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

References

Further reading

  • Mathew, A.; Meier, L. (2015). "Affineness and chromatic homotopy theory". Journal of Topology 8 (2): 476–528. doi:10.1112/jtopol/jtv005.