Morava stabilizer group in nLab

Contents

Idea

The moduli stack of formal groups ℳ FG\mathcal{M}_{FG} (1-dimensional commutative formal groups) admits a natural stratification whose open strata are labeled by a natural number called the height of formal groups.

For pp a prime number, write 𝔽¯ p\overline{\mathbb{F}}_{\mathrm{p}} for the algebraic closure of the prime field 𝔽 p\mathbb{F}_p.

The stratum ℳ FG n\mathcal{M}_{FG}^n can be identified with the homotopy quotient Spec(𝔽¯ p)//𝔾Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}, where the group 𝔾\mathbb{G} is the automorphism group over 𝔽 p\mathbb{F}_p of the unique formal group law ff of height nn,

𝔾≔Aut(𝔽¯ p,f) \mathbb{G} \coloneqq Aut(\overline{\mathbb{F}}_{\mathrm{p}}, f)

This is called the Morava stabilizer group. Essentially its group algebra (Hopf algebra) is called the Morava stabilizer algebra.

This is discussed around (Lurie 10, lect. 19, prop. 1), see also the beginning of Lurie 10, lect 21, and in (Ravenel, chapt. 6).

The deformation theory around these strata is Lubin-Tate theory.

References

• Jacob Lurie, Morava Stabilizer Groups (pdf), lecture 19 in Chromatic Homotopy Theory, Lecture series 2010 (web)

• Jacob Lurie, Lubin-Tate Theory (pdf), lecture 21 in Chromatic Homotopy Theory, Lecture series 2010 (web)

• Doug Ravenel, Complex cobordism and stable homotopy groups of spheres, Chapter 6 Morava stabilizer algebra (pdf)

• Niko Naumann, Torsors under smooth group-schemes and Morava stabilizer groups (web)