# 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.