Formal Groups 3


Today we move on to higher dimensional formal group laws over a ring {A}. Side note: later on we’ll care about the formal group attached to a Calabi-Yau variety in positive characteristic which is always one-dimensional, but to talk about Witt vectors we’ll need the higher dimensional ones.
An {n}-dimensional formal group law over {A} is just an {n}-tuple of power series, each of {2n} variables and no constant term, satisfying certain relations. We’ll write {F(X,Y)=(F_1(X,Y), F_2(X,Y), \ldots , F_n(X,Y))} where {X=(x_1, \ldots , x_n)} and {Y=(y_1, \ldots , y_n)} to simplify notation.

There are a few natural guesses for the conditions, but the ones we actually use are that {F_i(X,Y)=x_i+y_i+}(higher degree) and for all {i} {F_i(F(X,Y),Z)=F_i(X, F(Y,Z))}. We call the group law commutative if {F_i(X,Y)=F_i(Y,X)} for all {i}.

We still have our old examples that are fairly trivial {\widehat{\mathbb{G}}_a^n(X,Y)=X+Y}, meaning the i-th one is just {x_i+y_i}. For a slightly less trivial example, let’s explicitly write a four-dimensional one

{F_1=x_1+y_1+x_1y_1+x_2y_3}

{F_2=x_2+y_2+x_1y_2+x_2y_4}

{F_3=x_3+y_3+x_3y_1+x_4y_3}

{F_4=x_4+y_4+x_3y_2+x_4y_4}

It would beastly to fully check even one of those associative conditions. I should probably bring this to your attention, but the condition {F_i(F(X,Y), Z)} has you input as the first four variables those four equations, so in this case checking the first condition amounts to {F_1(F_1(X,Y),F_2(X,Y),F_3(X,Y),F_4(X,Y),z_1,z_2,z_3,z_4)}

{=F_1(x_1,x_2,x_3,x_4, F_1(Y,Z),F_2(Y,Z),F_3(Y,Z),F_4(Y,Z))}.

But on the other hand, the fact that {\widehat{\mathbb{G}}_a^n} satisfies it is trivial since you are only adding everywhere which is associative.

Now we can do basically everything we did with the one-dimensional case now. For one thing we can form the set {\mathcal{C}(F)} of {n}-tuples of power series in one indeterminant and no constant term. This set has an honest group structure on it given by {\gamma_1(t)+_F \gamma_2(t)=F(\gamma_1(t), \gamma_2(t))}.

We can define a homomorphism between an {n}-dimensional group law {F} and {m}-dimensional group law {G} to be an {m}-tuple of power series in {n}-indeterminants {\alpha(X)} with no constant term and satisfying {\alpha(F(X,Y))=G(\alpha(X), \alpha(Y))}.

From here we still have the same inductively defined endomorphisms for any {n} (not just the dimension) from the one-dimensional case {[0]_F(X)=0}, {[1]_F(X)=X} and {[n]_F(X)=F(X, [n-1]_F(X))}. That’s a lot of information to absorb, so we’ll end here for today.

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