Moduli stack of formal group laws

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In algebraic geometry, the moduli stack of formal group laws is a stack classifying formal group laws and isomorphisms between them. It is denoted by M FG {\displaystyle {\mathcal {M}}_{\text{FG}}} {\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.

Currently, it is not known whether M FG {\displaystyle {\mathcal {M}}_{\text{FG}}} {\displaystyle {\mathcal {M}}_{\text{FG}}} is a derived stack or not. Hence, it is typical to work with stratifications. Let M FG n {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} be given so that M FG n ( R ) {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}(R)} {\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 M FG {\displaystyle {\mathcal {M}}_{\text{FG}}} {\displaystyle {\mathcal {M}}_{\text{FG}}}. Spec ⁡ F p ¯ → M FG n {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}\to {\mathcal {M}}_{\text{FG}}^{n}} {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}\to {\mathcal {M}}_{\text{FG}}^{n}} is faithfully flat. In fact, M FG n {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} is of the form Spec ⁡ F p ¯ / Aut ⁡ ( F p ¯ , f ) {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}/\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)} {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}/\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)} where Aut ⁡ ( F p ¯ , f ) {\displaystyle \operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)} {\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 M FG n {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}} fit together.

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