One-dimensional commutative formal group law

Definition

Let $R$ be a commutative ring. A one-dimensional commutative formal group law over $R$ is a series $F(X,Y)\in R[[X,Y]]$ satisfying

F(X,0)=X,\qquad F(0,Y)=Y,

F(F(X,Y),Z)=F(X,F(Y,Z)),\qquad F(X,Y)=F(Y,X).

There is then a unique inverse series $i(X)\in XR[[X]]$ such that $F(X,i(X))=0$ .

Convention and scope

In algebraic topology and much of arithmetic geometry, “one-dimensional formal group law” conventionally includes commutativity. This knowl follows that convention. The general $n$ -dimensional formal group law used elsewhere in this collection does not impose commutativity, even when $n=1$ .

Dimension counts formal parameters, not the number of variables appearing in the multiplication: a one-dimensional law uses the two inputs $X$ and $Y$ , but its underlying formal disc has one coordinate.

The

m

-series

For $m\in\mathbb Z$ , define $[m]_F(X)$ by repeated formal addition, using the inverse for negative $m$ . The series $[p]_F$ in characteristic $p$ determines the height of $F$ , a central invariant that helps explain why tangent Lie algebras do not classify formal groups in positive characteristic.

Characteristic zero

Over a $\mathbb Q$ -algebra, every such law has a unique strict logarithm to the additive law. This does not trivialize the integral or positive-characteristic theory: denominators in the logarithm generally prevent descent to the original coefficient ring.

References

Michiel Hazewinkel, Formal Groups and Applications, AMS Chelsea Publishing, 2012. AMS book record. Relevant: Chapters 1 and 3, one-dimensional commutative laws and $p$ -typical theory.
J. F. Adams, Stable Homotopy and Generalised Homology, University of Chicago Press, 1974. Relevant: Part II, formal groups in complex-oriented cohomology.