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Formal group

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Definition ^[?]

Definition

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

$F (X,0) = F (0, X) = X$
$F (X, Y) = F (Y, X)$
$F (F (X, Y), Z) = F (X, F (Y, Z))$ in $A [[X, Y, Z]]$
There is a series $\sigma\in A[[X]]$ such that $F (X,σ(X)) = 0$

Examples

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