## Invertible spectra in the E(n)-local stable homotopy category

Venue: | J. London Math. Soc |

Citations: | 28 - 7 self |

### BibTeX

@ARTICLE{Hovey_invertiblespectra,

author = {Mark Hovey and Hal Sadofsky},

title = {Invertible spectra in the E(n)-local stable homotopy category},

journal = {J. London Math. Soc},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

Suppose C is a category with a symmetric monoidal structure, which we will refer to as the smash product. Then the Picard category is the full subcategory of objects which have an inverse under the smash product in C, and the Picard group Pic(C) is the collection of isomorphism classes of such invertible objects. The

### Citations

203 | Complex cobordism and stable homotopy groups of spheres - Ravenel - 1986 |

144 |
Stable Homotopy and Generalised Homology
- Adams
- 1974
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Citation Context ...E(n)sX �� = E(n)sas a comodule over E(n)sE(n). This suggests considering the Adams spectral sequence based on E(n)- homology. The E 2 term of this spectral sequence is Ext ; E(n)sE(n) (E(n)s; E(n)=-=sX) [Ada74]-=-, since E(n)sE(n) is flat over E(n)s. The differential d r follows the usual Adams pattern, lowering the t \Gamma s degree by 1 and raising the filtration s by r. In order to understand this E 2 -term... |

87 | Localization with respect to certain periodic homology theories - Ravenel - 1984 |

65 |
Nilpotence and periodicity in stable homotopy theory, Annals of Maths studies 128
- Ravenel
- 1992
(Show Context)
Citation Context ... from BP to a wedge of suspensions of E(j) and a similar K(j)-equivalence from E(n) to a different wedge of suspensions of E(j) when nsj. Let X be a type j finite spectrum with v j -self map v, as in =-=[Rav92]-=-. It follows that there is a K(j)-equivalence from BP v \Gamma1 X to a wedge of suspensions of E(j)sv \Gamma1 X and a similar K(j)-equivalence from E(n)sv \Gamma1 X to a wedge of suspensions of E(j)sv... |

49 |
Bousfield, The localization of spectra with respect to homology, Topology 18
- K
- 1979
(Show Context)
Citation Context ...y category S consisting of E(n)-local spectra. A spectrum X is E(n)-local if and only if [W; X ] = 0 for all spectra W such that E(n)sW = 0. There is a Bousfield localization functor Ln : S \Gamma! L =-=[Bou79] adj-=-oint to the inclusion functor L \Gamma! S. By the smashing theorem of Hopkins-Ravenel [Rav92, Chapter 8], LnX �� = LnS 0 X . Thus L is closed under the smash product in S. Furthermore, in the term... |

39 |
properties of comodules over M
- Landweber, Homological
- 1976
(Show Context)
Citation Context ..., where Pic(L) 0 = ker(d). 2. Certain E(n)sE(n)-comodules In this section, we investigate the structure of E(n)sX , when X is in Pic(L). We will use the fact that E(n) is Landweber exact. Recall from =-=[Lan76]-=- that this means that the functor F that takes a BPsBP-comodule M to the E(n)s- module E(n) \Omega BPsM is exact. We also need to recall that F (M) in fact admits a natural structure as an E(n)sE(n)-c... |

33 | Bousfield localization functors and Hopkins’ chromatic splitting conjecture
- Hovey
- 1995
(Show Context)
Citation Context ...which are 0, and where r(I) = P k 2p n i k (p k \Gamma 1). Note that the K(n)-localization of BP is the completion of v \Gamma1 n BP at the ideal I n = (p; v 1 ; : : : ; v n\Gamma1 ), as is proved in =-=[Hov95]-=-. Theorem B was originally proved as The first author was partially supported by an NSF Postdoctoral Fellowship. The second author was partially supported by the NSF. 2 M. HOVEY AND H. SADOFSKY part o... |

27 | Sur les groupes de Lie formels à un paramétre - Lazard - 1955 |

23 |
Morava stabilizer algebras and the localization of Novikov’s Ez-term
- Miller, Ravenel
- 1977
(Show Context)
Citation Context ... For our purposes here we need a similar splitting for the K(n)-localization of E(m) when msn. We use Theorem B to give a simple proof of a generalization of the MillerRavenel change of rings theorem =-=[MR77]-=-. We also present another proof of this change of rings theorem using an algebraic result of Hopkins that deserves to be more widely known. We prove that every spectrum in L is E(n)-nilpotent. These f... |

