On the moduli stack of commutative, 1-parameter formal groups

Smithling, Brian

doi:10.1016/j.jpaa.2010.04.024

Cited by 7 publications

(8 citation statements)

References 35 publications

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“…For example, Goerss' chromatic convergence theorem [, Theorem 8.22] translates into a special case of Theorem . Similar geometric approaches have been studied by Hollander , Naumann , Pribble , Sitte and Smithling .…”

Section: Introductionmentioning

confidence: 77%

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Algebraic chromatic homotopy theory for BP∗BP-comodules

Barthel

Heard

2018

Proc. London Math. Soc.

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In this paper, we study the global structure of an algebraic avatar of the derived category of ind-coherent sheaves on the moduli stack of formal groups. In analogy with the stable homotopy category, we prove a version of the nilpotence theorem as well as the chromatic convergence theorem, and construct a generalized chromatic spectral sequence. Furthermore, we discuss analogs of the telescope conjecture and chromatic splitting conjecture in this setting, using the local duality techniques established earlier in joint work with Valenzuela.Contents Recall that, working in the p-complete setting, the motivic cohomology of a point over Spec(C) is isomorphic to Fp[τ ], where τ has bidegree (0,1), and that this gives rise to an essential map τ : S 0,−1 → S 0,0 .Theorem (The chromatic spectral sequence). For any spectra X, Y , there is a natural convergent spectral sequence E n,s,t 1 = Ext s,t BP Furthermore, if BP * Y satisfies the conditions of Section 1, then the spectral sequence converges to Ext BP * BP (BP * X, BP * Y ).By truncating the chromatic tower, we can also build a height n analog of the chromatic spectral sequence which, as a special case, recovers the truncated chromatic spectral sequence constructed by Hovey and Sadofsky [29, Theorem 5.1].As a concrete application of our results, we obtain the following transchromatic comparison between the E 2 -terms of the BP -Adams spectral sequence and the E-Adams spectral sequence at height n, see Corollary 7.19.Corollary. If X is a p-local bounded below spectrum such that BP * X has projective BP * -dimension pdim(BP * X) r, then the natural map Ext s BPis an isomorphism if s < n − r − 1 and injective for s = n − r − 1.A related result can be found in work of Goerss [20, Theorem 8.24], however the authors are unaware of a result of this generality in the literature.

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Section: Introductionmentioning

confidence: 77%

“…In an earlier version of Goerss' manuscript , he uses the functors

q_{n}

to compare

{Comod}_{B P_{*} B P}

to an appropriately defined colimit of the categories

{Comod}_{W_{n}}

, see also . In order to prove the algebraic chromatic convergence theorem, we will use a derived version of this theory.…”

Section: The Algebraic Chromatic Convergence Theoremmentioning

confidence: 99%

Algebraic chromatic homotopy theory for BP∗BP-comodules

Barthel

Heard

2018

Proc. London Math. Soc.

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“…8.22] translates into a special case of Theorem 7.12. Similar geometric approaches have been studied by Hollander [Hol09], Naumann [Nau07], Pribble [Pri04], Sitte [Sit14], and Smithling [Smi11].…”

Section: Theorem (Algebraic Nilpotence Theorem -Weak Version)mentioning

confidence: 84%

“…It is proven in [Goe08,Prop. 3.25] that q * is faithful and that it induces an equivalence q ω * : colim n Comod ω Wn ∼ / / Comod ω BP * BP , see also [Smi11]. The next result relates two important properties of a comodule to the categories Comod Wn .…”

Section: The Algebraic Chromatic Convergence Theoremmentioning

confidence: 88%

Algebraic chromatic homotopy theory for $BP_*BP$-comodules

Barthel,

Heard

2017

Preprint

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“…This difficulty can be surmounted in two ways: enlarge the notion of an algebraic stack to include flat presentations or note that M fg can be written as the 2-category inverse limit of a tower of the algebraic stacks of "buds" of formal groups and is, thus, pro-algebraic. See [37].…”

Section: Remarkmentioning

confidence: 99%

Realizing families of Landweber exact homology theories

Goerss¹

2009

Geometry &Amp; Topology Monographs

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I discuss the problem of realizing families of complex orientable homology theories as families of E ∞ -ring spectra, including a recent result of Jacob Lurie emphasizing the role of p-divisible groups. 1 55N22; 55N34, 14H10 A few years ago, I wrote a paper [10] discussing a realization problem for families of Landweber exact spectra. Since Jacob Lurie [27] now has a major positive result in this direction, it seems worthwhile to revisit these ideas.In brief, the realization problem can be stated as follows. Suppose we are given a flat morphism g : Spec(R) −→ M fg from an affine scheme to the moduli stack of smooth 1-dimensional formal groups.Then we get a two-periodic homology theory E(R, G) with E(R, G) 0 ∼ = R and associated formal groupisomorphic to the formal group classified by g. The higher homotopy groups of E(R, G) are zero in odd degrees andwhere ω G is the module of invariant differentials for G. The module ω G is locally free of rank 1 over R, and free of rank 1 if G has a coordinate. In this case ) 2 is a generator. The fact that g was flat implies E(R, G) is Landweber exact, even if G doesn't have a coordinate. Now suppose we are given a flat morphism of stacks g : X −→ M fg .

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On the moduli stack of commutative, 1-parameter formal groups

Cited by 7 publications

References 35 publications

Algebraic chromatic homotopy theory for BP∗BP-comodules

Algebraic chromatic homotopy theory for BP∗BP-comodules

Algebraic chromatic homotopy theory for $BP_*BP$-comodules

Realizing families of Landweber exact homology theories

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