User:TakuyaMurata/Moduli stack of formal group laws

In algebraic geometry, the moduli stack of formal group laws classifies formal group laws and isomorphisms between them. It is denoted by ${\mathcal {M}}_{\text{FG}}$. It is a "geometric “object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology. This moduli stack is closely related to the moduli stack of elliptic curves.Template:Vague

In characteristic zero, this moduli stack is very easy to understand: the quotient stack ${\mathcal {M}}_{\text{FG}}\times _{\operatorname {Spec} \mathbb {Z} }\operatorname {Spec} \mathbb {Q}$ is isomorphic to $B{\mathfrak {G}}_{m}$, the classifying stack. This can be thought of as a manifestation of the "thesis" of differential graded Lie algebras.

Currently, it is not known whether ${\mathcal {M}}_{\text{FG}}$ is a derived stack or not. Hence, it is typical to work with stratifications. Let ${\mathcal {M}}_{\text{FG}}^{n}$ be given so that ${\mathcal {M}}_{\text{FG}}^{n}(R)$ consists of formal group laws over R of height exactly n. They form a stratification of the moduli stack ${\mathcal {M}}_{\text{FG}}$. $\operatorname {Spec} {\overline {\mathbb {F} _{p}}}\to {\mathcal {M}}_{\text{FG}}^{n}$ is faithfully flat. In fact, ${\mathcal {M}}_{\text{FG}}^{n}$ is of the form $\operatorname {Spec} {\overline {\mathbb {F} _{p}}}/\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)$ where $\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)$ is a profinite group called the Morava stabilizer group. The Lubin–Tate theory describes how the strata ${\mathcal {M}}_{\text{FG}}^{n}$ fit together.

Moduli stack of p-divisible groups

See also p-divisible groups.

References

• J. Lurie, Chromatic Homotopy Theory (252x)
• P. Goerss, Realizing families of Landweber exact homology theories

Further reading

• http://arxiv.org/pdf/1311.0514.pdf