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

References

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Further reading

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