## HZ-algebra spectra are differential graded algebras (2004)

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Venue: | Amer. Jour. Math |

Citations: | 32 - 10 self |

### BibTeX

@ARTICLE{Shipley04hz-algebraspectra,

author = {Brooke Shipley},

title = {HZ-algebra spectra are differential graded algebras},

journal = {Amer. Jour. Math},

year = {2004},

volume = {129},

pages = {351--379}

}

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### Abstract

Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 1.

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Citation Context ...ules over an Eilenberg-Mac Lane spectrum associated to a discrete ring R is equivalent to the unbounded derived category of R, or the homotopy category of (unbounded) di#erential graded R-modules. In =-=[EKMM]-=-, this result was reproved in the context of a symmetric monoidal category of spectra where A# modules are replaced by modules. In [SS1, 5.1.6], we strengthened this result by showing that the associa... |

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Citation Context ...in the ordinary underlying model category for Sp # (ChQ ) rather than the positive model category. 3. Monoidal model categories and the monoid axiom In this section we prove Proposition 2.5. We use [=-=SS00]-=- to establish model categories and Quillen invariance on the categories of modules and algebras. The monoid axiom is the main property needed to extend a model structure on a monoidal model category t... |

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Citation Context ...ive a way around this problem. A Quillen model structure on a category C is a choice of three subcategories called weak equivalences, cofibrations and fibrations which satisfy certain axioms [Qui67]. =-=[DS95]-=- is a good introduction to model categories; our standard reference though is [Hov99]. Given a Quillen model structure, inverting the weak equivalences, W, produces a well-defined homotopy category Ho... |

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Citation Context ...S2, 1.1]. In this paper we use symmetric spectra as our symmetric monoidal category of spectra; the results in Theorem 1.1 can be translated to other symmetric monoidal categories of spectra by using =-=[MMSS]-=-, [Sch1] or [S1]. The comparison between HZ-module spectra and (unbounded) di#erential graded Z-modules, Ch in [SS1, Appendix B] is given by comparing each of these model categories to an intermediary... |

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Citation Context ...sewhere we could consider either right or left modules. We mention left modules in general because they are more standard, but right modules are what appear naturally in the classification results in =-=[SS1]-=-. [SS1, 3.3.3] shows that any stable model category C with a set of small generators G is Quillen equivalent to (right) modules over a symmetric ring spectrum with many objects, the endomorphism ring ... |

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Citation Context ...ras and modules with the weak equivalences and fibrations being determined on the underlying category. If C is a monoidal model category, [Hov00, 8.11] shows that the stable model category defined in =-=[Hov00]-=- on Sp # (C) is also a monoidal model category. [Hov00] was not able to verify the monoid axiom in general though. In Proposition 3.3 we show that the monoid axiom does hold for Sp # (sAb), Sp # (ch +... |

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Citation Context ...⊗ Z[1]) −→ FnA}n≥0 for one object A from each isomorphism class of countable objects in Sp Σ (ch + ). See [Hov01, Section 2] for a brief summary of Bousfield localization; the definitive reference is =-=[Hir00]-=-. One could also use the machinery developed by Jeff Smith [Sm] to establish the existence of these local model categories because these categories are also left proper and combinatorial (cofibrantly ... |

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Citation Context ...# Z[1]) -# FnA}n#0 for one object A from each isomorphism class of countable objects in Sp # (ch + ). See [Hov00, Section 2] for a brief summary of Bousfield localization; the definitive reference is =-=[Hir00]-=-. One could also use the machinery developed by Je# Smith [Sm] to establish the existence of these local model categories because these categories are also left proper and combinatorial (cofibrantly g... |

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Citation Context ...ent comparison between HZ-Mod and Ch via three intermediary categories. The functors and categories involved in each of these Quillen equivalences are monoidal and satisfy the hypotheses developed in =-=[SS2]-=- for lifting Quillen equivalences to categories of algebras and modules. In [SS2, Section 6] we consider monoids with many objects (or enriched categories) and their categories of (right) modules. In ... |

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Citation Context .... In this paper we use symmetric spectra as our symmetric monoidal category of spectra; the results in Theorem 1.1 can be translated to other symmetric monoidal categories of spectra by using [MMSS], =-=[Sch1]-=- or [S1]. The comparison between HZ-module spectra and (unbounded) di#erential graded Z-modules, Ch in [SS1, Appendix B] is given by comparing each of these model categories to an intermediary categor... |

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Citation Context ... is a Quillen equivalence between di#erential graded #B-modules and B-module spectra. #B-Mod #Q B-Mod Special cases of Theorem 1.1, Parts 2 and 3 have already appeared in the literature. Robinson, in =-=[Rob87]-=-, showed that the homotopy category of A# modules over an Eilenberg-Mac Lane spectrum associated to a discrete ring R is equivalent to the unbounded derived category of R, or the homotopy category of ... |

