Today we move on to higher dimensional formal group laws over a ring . 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 -dimensional formal group law over is just an -tuple of power series, each of variables and no constant term, satisfying certain relations. We’ll write where and to simplify notation.

There are a few natural guesses for the conditions, but the ones we actually use are that (higher degree) and for all . We call the group law commutative if for all .

We still have our old examples that are fairly trivial , meaning the i-th one is just . For a slightly less trivial example, let’s explicitly write a four-dimensional one

It would beastly to fully check even one of those associative conditions. I should probably bring this to your attention, but the condition has you input as the first four variables those four equations, so in this case checking the first condition amounts to

.

But on the other hand, the fact that 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 of -tuples of power series in one indeterminant and no constant term. This set has an honest group structure on it given by .

We can define a homomorphism between an -dimensional group law and -dimensional group law to be an -tuple of power series in -indeterminants with no constant term and satisfying .

From here we still have the same inductively defined endomorphisms for any (not just the dimension) from the one-dimensional case , and . That’s a lot of information to absorb, so we’ll end here for today.

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