Formal group

Definition

Let $A$ be a commutative ring. A formal group in one parameter is a formal power series $F\in A[[X,Y]]$ such that

1. $F(X,0)=F(0,X)=X$
2. $F(X,Y)=F(Y,X)$
3. $F(F(X,Y),Z)=F(X,F(Y,Z))$ in $A[[X,Y,Z]]$
4. There is a series $\sigma \in A[[X]]$ such that $F(X,\sigma (X))=0$

Examples

1. The additive formal group: $F(X,Y)=X+Y$
2. The multiplicative formal group: $F(X,Y)=(X+1)(Y+1)-1$. In this case, $\sigma ={\frac {1}{X+1}}-1=\sum _{k=1}^{\infty }(-1)^{k}X^{k}$.