nLab
Morava stabilizer group

Contents

Idea

The moduli stack of formal groups 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 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,

𝔾Aut(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

Last revised on February 5, 2016 at 16:16:35. See the history of this page for a list of all contributions to it.