Today we’ll check that the ring of Witt vectors is actually a ring. Let be a ring, then as a set is the collection of infinite sequences of . Recall that our construction involves lots of various polynomials and a strange definition addition and multiplication. I won’t rewrite those, since it was the entirety of the last post.
Now there is a nice trick to prove that is a ring when is a -algebra. Just define by . This is a bijection and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know is a ring. Also, , so is the additive identity, which shows is the multiplicative identity, and , so we see is the additive inverse.
We can actually get this idea to work for any characteristic ring by considering the embedding . We have an induced injective map . The addition and multiplication is defined by polynomials over , so these operations are preserved upon tensoring with . We just proved above that is a ring, so since and and the map preserves inverses we get that the image of the embedding is a subring and hence is a ring.
Lastly, we need to prove this for positive characteristic rings. Choose a characteristic ring that surjects onto , say . Then since the induced map again preserves everything and is surjective, the image is a ring and hence is a ring.
So where does all this formal group stuff we started with come into play? Well, notice that what we were really implicitly using is that is a functor. It takes a ring and gives a new ring . If is a ring map, then by is still a ring map. We also have by are ring maps for all .
Some people think it is cleaner to define the ring of Witt vectors as the unique functor that satisfies these three properties. From a functorial point of view it turns out that is representable. The representing ring via the ring axioms gives a Hopf algebra structure, and hence we get an affine group scheme out of it. Then as in the formal group discussion, we can complete this to get a formal group. This will be the discussion of next time.