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Citation Context ... it would require different techniques. The following related statement, however, is simple to prove here and is used in the construction of an algebraic model of rational T n -equivariant spectra in =-=[GS]-=-. Its proof appears near the end of Section 4. Recall Θ from Theorem 1.1. Theorem 1.2. For C any commutative HQ-algebra, ΘC is naturally weakly equivalent to a commutative differential graded Q-algebr... |

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Topological André-Quillen cohomology and E∞ André-Quillen cohomology Adv
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Citation Context ...en the categories of modules over these algebras; see Corollary 2.15. We then use these Quillen equivalences to construct algebraic models for rational stable model categories; see Corollary 2.16. In =-=[Man03]-=-, an analogue of Theorem 1.1 is proved for E∞-algebras, the commutative analogue of A∞-algebras which are associative and commutative up to coherent homotopies. Mandell shows that the homotopy categor... |

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Citation Context ... I × I −→ I given by p(n, m) = n + m does not agree with the functor pτ : I × I −→ I given by pτ(n, m) = m+n where τ(n, m) = (m, n). See also the discussion of commutative I-monoids in section 2.2 of =-=[Sc]-=-. Since the paper [S1] is mainly concerned with associative algebras, the only place this mistaken claim was used was in the proof of Theorem 1.2. Since D is not symmetric monoidal, it does not preser... |

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Citation Context ...te large. This model can be used as a stepping stone to a practical model though. For example, this corollary applies to the category of rational G-equivariant spectra for any compact Lie group G. In =-=[Shi02]-=- and [GS] this large model is used to show that there is an explicit algebraic model Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-dimensional torus. For G fini... |

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Citation Context ...his corollary applies to the category of rational G-equivariant spectra for any compact Lie group G. In [S2] and [GS] this algebraicization is used to show that the explicit algebraic models given in =-=[Gr99]-=- and [Gr] are Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-dimensional torus. For G finite this extends the results of [GM95, Appendix A] to incomplete univers... |

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Citation Context ...H∗A and the full graded subcategory of Ho(C) with objects G. C≃QMod-E(G) ≃Q Mod-A For example, this corollary applies to the category of rational G-equivariant spectra for any compact Lie group G. In =-=[S2]-=- and [GS] this algebraicization is used to show that the explicit algebraic models given in [Gr99] and [Gr] are Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-di... |

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Citation Context ...ebras, in the author’s paper [American Journal of Mathematics 129 (2007) 351-379 (arxiv:math/0209215v4)] is correct as originally stated. Neil Strickland carefully proved that D is symmetric monoidal =-=[St1]-=-; so Proposition 4.7 and hence also Theorem 1.2 hold as stated. Strickland’s proof will appear in joint work with Stefan Schwede [ScSt]; see related work in [arXiv:0810.1747] [St2]. Note here D is def... |

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Citation Context ...A and the full graded subcategory of Ho(C) with objects G. C #Q Mod-E(G) #Q Mod-A For example, this corollary applies to the category of rational G-equivariant spectra for any compact Lie group G. In =-=[S2]-=- and [GS] this algebraicization is used to show that the explicit algebraic models given in [Gr99] and [Gr] are Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-di... |

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Citation Context ...his corollary applies to the category of rational G-equivariant spectra for any compact Lie group G. In [S2] and [GS] this algebraicization is used to show that the explicit algebraic models given in =-=[Gr99]-=- and [Gr] are Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-dimensional torus. For G finite this extends the results of [GM95, Appendix A] to incomplete univers... |

2 | Rational torus-equivariant stable homotopy III: the Quillen equivalence, in preparation
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Citation Context ...ry applies to the category of rational G-equivariant spectra for any compact Lie group G. In [S2] and [GS] this algebraicization is used to show that the explicit algebraic models given in [Gr99] and =-=[Gr]-=- are Quillen equivalent to the category of rational T n -equivariant spectra for T n the n-dimensional torus. For G finite this extends the results of [GM95, Appendix A] to incomplete universes; see [... |

2 |
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Citation Context ...ull graded subcategory of Ho(C) with objects G. C �Q Mod-E(G) �Q Mod-A To specify A, recall the functor Θ from Theorem 1.1. This functor can be extended to rings with many objects; see [SS03b, 6] and =-=[DS, 6]-=-. Then A ∼ = Θ(HQ ∧ cE(G)) where c is a cofibrant replacement functor for ring spectra (with many objects) from [SS03b, 6.3]. In general, Mod-A is not a practical algebraic model; it is difficult to m... |

2 |
Topological André-Quillen cohomology and
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Citation Context ...en the categories of modules over these algebras; see Corollary 2.15. We then use these Quillen equivalences to construct algebraic models for rational stable model categories; see Corollary 2.16. In =-=[Man03]-=-, an analogue of Theorem 1.1 is proved for E∞-algebras, the commutative analogue of A∞-algebras which are associative and commutative up to coherent homotopies. Mandell shows that the homotopy categor... |

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