nLab height of a formal group



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The moduli stack of formal groups FG\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 RR be a commutative ring and fix

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

a formal group law over RR.


For every nn \in \mathbb{N} the nn-series of ff

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

is defined recursively by

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

  2. if n>0n \gt 0 then [n](t)=f([n1](t),t)[n](t) = f([n-1](t),t).

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


Write v nv_n for the coefficient of t p nt^{p^n} in the pp-series [p][p] of ff.


Say that ff

  • has height n\geq n if v i=0v_i = 0 for i<ni \lt n;

  • has height exactly nn if it has height n\geq n and v nRv_n \in R is invertible.

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



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

[n](t)=(1+t) n1. [n](t) = (1+t)^n - 1 \,.

If p=0p = 0 in RR then

[p](t)=(1+t) p1=t p [p](t) = (1+t)^p - 1 = t^p

and thus ff 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.


Chromatic height filtation

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

tower diagram/filteringspectral sequence of a filtered stable homotopy type
filtered chain complexspectral sequence of a filtered complex
Postnikov towerAtiyah-Hirzebruch spectral sequence
chromatic towerchromatic spectral sequence
skeleta of simplicial objectspectral sequence of a simplicial stable homotopy type
skeleta of Sweedler coring of E-∞ algebraAdams spectral sequence
filtration by support
slice filtrationslice spectral sequence


Last revised on December 11, 2020 at 08:56:21. See the history of this page for a list of all contributions to it.