Moduli stack of formal group laws

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

• Lurie, J. (2010). "Chromatic Homotopy Theory". 252x (35 lectures). Harvard University. http://www.math.harvard.edu/~lurie/252x.html. 
• Goerss, P.G. (2009). "Realizing families of Landweber exact homology theories". New topological contexts for Galois theory and algebraic geometry (BIRS 2008). Geometry & Topology Monographs. 16. pp. 49–78. doi:10.2140/gtm.2009.16.49. http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf. 

Further reading

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