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 $\mathcal{M}^n_{\text{FG}}$ be given so that $\mathcal{M}^n_{\text{FG}}(R)$ consists of formal group laws over R of height exactly n. They form a stratification of the moduli stack $\mathcal{M}_{\text{FG}}$ . $\operatorname{Spec} \overline{\mathbb{F}_p} \to \mathcal{M}^n_{\text{FG}}$ is faithfully flat. In fact, $\mathcal{M}^n_{\text{FG}}$ 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 $\mathcal{M}^n_{\text{FG}}$ fit together.

References

{{cite web |first=J. |last=Lurie |url=http://www.math.harvard.edu/~lurie/252x.html |title=Chromatic Homotopy Theory |work=252x (35 lectures) |date=2010 |publisher=Harvard University}}
{{cite book |first=P.G. |last=Goerss |chapter=Realizing families of Landweber exact homology theories |chapter-url=http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf |title=New topological contexts for Galois theory and algebraic geometry (BIRS 2008) |publisher= |series=Geometry & Topology Monographs |volume=16 |date=2009 |isbn= |pages=49–78 |doi=10.2140/gtm.2009.16.49 |arxiv=0905.1319}}

moduli stack of formal group laws

References

Further reading