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

Andrew Salch

Algebraic & Geometric Topology 18 (2018) 797–826

We develop methods for computing the restriction map from the cohomology of the automorphism group of a height dn formal group law (ie the height dn 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(KabKnr) on the Lubin–Tate/Morava E–theory spectrum E2 for each quadratic extension Kp. 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.

formal groups, class field theory, stable homotopy groups, Lubin–Tate theory, formal modules, formal groups with complex multiplication, Morava stabilizer groups
Mathematical Subject Classification 2010
Primary: 11S31, 14L05, 55N22, 55P42, 55Q10
Received: 14 July 2016
Revised: 25 July 2017
Accepted: 26 September 2017
Published: 12 March 2018
Andrew Salch
Department of Mathematics
Wayne State University
Detroit, MI
United States