Homomorphism of formal groups

Wei Yue asks the following question: In the definition of a homomorphism f(T) between two formal groups F and G, what is the reason we required f\in TR[[T]]? The short reason is that we want f to preserve the origin, that is, f(0)=0. A more complicated reason is that in general, the expression G(f(X), f(Y)) may not make sense for a power series G(X,Y) if f has a non-zero constant term. This is a point we have been somewhat cavalier about: When can we substitute one power series in another and get a well-defined power series as the answer? You should ponder this for yourself a bit, and then try to formulate some conditions precisely using the definition

R[[T]]= \lim_{\leftarrow} R[T]/(T^n)

of the ring of power series with coefficients in R.

Here is an exercise worth trying out: Suppose F(X,Y) is a polynomial that defines an algebraic group law on the field \mathbb{C} with 0 as the origin. Then F(X,Y)=X+Y is the only choice.
Hang on, what about

X+Y+XY?

Well, you have to formulate a bit carefully what is meant by an algebraic group law. Anyways, the conclusion is that deforming the usual group structure in any reasonable sense requires us to move out of the realm of polynomials.

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