Moduli stack of formal group laws explained

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

l{M} _{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

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

References

• Web site: J. . Lurie . Chromatic Homotopy Theory . 252x (35 lectures) . 2010 . Harvard University.
• Book: Goerss, P.G. . Realizing families of Landweber exact homology theories . http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf . New topological contexts for Galois theory and algebraic geometry (BIRS 2008) . Geometry & Topology Monographs . 16 . 2009 . 49–78 . 10.2140/gtm.2009.16.49 . 0905.1319.

Further reading

• Mathew . A. . Meier . L. . Affineness and chromatic homotopy theory . Journal of Topology . 8 . 2 . 476–528 . 2015 . 10.1112/jtopol/jtv005 . 1311.0514.