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

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.

Further reading

• Mathew, A.; Meier, L. (2015). "Affineness and chromatic homotopy theory". Journal of Topology. 8 (2): 476–528. arXiv:1311.0514. doi:10.1112/jtopol/jtv005.