Algebraic & Geometric Topology

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

Andrew Salch

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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1521684021

Digital Object Identifier
doi:10.2140/agt.2018.18.797

Mathematical Reviews number (MathSciNet)
MR3773739

Zentralblatt MATH identifier
06859605

Subjects
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

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

Citation

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


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References

  • M Behrens, D G Davis, The homotopy fixed point spectra of profinite Galois extensions, Trans. Amer. Math. Soc. 362 (2010) 4983–5042
  • J W S Cassels, A Fröhlich (editors), Algebraic number theory, 2nd edition, London Mathematical Society (2010)
  • J W Cogdell, On Artin $L$–functions, preprint (2014) Available at \setbox0\makeatletter\@url https://people.math.osu.edu/cogdell.1/artin-www.pdf {\unhbox0
  • E S Devinatz, M J Hopkins, The action of the Morava stabilizer group on the Lubin–Tate moduli space of lifts, Amer. J. Math. 117 (1995) 669–710
  • E S Devinatz, M J Hopkins, Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups, Topology 43 (2004) 1–47
  • P Duren, U C Merzbach (editors), A century of mathematics in America, II, History of Mathematics 2, Amer. Math. Soc., Providence, RI (1989)
  • P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, from “Structured ring spectra” (A Baker, B Richter, editors), London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151–200
  • M Harris, R Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton Univ. Press (2001)
  • M Hazewinkel, Formal groups and applications, AMS Chelsea, Providence, RI (2012)
  • B Klopsch, N Nikolov, C Voll, Lectures on profinite topics in group theory, London Mathematical Society Student Texts 77, Cambridge Univ. Press (2011)
  • M Lazard, Groupes analytiques $p$–adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965) 389–603
  • M Lazard, Commutative formal groups, Lecture Notes in Mathematics 443, Springer (1975)
  • J I Manin, Theory of commutative formal groups over fields of finite characteristic, Uspehi Mat. Nauk 18 (1963) 3–90 In Russian; translated in Russian Math. Surveys 18 (1963) 1–83
  • H R Miller, D C Ravenel, W S Wilson, Periodic phenomena in the Adams–Novikov spectral sequence, Ann. of Math. 106 (1977) 469–516
  • J Neukirch, Algebraic number theory, Grundl. Math. Wissen. 322, Springer (1999)
  • N Nikolov, D Segal, On finitely generated profinite groups, I: Strong completeness and uniform bounds, Ann. of Math. 165 (2007) 171–238
  • M Rapoport, T Zink, Period spaces for $p$–divisible groups, Annals of Mathematics Studies 141, Princeton Univ. Press (1996)
  • D C Ravenel, Formal $A$–modules and the Adams–Novikov spectral sequence, J. Pure Appl. Algebra 32 (1984) 327–345
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press, Orlando, FL (1986)
  • D C Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128, Princeton Univ. Press (1992)
  • A M Robert, A course in $p$–adic analysis, Graduate Texts in Mathematics 198, Springer (2000)
  • A Salch, Computation of the classifying ring of formal groups with complex multiplication, preprint (2015)
  • A Salch, The structure of the classifying ring of formal groups with complex multiplication, preprint (2015)
  • A Salch, Height four formal groups with quadratic complex multiplication, preprint (2016)
  • A Salch, Ravenel's algebraic extensions of the sphere spectrum do not exist, preprint (2016)
  • W C Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics 66, Springer (1979)