User:TakuyaMurata/Moduli stack of formal group laws

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In algebraic geometry, the moduli stack of formal group laws classifies formal group laws and isomorphisms between them. It is denoted by ${\displaystyle {\mathcal {M}}_{\text{FG}}}$. 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 ${\displaystyle {\mathcal {M}}_{\text{FG}}\times _{\operatorname {Spec} \mathbb {Z} }\operatorname {Spec} \mathbb {Q} }$ is isomorphic to ${\displaystyle B{\mathfrak {G}}_{m}}$, 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 ${\displaystyle {\mathcal {M}}_{\text{FG}}}$ is a derived stack or not. Hence, it is typical to work with stratifications. Let ${\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}$ be given so that ${\displaystyle {\mathcal {M}}_{\text{FG}}^{n}(R)}$ consists of formal group laws over R of height exactly n. They form a stratification of the moduli stack ${\displaystyle {\mathcal {M}}_{\text{FG}}}$. ${\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}\to {\mathcal {M}}_{\text{FG}}^{n}}$ is faithfully flat. In fact, ${\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}$ is of the form ${\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}/\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)}$ where ${\displaystyle \operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)}$ is a profinite group called the Morava stabilizer group. The Lubin–Tate theory describes how the strata ${\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}$ fit together.