Moduli stack of formal group laws

In algebraic geometry, the moduli stack of formal group laws is a <a href="/facts/Stack_(mathematics)/MkASN1rA">stack</a> classifying <a href="/facts/Formal_group/xUnVSCC5">formal group</a> laws and isomorphisms between them. It is denoted by 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}}
 
. It is a "geometric object" that underlies the <a href="/facts/Chromatic_homotopy_theory/gtcvY7LU">chromatic</a> approach to the <a href="/facts/Stable_homotopy_theory/wnedyf7K">stable homotopy theory</a>, a branch of <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.
Currently, it is not known whether 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}}
 
 is a <a href="/facts/Derived_stack/UzP9BsnQ">derived stack</a> or not. Hence, it is typical to work with stratifications. Let 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 n
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}
 
 be given so that 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 n
 
 
 (
 R
 )
 
 
 {\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 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}}
 
. 
 
 
 
 Spec
 ⁡
 
 
 
 
 F
 
 
 p
 
 
 ¯
 
 
 →
 
 
 
 M
 
 
 
 FG
 
 
 n
 
 
 
 
 {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}\to {\mathcal {M}}_{\text{FG}}^{n}}
 
 is <a href="/facts/Faithfully_flat_morphism/CLM7e8d6">faithfully flat</a>. In fact, 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 n
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}
 
 is of the form 
 
 
 
 Spec
 ⁡
 
 
 
 
 F
 
 
 p
 
 
 ¯
 
 
 
 /
 
 Aut
 ⁡
 (
 
 
 
 
 F
 
 
 p
 
 
 ¯
 
 
 ,
 f
 )
 
 
 {\displaystyle \operatorname {Spec} {\overline {\mathbb {F} _{p}}}/\operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)}
 
 where 
 
 
 
 Aut
 ⁡
 (
 
 
 
 
 F
 
 
 p
 
 
 ¯
 
 
 ,
 f
 )
 
 
 {\displaystyle \operatorname {Aut} ({\overline {\mathbb {F} _{p}}},f)}
 
 is a <a href="/facts/Profinite_group/0RX9Da9C">profinite group</a> called the Morava stabilizer group. The <a href="/facts/Lubin%E2%80%93Tate_theory/I4rCphUm">Lubin–Tate theory</a> describes how the strata 
 
 
 
 
 
 
 M
 
 
 
 FG
 
 
 n
 
 
 
 
 {\displaystyle {\mathcal {M}}_{\text{FG}}^{n}}
 
 fit together.

<ul><li>Lurie, J. (2010). <a href="http://www.math.harvard.edu/~lurie/252x.html">"Chromatic Homotopy Theory"</a>. 252x (35 lectures). Harvard University.</li>
<li>Goerss, P.G. (2009). <a href="http://www.math.northwestern.edu/~pgoerss/papers/banff.pdf">"Realizing families of Landweber exact homology theories"</a> (PDF). New topological contexts for Galois theory and algebraic geometry (BIRS 2008). Geometry & Topology Monographs. Vol. 16. pp. 49–78. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/0905.1319">0905.1319</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fgtm.2009.16.49">10.2140/gtm.2009.16.49</a>.</li></ul>

Moduli stack of formal group laws open-in-new

Moduli stack of formal group laws