Comodules and Landweber exact homology theories

Hovey, Mark; Strickland, Neil P.

doi:10.1016/j.aim.2004.04.011

Cited by 39 publications

(80 citation statements)

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“…Hopkins has covered a considerable amount of the theory in [17] and in other courses at MIT, and Pribble's thesis [29] has also covered some of the basic theory, including some aspects of the height stratification and an algebraic analog of the chromatic convergence theorem of Hopkins-Ravenel [32,Section 8.6]. Naumann [28] has given the first published account of some of the basic moduli theory and has used it to prove generalizations of results of Hovey [18] and Hovey and Strickland [19]. Our paper takes another step towards filling the gap in the literature, but we don't go so far as to study the important topics of quasi-coherent sheaves on M or its deformation theory: these are where the essential connections to topology are found.…”

Section: Theorem (448) B N Is Smooth Overmentioning

confidence: 98%

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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Section: Theorem (448) B N Is Smooth Overmentioning

confidence: 98%

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

Smithling

2011

Journal of Pure and Applied Algebra

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“…be a Landweber filtration of M [11,Theorem D]; that is, each M i is an E n * E n -comodule and M i+1 /M i is isomorphic to either a suspension of F p n Ju n−1 K[u, u −1 ] or a suspension of F p n [u, u −1 ]. In the first case, H s (M i+1 /M i ) is of essentially finite rank by Theorem 2 together with Proposition 1.6, and in the second case, H s (M i+1 /M i ) is a module of finite type annihilated by ν and therefore of essentially finite rank.…”

Section: (B/t B )/(A/t a )mentioning

confidence: 99%

Towards the finiteness of π*LK(n)S0

Devinatz

2008

Advances in Mathematics

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Let G be a closed subgroup of the nth Morava stabilizer group S n , n 2, and let E hG n denote the continuous homotopy fixed point spectrum of Devinatz and Hopkins. If G = z , the subgroup topologically generated by an element z in the p-Sylow subgroup S 0 n of S n , and z is non-torsion in the quotient of S 0 n by its center, we prove that the E h z n -homology of any K(n − 2) * -acyclic finite spectrum annihilated by p is of essentially finite rank. We also show that the units in E n * fixed by z are just the units in the Witt vectors with coefficients in the field of p n elements. If n = 2 and p 5, we show that, if G is a closed subgroup of S 0 n not contained in the center, then G contains an open subnormal subgroup U such that the mod(p) homotopy of E h(U ×F × p ) n is of essentially finite rank.

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“…To relate this to E(n) * E(n)-comodules, we recall from [9] and [10] the exact functor * from BP * BP-comodules to E(n) * E(n)-comodules defined by * M = E(n) * ⊗ BP * M. The functor * has a fully faithful right adjoint * , the composite * * is naturally isomorphic to the identity, and the composite L n = * * is the localization functor on the category of BP * BP-comodules with respect to the hereditary torsion theory of v n -torsion comodules. The functor L n is left exact, but has right derived functors L q n for 0 ≤ q ≤ n, studied in [10].…”

Section: Relatively Injective Resolutions and An Applicationmentioning

confidence: 99%

“…B P * BP-comodules and E(n) * E(n)-comodules. In this section, we exploit the close relationship between BP * BP-comodules and E(n) * E(n)-comodules studied in [9] to get some partial understanding of the product of comodules.…”

Section: Relatively Injective Resolutions and An Applicationmentioning

confidence: 99%

The Generalized Homology of Products

Hovey¹

2007

Glasgow Math. J.

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Abstract. We construct a spectral sequence that computes the generalized homology E * ( X α ) of a product of spectra. The E 2 -term of this spectral sequence consists of the right derived functors of product in the category of E * E-comodules, and the spectral sequence always converges when E is the Johnson-Wilson theory E(n) and the X α are L n -local. We are able to prove some results about the E 2 -term of this spectral sequence; in particular, we show that the E(n)-homology of a product of E(n)-module spectra X α is just the comodule product of the E(n) * X α . This spectral sequence is relevant to the chromatic splitting conjecture.2000 Mathematics Subject Classification. 55T25, 55N22, 55P60, 18G10, 16W30.Introduction. The basic tools of computation in algebraic topology are homology theories. Homology theories preserve coproducts, but can behave very badly on products. There are examples of homology theories E and sets of spectra (generalized spaces) {X α }, for which E * X α = 0 for all i and yet E * ( α X α ) = 0. Indeed, we can take E = H‫,ޑ‬ rational homology, where we have (H‫)ޑ‬ * (H‫/ޚ‬p k ) = 0 for all k, but

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Comodules and Landweber exact homology theories

Cited by 39 publications

References 20 publications

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

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

Towards the finiteness of π*LK(n)S0

The Generalized Homology of Products

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