Height of a one-dimensional formal group law

Definition

Let $F$ be a one-dimensional commutative formal group law over a field $k$ of characteristic $p>0$ . Its height is defined from the $p$ -series $[p]_F(X)$ . If $[p]_F(X)=0$ , then

\operatorname{ht}(F)=\infty.

Otherwise there is a unique integer $h\geq1$ such that

[p]_F(X)=aX^{p^h}+\text{terms of higher degree}, \qquad a\in k^\times,

and $\operatorname{ht}(F)=h$ .

Invariance

A formal-group-law isomorphism $f:F\to G$ intertwines the $p$ -series:

f([p]_F(X))=[p]_G(f(X)).

Because $f$ has an invertible linear coefficient, the first nonzero exponent is unchanged. Height is therefore an isomorphism invariant and is preserved by field extension.

Standard examples

For the additive and multiplicative laws,

[p]_{F_a}(X)=0, \qquad [p]_{F_m}(X)=X^p.

Thus $F_a$ has height $\infty$ and $F_m$ has height $1$ . Their tangent Lie algebras are nevertheless isomorphic, giving the basic failure of tangent classification in positive characteristic.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 3–5.
J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 1974. Relevant: Part II.