∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
$\infty$-Lie groupoids
$\infty$-Lie groups
$\infty$-Lie algebroids
$\infty$-Lie algebras
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.
Let $R$ be a commutative ring and fix
a formal group law over $R$.
For every $n \in \mathbb{N}$ the $n$-series of $f$
is defined recursively by
if $n = 0$ then $[n](t) = 0$;
if $n \gt 0$ then $[n](t) = f([n-1](t),t)$.
Now fix $p \in \mathbb{N}$ a prime number,
Write $v_n$ for the coefficient of $t^{p^n}$ in the $p$-series $[p]$ of $f$.
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).
For $f(x,y) = x + y + x y$ the formal multiplicative group the $n$-series is
If $p = 0$ in $R$ then
and thus $f$ has height exactly 1.
For instance (Lurie 10, lecture 12, example 16).
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.
The height of formal groups induces the height filtration on the moduli stack of formal groups.
Last revised on December 11, 2020 at 08:56:21. See the history of this page for a list of all contributions to it.