Contents

# Contents

## Idea

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.

## Definition

Let $R$ be a commutative ring and fix

$f(x,y) \in R [ [ x,y ] ]$

a formal group law over $R$.

###### Definition

For every $n \in \mathbb{N}$ the $n$-series of $f$

$[n](t) \in R[ [ t ] ]$

is defined recursively by

1. if $n = 0$ then $[n](t) = 0$;

2. if $n \gt 0$ then $[n](t) = f([n-1](t),t)$.

Now fix $p \in \mathbb{N}$ a prime number,

###### Definition

Write $v_n$ for the coefficient of $t^{p^n}$ in the $p$-series $[p]$ of $f$.

###### Definition

Say that $f$

• has height $\geq n$ if $v_i = 0$ for $i \lt n$;

• has height exactly $n$ if it has height $\geq n$ and $v_n \in R$ is invertible.

For instance (Lurie 10, lecture 12, def. 13).

## Examples

###### Example

For $f(x,y) = x + y + x y$ the formal multiplicative group the $n$-series is

$[n](t) = (1+t)^n - 1 \,.$

If $p = 0$ in $R$ then

$[p](t) = (1+t)^p - 1 = t^p$

and thus $f$ has height exactly 1.

For instance (Lurie 10, lecture 12, example 16).

###### Example

An elliptic curve over a field of positive characteristic whose formal group law has height of a formal group equal to 2 is called a supersingular elliptic curve. Otherwise the height equals 1 and the elliptic curve is called ordinary.

## Properties

### Chromatic height filtation

The hight of formal groups induces the height filtration on the moduli stack of formal groups.