## Chromatic phenomena in the algebra of BP∗BP -comodules (2002)

Citations: | 2 - 1 self |

### BibTeX

@TECHREPORT{Hovey02chromaticphenomena,

author = {Mark Hovey},

title = {Chromatic phenomena in the algebra of BP∗BP -comodules},

institution = {},

year = {2002}

}

### OpenURL

### Abstract

Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP-

### Citations

202 | Model Categories - Hovey - 1999 |

179 |
Model categories and their localizations
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Citation Context ...ousfield localization of the homotopy model structure with respect to the maps 0 −→ s k A/In+1 for all k, where s k A/In+1 denotes the complex which is A/In+1 in degree k and 0 elsewhere. Recall from =-=[Hir03]-=- that this means that a left Quillen functor defines a left Quillen functor Φ∗Y F : Ch(Γ) −→ M F : L f nCh(Γ) −→ M if and only if 0 −→ (LF )(skA/In+1) is an isomorphism in ho M, where LF denotes the t... |

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Citation Context ...y a corollary of Theorem 3.3 and general facts about model categories. It is proved in [Hov02a]; another way to say it is that Stable(Γ) is a unital algebraic stable homotopy category in the sense of =-=[HPS97]-=-. One drawback of Stable(Γ) is that it is not in general monogenic. That is, A and its suspensions are not generally enough to generate the whole category. This is unavoidable even for Hopf algebras. ... |

99 |
Representations and cohomology I
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Citation Context ...finite group and k is a field, the stable homotopy category of the Hopf algebra of functions from G to k is closely related to the stable module category much studied in modular representation theory =-=[Ben98]-=-, as explained in [HPS97]. If G is a p-group, the stable module category is monogenic, but not in general. However, one certainly expects Stable(BP∗BP ) and Stable(E(n)∗E(n)) to be monogenic, so we ne... |

87 |
Localization with respect to certain periodic homology theories
- Ravenel
- 1984
(Show Context)
Citation Context ... of type n + 1, and Ln is localization with respect to the homology theory E(n). These functors are probably different in the ordinary stable homotopy category (because Ravenel’s telescope conjecture =-=[Rav84]-=- is widely expected to be false), but they turn out to agree on the abelian category of BP∗BP -comodules. We then have the following theorem. Theorem A. Let Ln denote localization away from BP∗/In+1 i... |

60 |
Sheafifiable homotopy model categories
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(Show Context)
Citation Context ...to do to form Stable(Γ). To see this, note that there is a model structure on Ch(Γ) whose homotopy category is the derived category, as there is on Ch(A) for any Grothendieck14 MARK HOVEY category A =-=[Bek00]-=-. The cofibrations in this model structure are the monomorphisms, the fibrations are the epimorphisms with DG-injective kernel, and the weak equivalences are the homology isomorphisms. Here a complex ... |

49 |
Bousfield, The localization of spectra with respect to homology, Topology 18
- K
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Citation Context ...entrated where t = 0. However, it is not entirely clear to what it converges. The obvious guess is π∗LHBS, where LHB denotes Bousfield localization with respect to HB. Bousfield’s convergence results =-=[Bou79]-=- should be re-examined to see if they apply in a more general setting to answer this question. Note that if B is an A-algebra that is also a field, then HB will be a field object of Stable(Γ). In part... |

39 |
properties of comodules over M
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- 1976
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Citation Context ... equivalent categories of E∗E-comodules, even though the categories of E∗-modules can be drastically different. It also leads to structural results about E∗E-comodules analogous to those of Landweber =-=[Lan76]-=- for BP∗BP-comodules. To extend this result to the derived category setting, we first must decide what we mean by the derived category. The ordinary derived category, obtained by inverting homology is... |

29 | A Homotopy Theory for Stacks
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(Show Context)
Citation Context ...b ′ ) is a ring isomorphism. The “if”half of this theorem is the main result of [Hov02b]. The “only if” half is much easier and was proven in [HS03a]. This theorem has a better formulation. Hollander =-=[Hol01]-=- has constructed a model structure on presheaves of groupoids on a Grothendieck topology C; the fibrant objects are stacks; see also [Jar01]. In particular, we can take our Grothendieck site to be the... |

28 | Invertible spectra in the E(n)-local stable homotopy category
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(Show Context)
Citation Context ...M, N) ∼ = Ext E(n)∗E(n)(E(n)∗ ⊗BP∗ M, E(n)∗ ⊗BP∗ N). This change of rings theorem includes the Miller-Ravenel change of rings theorem [MR77] and the change of rings theorem of the author and Sadofsky =-=[HS99]-=- as special cases. Also, E(n) can be replaced by any Landweber exact commutative ring spectrum E with E∗/In ̸= 0 and E∗/In+1 = 0. Because the basic structure of the abelian category of comodules over ... |

24 |
Homological Properties of Comodules over MU∗(MU) and BP∗(BP
- Landweber
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Citation Context ... equivalent categories of E∗E-comodules, even though the categories of E∗-modules can be drastically different. It also leads to structural results about E∗E-comodules analogous to those of Landweber =-=[Lan76]-=- for BP∗BP -comodules. To extend this result to the derived category setting, we first must decide what we mean by the derived category. The ordinary derived category, obtained by inverting homology i... |

23 |
Morava stabilizer algebras and the localization of Novikov’s Ez-term
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- 1977
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Citation Context ... i > 0. Then there is a change of rings isomorphism Ext ∗∗ BP∗BP (M, N) ∼ = Ext E(n)∗E(n)(E(n)∗ ⊗BP∗ M, E(n)∗ ⊗BP∗ N). This change of rings theorem includes the Miller-Ravenel change of rings theorem =-=[MR77]-=- and the change of rings theorem of the author and Sadofsky [HS99] as special cases. Also, E(n) can be replaced by any Landweber exact commutative ring spectrum E with E∗/In ̸= 0 and E∗/In+1 = 0. Beca... |

