The Action of the Morava Stabilizer Group on the Lubin-Tate Moduli Space of Lifts

Devinatz, Ethan S.; Hopkins, Michael

doi:10.2307/2375086

Cited by 64 publications

(53 citation statements)

References 10 publications

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“…We note that -as in [4] -there is a choice of the universal deformation G of Γ such that the p-series of the associated graded formal group law G 0 over…”

Section: The K(n)-local Category and The Lubin-tate Theories E Nmentioning

confidence: 99%

A resolution of the K(2)-local sphere at the prime 3

et al. 2005

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We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L K(2) S 0 as the inverse limit of a tower of fibrations with four layers. The successive fibers are of the form E hF 2 where F is a finite subgroup of the Morava stabilizer group and E 2 is the second Morava or Lubin-Tate homology theory. We give explicit calculation of the homotopy groups of these fibers. The case n = 2 at p = 3 represents the edge of our current knowledge: n = 1 is classical and at n = 2, the prime 3 is the largest prime where the Morava stabilizer group has a p-torsion subgroup, so that the homotopy theory is not entirely algebraic.The problem of understanding the homotopy groups of spheres has been central to algebraic topology ever since the field emerged as a distinct area of mathematics. A period of calculation beginning with Serre's computation of the cohomology of Eilenberg-MacLane spaces and the advent of the Adams spectral sequence culminated, in the late 1970s, with the work of Miller, Ravenel, and Wilson on periodic phenomena in the homotopy groups of spheres and Ravenel's nilpotence conjectures. The solutions to most of these conjectures by Devinatz, Hopkins, and Smith in the middle 1980s established the primacy of the "chromatic" point of view and there followed a period in which the community absorbed these results and extended the qualitative picture of stable homotopy theory. Computations passed from center stage, to some extent, although there has been steady work in the wings

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“…We note that -as in [4] -there is a choice of the universal deformation G of Γ such that the p-series of the associated graded formal group law G 0 over…”

Section: The K(n)-local Category and The Lubin-tate Theories E Nmentioning

confidence: 99%

A resolution of the K(2)-local sphere at the prime 3

et al. 2005

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“…Part (a) is immediate from the definition of the action. Part (b) follows easily from the formulas of [6]; however, a more elementary proof is possible. First observe that if F is any formal group law over a p-adically complete ring, there is a unique continuous ring homomorphism Z p → End(F ) extending the homomorphism Z → End(F ) mapping n to [n] F (x).…”

Section: Given Constants a And B G Then Contains An Open Subnormal Smentioning

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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“…The profinite group G n acts on the ring E n * (cf. [7]). By Brown representability, this induces an action of G n by maps of rings in the stable homotopy category.…”

Section: Introductionmentioning

confidence: 99%

Continuous homotopy fixed points for Lubin-Tate spectra

Quick

2013

Homology, Homotopy and Applications

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We provide a new and conceptually simplified construction of continuous homotopy fixed point spectra for Lubin-Tate spectra under the action of the extended Morava stabilizer group. Moreover, our new construction of a homotopy fixed point spectral sequence converging to the homotopy groups of the homotopy fixed points of Lubin-Tate spectra is isomorphic to an Adams spectral sequence converging to the homotopy groups of the spectra constructed by Devinatz and Hopkins. The new idea is built on the theory of profinite spectra with a continuous action by a profinite group.

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The Action of the Morava Stabilizer Group on the Lubin-Tate Moduli Space of Lifts

Cited by 64 publications

References 10 publications

A resolution of the K(2)-local sphere at the prime 3

A resolution of the K(2)-local sphere at the prime 3

Towards the finiteness of π*LK(n)S0

Continuous homotopy fixed points for Lubin-Tate spectra

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