nLab height of a formal group

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

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

Related pages

• moduli stack of formal groups

• chromatic homotopy theory

• Morava K-theory

References

• Jacob Lurie, Chromatic Homotopy Theory, Lecture series (2010) (lecture notes), Lecture 12 Heights and formal groups (pdf)