Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 18, Number 2 (2018), 797-826.
Moduli of formal $A$–modules under change of $A$
We develop methods for computing the restriction map from the cohomology of the automorphism group of a height formal group law (ie the height Morava stabilizer group) to the cohomology of the automorphism group of an –height formal –module, where is the ring of integers in a degree field extension of . We then compute this map for the quadratic extensions of and the height Morava stabilizer group at primes . 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 –height –formal –module with the ramified part of the abelianization of the absolute Galois group of , yielding an action of on the Lubin–Tate/Morava –theory spectrum for each quadratic extension . Finally, we run the associated descent spectral sequence to compute the –homotopy groups of the homotopy fixed-points of this action; one consequence is that, for each element in the –local homotopy groups of , either that element or an appropriate dual of it is detected in the Galois cohomology of the abelian closure of some quadratic extension of .
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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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
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. https://projecteuclid.org/euclid.agt/1521684021