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 ${\mathcal {M}}_{\text{FG}}$ . 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 ${\mathcal {M}}_{\text{FG}}$ 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.

References

Lurie, J. (2010). "Chromatic Homotopy Theory". 252x (35 lectures). Harvard University.
Goerss, P.G. (2009). "Realizing families of Landweber exact homology theories" (PDF). New topological contexts for Galois theory and algebraic geometry (BIRS 2008). Geometry & Topology Monographs. Vol. 16. pp. 49–78. arXiv:0905.1319. doi:10.2140/gtm.2009.16.49.

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Further reading