Contents

cohomology

# Contents

## 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

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

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

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

## 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.

### 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.

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

coveringby 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)
of1d toriTate curveselliptic curvescubic curves1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$KU$KU[ [q] ]$elliptic spectrumcomplex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf(KO $\hookrightarrow$ KU) = KR-theoryTate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)tmf$\mathbb{S}$

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