Algebraic & Geometric Topology

Moduli of formal $A$–modules under change of $A$

Andrew Salch

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We develop methods for computing the restriction map from the cohomology of the automorphism group of a height d n formal group law (ie the height d n Morava stabilizer group) to the cohomology of the automorphism group of an A –height n formal A –module, where A is the ring of integers in a degree d field extension of  p . We then compute this map for the quadratic extensions of p and the height  2 Morava stabilizer group at primes p > 3 . We show that the these automorphism groups of formal modules are closed subgroups of the Morava stabilizer groups, and we use local class field theory to identify the automorphism group of an A –height 1 –formal A –module with the ramified part of the abelianization of the absolute Galois group of K , yielding an action of Gal ( K ab K nr ) on the Lubin–Tate/Morava E –theory spectrum E 2 for each quadratic extension K p . Finally, we run the associated descent spectral sequence to compute the V ( 1 ) –homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the K ( 2 ) –local homotopy groups of V ( 1 ) , either that element or an appropriate dual of it is detected in the Galois cohomology of the abelian closure of some quadratic extension of p .

Article information

Algebr. Geom. Topol., Volume 18, Number 2 (2018), 797-826.

Received: 14 July 2016
Revised: 25 July 2017
Accepted: 26 September 2017
First available in Project Euclid: 22 March 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11S31: Class field theory; $p$-adic formal groups [See also 14L05] 14L05: Formal groups, $p$-divisible groups [See also 55N22] 55N22: Bordism and cobordism theories, formal group laws [See also 14L05, 19L41, 57R75, 57R77, 57R85, 57R90] 55P42: Stable homotopy theory, spectra 55Q10: Stable homotopy groups

formal groups class field theory stable homotopy groups Lubin–Tate theory formal modules formal groups with complex multiplication Morava stabilizer groups


Salch, Andrew. Moduli of formal $A$–modules under change of $A$. Algebr. Geom. Topol. 18 (2018), no. 2, 797--826. doi:10.2140/agt.2018.18.797.

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