19 |
Constructions of elements in Picard groups. Topology and representation theory
- Hopkins, Mahowald, et al.
- 1992
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Citation Context ...E-local spectra need not be E-local, so one must relocalize the result by applying the Bousfield localization functor LE . The most well-known case is E = K(n), the nth Morava K-theory, considered in =-=[HMS94]-=-. In this paper we study the case E = E(n), where E(n) is the Johnson-Wilson spectrum. In this case the E-localization functor is universally denoted Ln , and we denote the category of E-local spectra... |

12 |
Completions of Z/(p)-Tate cohomology of periodic spectra
- Ando, Morava, et al.
- 1998
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Citation Context ...B was originally proved as The first author was partially supported by an NSF Postdoctoral Fellowship. The second author was partially supported by the NSF. 2 M. HOVEY AND H. SADOFSKY part of work on =-=[AMS]-=-. It is included here since [AMS] has not yet been published. That such a splitting should exist was conjectured by the first author in [Hov94, Conjecture 2.3.5]. Note that LK(n) does not commute with... |

12 |
Liftings of formal groups and the Artinian completion of v −1 n
- Baker, Würgler
- 1989
(Show Context)
Citation Context ...ked whether E(n) is a summand in v \Gamma1 n BP after applying some completion functor. Baker and Wurgler proved that the Artinian completion of v \Gamma1 n BP is a product of copies of LK(n) E(n) in =-=[BW89]. Theore-=-m B. We have isomorphisms of spectra LK(n) BP �� = LK(n) ( I \Sigma r(I) E(n)) �� = LK(n) ( I \Sigma r(I) LK(n) E(n)) where I runs through all sequences of nonnegative integers (i 1 ; i 2 ; : ... |

9 | On the p-adic interpolation of stable homotopy groups, Adams Memorial Symposium on Algebraic Topology - Strickland - 1990 |

5 |
Cohomology theory of unitary manifolds with singularities and formal group laws
- Würgler
- 1976
(Show Context)
Citation Context ...there is no finite spectrum M(I) with BPsM(I) �� = BPs=I . In this case, it is still possible to prove that v \Gamma1 j BP=I splits as a wedge of suspensions of E(j)=I using the methods of [JW75] =-=and [Wur76]-=-. If I = I n , this is the result in [Wur76] that B(n) splits additively into a wedge of suspensions of K(n). We would like to prove an analogous splitting for LK(n) E(n) when nsj. We begin with the s... |

4 | vn-elements in ring spectra and applications to bordism theory - Hovey - 1997 |

3 |
Hopf-algebroids and a new proof of the Morava-Miller-Ravenel change of rings theorem
- Hopkins
- 1995
(Show Context)
Citation Context ... 0 ; v i 1 1 ; : : : ; v i j\Gamma1 j \Gamma1 ), we find that condition (b) of Lemma 3.2 holds, completing the proof of Theorem 3.1. We now give the algebraic proof of the change-of-rings theorem. In =-=[Hop95]-=-, Hopkins considers the following general situation. Suppose (A; \Gamma) is a Hopf algebroid over a commutative ring K such that \Gamma is flat as a left (and hence also as a right) A-module. Let f : ... |

1 |
The generating hypothesis revisited, preprint
- Devinatz
(Show Context)
Citation Context ...(n)-nilpotent. These facts give us some control over the E 2 -term of the E(n)-based Adams spectral sequence (Theorem 5.1) and its convergence (Theorem 5.3) respectively. These properties are used in =-=[Dev96]. We-=- also investigate what happens when p = 2 and n = 1, the first case not covered by Theorem A. We find Pic(L) �� = Z \Phi Z=2 in this case. We would like to thank Mike Hopkins, Haynes Miller, Neil ... |

1 | Minimal atlases of real projective spaces - Hopkins |

1 |
Axiomatic stable homotopy theory, preprint
- Hovey, Palmieri, et al.
- 1995
(Show Context)
Citation Context ... the inclusion functor L \Gamma! S. By the smashing theorem of Hopkins-Ravenel [Rav92, Chapter 8], LnX �� = LnS 0 X . Thus L is closed under the smash product in S. Furthermore, in the terminology=-= of [HPS95]-=-, L is a monogenic Brown category. That is, L has almost all of the same formal properties as the ordinary stable homotopy category S, where the unit of the smash product is no longer S 0 but LnS 0 . ... |