19 |
André-Quillen (co)homology for simplicial algebras over simplicial operads, Une Dégustation Topologique [Topological Morsels]: Homotopy Theory
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Citation Context ...alizable comodules. This hypothesis is really due to Adams [Ada74, Section III.13], who used it for the Hopf algebroid (E∗, E∗E) to set up universal coefficient spectral sequences. We learned it from =-=[GH00]-=-, as well as the following lemma. Lemma 1.6. Suppose (A, Γ) is an Adams Hopf algebroid. Then it is flat, and the dualizable comodules generate the category of Γ-comodules. In categorical language, the... |

19 | Stacks and the homotopy theory of simplicial sheaves, in Equivariant stable homotopy theory and related areas
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Citation Context ...n [HS03a]. This theorem has a better formulation. Hollander [Hol01] has constructed a model structure on presheaves of groupoids on a Grothendieck topology C; the fibrant objects are stacks; see also =-=[Jar01]-=-. In particular, we can take our Grothendieck site to be the flat topology on Aff, the opposite category of commutative rings (with a cardinality bound so we get a small category). In this topology, a... |

14 | Morita theory for Hopf algebroids and presheaves of groupoids
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Citation Context ...S 23 (c) We show that, although Φ ∗ (LB) is not cofibrant, it is still nice enough that it suffices to check that v −1 k is a homotopy isomorphism. A/Ik −→ v −1 k A/Ik ∧ Φ ∗ (LB) (d) It was proved in =-=[Hov02b]-=- that the Hopf algebroid (v −1 k weakly equivalent to (v −1 k v −1 k B/Ik −→ v −1 k B/Ik, v −1 k Γ/Ik) is ΓB/Ik). Hence it suffices to show that B/Ik ∧ Φ∗Φ ∗ (LB) ∼ = v −1 k B/Ik ∧ LB A/Ik, v −1 k is ... |

14 | Cobordism and the Stable Homotopy Groups of Spheres - Complex - 1986 |

13 | Homotopy theory of comodules over a Hopf algebroid. In Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory, volume
- Hovey
- 2004
(Show Context)
Citation Context ...the abelian sense, of the category of BP∗BP - comodules. This gives analogues of the usual structure theorems, such as the Landweber filtration theorem, for E(n)∗E(n)-comodules. We recall the work of =-=[Hov02a]-=-, where an improved version Stable(Γ) of the derived category of comodules over a well-behaved Hopf algebroid (A, Γ) is constructed. The main new result of the paper is that Stable(E(n)∗E(n)) is a Bou... |

12 |
Completions in algebra and topology.” Handbook of algebraic topology
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Citation Context ...n E 0,∗ 2 that comes from BP∗X is a permanent cycle. This theorem is proved in [HS03b]. It is very closely related to the local cohomology spectral sequence of Greenlees [Gre93] and Greenlees and May =-=[GM95]-=-. One way of putting it is that we show that the Greenlees spectral sequence is a spectral sequence of comodules in this case. When X = S 0 , this implies that the spectral sequence collapses with no ... |

12 | Comodules and Landweber exact homology theories
- Hovey, Strickland
(Show Context)
Citation Context ...BP -COMODULES MARK HOVEY Abstract. We describe the author’s research with Neil Strickland on the global algebra and global homological algebra of the category of BP∗BP - comodules. We show, following =-=[HS03a]-=-, that the category of E(n)∗E(n)comodules is a localization, in the abelian sense, of the category of BP∗BP - comodules. This gives analogues of the usual structure theorems, such as the Landweber fil... |

11 |
K-homology of universal spaces and local cohomology of the representation ring, Topology 32
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Citation Context .... Furthermore, every element in E 0,∗ 2 that comes from BP∗X is a permanent cycle. This theorem is proved in [HS03b]. It is very closely related to the local cohomology spectral sequence of Greenlees =-=[Gre93]-=- and Greenlees and May [GM95]. One way of putting it is that we show that the Greenlees spectral sequence is a spectral sequence of comodules in this case. When X = S 0 , this implies that the spectra... |

7 |
Finite localizations, Boletin de la Sociedad Matematica Mexicana 37
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Citation Context ...−→ (LF )(skA/In+1) is an isomorphism in ho M, where LF denotes the total left derived functor of F . The homotopy category of Lf nCh(Γ) is the finite localization Lf nStable(Γ) in the sense of Miller =-=[Mil92]-=- of Stable(Γ) away from A/In+1. The total left derived functor of the identity, thought of as a functor from Ch(Γ) to Lf nCh(Γ), is the finite localization functor Lf n on Stable(Γ). Proposition 4.5. ... |

4 | Johnson and Zen-ichi Yosimura, Torsion in Brown-Peterson homology and Hurewicz homomorphisms - Copeland - 1980 |

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Model categories and their localizations, preprint, available at http://www-math.mit.edu/∼psh
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(Show Context)
Citation Context ...he Bousfield localization of the homotopy model structure with respect to the maps 0 −→ skA/In+1 for all k, where skA/In+1 denotes the complex which is A/In+1 in degree k and 0 elsewhere. Recall from =-=[Hir02]-=- that this means that a left Quillen functor defines a left Quillen functor F : Ch(Γ) −→ M F : L f nCh(Γ) −→ M if and only if 0 −→ (LF)(skA/In+1) is an isomorphism in ho M, where LF denotes the total ... |