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.
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