nLab moduli stack of formal groups

Idea

The moduli stack $\mathcal{M}_{FG}$ of all formal groups. Often meant are 1-dimensional commutative formal groups-

Definition

Let $L = \pi_\bullet MU$ be the Lazard ring.

Write $G^+$ for the group scheme given on a ring $R$ by

$$G^+(R) \;\coloneqq\; \big\{ g\in R[ [x] ] \;\big\vert\; g(t) = b_1 t + b_2 t^2 + \cdots \;\text{with}\; b_1 \in R^\times \big\} \,.$$

There is a canonical action of $G^+$ on $Spec(L)$. The quotient stack of this action is the moduli stack of (1d commutative) formal groups

$$\mathcal{M}_{fg} \;=\; \big(Spec(L)\big)\sslash G^+ \,.$$

(e.g. Lurie, lecture 11, def. 2)

Properties

Chromatic height stratification

The moduli stack of formal groups $\mathcal{M}_{FG}$ admits a natural stratification whose open strata are labeled by a natural number called the height of formal groups.

The complex oriented cohomology theories associated to these formal groups by the Landweber exact functor theorem accordingly also inherit such an integer label, called chromatic filtration. Studying this is the topic of chromatic homotopy theory.

tower diagram/filtering	spectral sequence of a filtered stable homotopy type
filtered chain complex	spectral sequence of a filtered complex
Postnikov tower	Atiyah-Hirzebruch spectral sequence
chromatic tower	chromatic spectral sequence
skeleta of simplicial object	spectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebra	Adams spectral sequence
filtration by support	…
slice filtration	slice spectral sequence

Morava stabilizer group

Write $\overline{\mathbb{F}_{\mathrm{p}}}$ for the algebraic closure of $\mathbb{F}_p$.

The stratum $\mathcal{M}_{FG}^n$ can be identified with the homotopy quotient $Spec (\overline{\mathbb{F}}_{\mathrm{p}})// \mathbb{G}$, where the group $\mathbb{G}$ is the Morava stabilizer group.

This is (Lurie 10, lect. 19, prop. 1) See also the beginning of Lurie 10, lect 21.

Deformation theory

The deformation theory around these strata is Lubin-Tate theory.

Relation to moduli of elliptic curves and tori

Inside the moduli stack of formal groups sit, in that order, that of cubic curves, the moduli stack of elliptic curves, the moduli stack of tori.

Related concepts

• Spec(S)

moduli spaces

• moduli space of complex structures

• moduli space of conformal structures

• moduli space of curves

• moduli space of Riemannian metrics

• moduli stack of elliptic curves

• moduli space of Calabi-Yau spaces

• moduli stack of tori

• moduli stack of formal groups

• moduli space of connections, moduli space of flat connections

• Hitchin moduli space

• moduli space of monopoles

• moduli space of framed manifolds

• moduli space of bundles

• moduli space of (higher) line bundles

• quiver variety

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering					by of level-n structures (modular curve)
	$\ast = Spec(\mathbb{Z})$	$\to$	$Spec(\mathbb{Z}[ [q] ])$	$\to$	$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering	$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack	$\mathcal{M}_{1dTori}$	$\hookrightarrow$	$\mathcal{M}_{Tate}$	$\hookrightarrow$	$\mathcal{M}_{\overline{ell}}$ (M_ell)	$\hookrightarrow$	$\mathcal{M}_{cub}$	$\to$	$\mathcal{M}_{fg}$ (M_fg)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)		(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)		tmf		$\mathbb{S}$

References

• Niko Naumann, Comodule categories and the geometry of the stack of formal groups, Advances in Mathematics 215 (2007) pp 569-600, doi:10.1016/j.aim.2007.04.007, arXiv:math/0503308.

• Brian D. Smithling, On the moduli stack of commutative, 1-parameter formal Lie groups (arXiv:0708.3326)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 11 Formal groups (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 14 Classification of formal groups (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 19 Morava stabilizer groups (pdf)

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series 2010, Lecture 21 Lubin-Tate theory (pdf)

• Jeroen van der Meer, A Stack-Theoretic Perspective on $\mathrm{KU}$-Local Stable Homotopy Theory, Chapter 3 The Moduli Stack of Formal Groups. 2019 (pdf)

On quasicoherent sheaves over $\mathcal{M}_{fg}$:

• Paul Goerss, Realizing Families of Landweber Exact Homology Theories (arXiv:0905.1319)

• Paul Goerss, Quasi-coherent sheaves on the moduli stack of formal groups (arXiv:0802.0996)