Geometric criteria for Landweber exactness

Hollander, Sharon

doi:10.1112/plms/pdp014

Cited by 6 publications

(8 citation statements)

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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%

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%

Algebraic chromatic homotopy theory for BP∗BP-comodules

Barthel

Heard

2018

Proc. London Math. Soc.

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“…For a comprehensive account of the stack of formal groups and its relation to stable homotopy theory, we refer to Goerss's forthcoming book [9]. Hollander has also done some recent work of note: in [15] she gives a simple proof of the Landweber exact functor theorem [21] in terms of the geometry of M , and in [16] she uses this stack to give a proof of the Miller-Ravenel-Morava change of rings theorem and another proof of the algebraic chromatic convergence theorem.…”

Section: Theorem (448) B N Is Smooth Overmentioning

confidence: 99%

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

Smithling

2011

Journal of Pure and Applied Algebra

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a b s t r a c tWe commence a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to various aspects of the height stratification relative to fixed prime p on the stacks of buds and formal groups.

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“…This note has several loosely connected aims, all modest and none very novel. The theory of formal groups and their occurence in topology has been expressed many times in quite sophisticated terms ( [4,5,8,10,11] are some examples). We hope here to achieve the level of generality and naturality one finds in those sources, without requiring as much infrastructure.…”

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confidence: 99%