Invertible Spectra in the <i>E</i>
(<i>n</i>
)-Local Stable Homotopy Category

Hovey, Mark; Sadofsky, Hal

doi:10.1112/s0024610799007784

Cited by 62 publications

(75 citation statements)

References 18 publications

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“…As an immediate corollary, we recover a strengthening of the change of rings theorem of [HS99], which itself is a strengthening of the well-known Miller-Ravenel change of rings theorem [MR77]. The precise change of rings theorem is prove is stated below.…”

Section: Introductionmentioning

confidence: 96%

“…This is often called the fpqc topology; in it, a cover of a ring R is a finite family {R − → S i } of flat extensions of R such that S i is faithfully flat over R. A strengthening of faithfully flat descent then leads to the following theorem, proved as Theorem 4.5. This condition has appeared before, in [Hop95] and [HS99]. We point out that if we used the more general notion of internal equivalence mentioned above, Theorem D would remain unchanged, since Spec A is already a sheaf in the flat topology by faithfully flat descent.…”

Section: Introductionmentioning

confidence: 97%

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Morita theory for Hopf algebroids and presheaves of groupoids

Hovey¹

2002

ajm

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Abstract. Comodules over Hopf algebroids are of central importance in algebraic topology. It is well-known that a Hopf algebroid is the same thing as a presheaf of groupoids on Aff , the opposite category of commutative rings. We show in this paper that a comodule is the same thing as a quasi-coherent sheaf over this presheaf of groupoids. We prove the general theorem that internal equivalences of presheaves of groupoids with respect to a Grothendieck topology T on Aff give rise to equivalences of categories of sheaves in that topology. We then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasicoherent sheaves. The corresponding statement for Hopf algebroids is that weakly equivalent Hopf algebroids have equivalent categories of comodules. We apply this to formal group laws, where we get considerable generalizations of the Miller-Ravenel [MR77] and Hovey-Sadofsky [HS99] change of rings theorems in algebraic topology.

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 97%

Morita theory for Hopf algebroids and presheaves of groupoids

Hovey¹

2002

ajm

Self Cite

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“…Hence by (4.3) n>o it suffices to show that ^(Cp; (Lj^.BP)' 16^) ) = 0. However, by [7] and [8] this will follow by showing that each H 1 (Cp; E(n^(XP)) = 0, which follows by the usual calculations from the hypotheses of the proposition.…”

Section: Proposition 44 -Suppose E(n) (X) Is a Completed Free E(n) mentioning

confidence: 99%

The complex oriented cohomology of extended powers

Hunton¹

1998

Annales De L’institut Fourier

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“…Then it is shown in [4] (cf. [3]) that Pic(L 1 ) 0 = Z/2, whose generator is represented by the E(1)-localization of the question mark complex QM = V (0) ∪ η e 3 , where V (0) = S 0 ∪ 2 e 1 is the mod 2 Moore spectrum.…”

Section: Invertible Spectra and The Smith-toda Spectramentioning

confidence: 99%

“…We call a spectrum X ∈ L n (E(n)-)invertible if there exists a spectrum Y ∈ L n such that X ∧ Y = L n S 0 . In [4], Hovey and Sadofsky showed that every E(n)-invertible spectrum is homotopy equivalent to a suspension of L n S 0 if n 2 + n < 2p − 2, and that every E(1)-invertible spectrum is homotopy equivalent to a suspension of L 1 S 0 or L 1 QM if p = 2. Here QM denotes the socalled question mark complex S 0 ∪ 2 e 1 ∪ η e 3 .…”

Section: Introductionmentioning

confidence: 99%

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

Ichigi

Shimomura

2004

Proc. Amer. Math. Soc.

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Abstract. Let L 2 denote the Bousfield localization functor with respect to the Johnson-Hovey and Sadofsky, Invertible spectra in the E(n)-local stable homotopy category, showed that every invertible spectrum is homotopy equivalent to a suspension of the E(2)-local sphere L 2 S 0 at a prime p > 3. At the prime 3, it is shown, A relation between the Picard group of the E(n)-local homotopy category and E(n)-based Adams spectral sequence, that there exists an invertible spectrum X that is not homotopy equivalent to a suspension of L 2 S 0 . In this paper, we show the homotopy equivalence v 3 2 : Σ 48 L 2 V (1) V (1) ∧ X for the Smith-Toda spectrum V (1). In the same manner as this, we also show the existence of the self-map β : Σ 144 L 2 V (1) → L 2 V (1) that induces v 9 2 on the E(2) * -homology.

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Invertible Spectra in the E (n )-Local Stable Homotopy Category

Cited by 62 publications

References 18 publications

Morita theory for Hopf algebroids and presheaves of groupoids

Morita theory for Hopf algebroids and presheaves of groupoids

The complex oriented cohomology of extended powers

𝐸(2)-invertible spectra smashing with the Smith-Toda spectrum 𝑉(1) at the prime 3

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