which is semi-free on each degree and is equipped with an algebra map, ɛ, to the constant simplicial algebra at A. We consider A as a B•-module with action a ⊗ b −→ aɛ(b) a... |

57 | André-Quillen cohomology of commutative S-algebras
- Basterra
- 1999
(Show Context)
Citation Context ...E∞ ring spectrum [5]. This provided the motivation for the first author’s thesis, and the construction of the category of E∞ ring spectra as a closed model category [3] allowed her to define homology =-=[2]-=- following the conceptual prescription given by Quillen. The equivalence between stabilization and Topological André-Quillen homology has been established in full generality by the first author in joi... |

56 |
Spectra in model categories and applications to the algebraic cotangent complex
- Schwede
- 1997
(Show Context)
Citation Context ...der of some basic facts about simplicial model categories which will allow us to define stabilization of functors. The standard reference for this material is [11] but there is also a nice account in =-=[14]-=-. Recall from Chapter II in [11] that a simplicial category C is endowed with a simplicial “function complex” Hom C(X, Y ) and for each simplicial set K, functors X � K ⊗ X and X � X K which satisfy t... |

27 |
Méthode simpliciale en algèbre homologique et algèbre commutative
- André
- 1967
(Show Context)
Citation Context ...e preimage of j where it is an order preserving bijection. Then, r∑ ∂q([fq|fq−1| · · · |f1] ⊗ x) = [f j j q−1 | · · · |f1 ] ⊗ lj∗(x) if q > 1 j=1 and ∂1([n → 1] ⊗ x) = ∑ n j=1 gj∗(x) where gj : [n] → =-=[1]-=- is the map which is 1 at j and 0 everywhere else. It is routine to check that with these definitions ∂i∂j = ∂j−1∂i for i < j so that d 2 = 0. When A and M are flat over k, the Γ−homology of A relativ... |

19 |
Operads and Γ-homology of commutative rings
- Robinson, Whitehouse
(Show Context)
Citation Context ...by ¯ hC. The first author would like to thank Mike Mandell and the second author would like to thank Ruth Kantorovitz for very helpful conversations. 1. Γ-homology: the Robinson-Whitehouse Complex In =-=[13]-=- A. Robinson and S. Whitehouse defined Γ-homology in the category of E∞ differential graded algebras. Since we are only concerned with the theory when applied to strictly commutative rings, in order t... |

14 | Robinson-Whitehouse complex and stable homotopy
- Pirashvili, Richter
(Show Context)
Citation Context ...hip to Γ-homology has not been determined. In this note we are able to identify the three constructions when we specialize to discrete commutative rings. We should mention that Pirashvili and Richter =-=[10]-=- have identified the complex that defines Γ-homology with the stabilization of a given functor from the category of finite sets to the category of differential graded modules but we make no use of suc... |

2 |
for the E∞ tensor product, in
- Mandell, Flatness
- 1999
(Show Context)
Citation Context .... Both categories above admit closed model structures where the weak-equivalences are the quasi-isomorphisms of their underlying differential graded modules and the fibrations are the surjective maps =-=[8]-=-. Hence, since the augmentation is split by the unit, all objects are fibrant. The augmentation ideal functor and the analogue of the functor KN from Section 2, which takes N, an object of E∞NA to an ... |

1 |
Towers of E∞-ring spectra with an application to BP
- Kriz
(Show Context)
Citation Context ...-algebras in terms of stabilization [4]. Also Kriz, using a similar approach to cohomology, developed a theory of Postnikov Towers for E∞ ring spectra towards the proof that BP is an E∞ ring spectrum =-=[5]-=-. This provided the motivation for the first author’s thesis, and the construction of the category of E∞ ring spectra as a closed model category [3] allowed her to define homology [2] following the co... |

1 |
The Unit of the Smash Product/Function Space Adjunction
- Lewis
- 1997
(Show Context)
Citation Context ... us the result. □ We should point out that the proof in the topological case above relies in the fact that the maps Ω n Σ n X −→ Ω n+1 Σ n+1 X over which we take the colimit are inclusions of spectra =-=[7]-=-. 4. Comparison of Γ-homology with T AQ-homology of discrete rings Let k be a commutative ring and A a commutative k-algebra. Let Hk and HA denote the corresponding Eilenberg-Mac Lane cofibrant commut... |

1 |
Topological André-Quillen Cohommology and E∞ André-Quillen Cohomology. Preprint available on line http://www.math.uchicago.edu/˜ mandell
- Mandell
(Show Context)
Citation Context ...mented differential graded algebra is, in particula,r an object of E∞CA/A. So, the normalization functor allow us to relate sCA/A, the category of simplicial augmented A-algebras to this category. In =-=[9]-=-, Mandell establishes an equivalence between the homotopy category of commutative HA-algebras and the homotopy category of E∞ differential graded A-algebras which extends the equivalence between the h... |