Formal Witt Vectors

Last time we checked that our explicit construction of the ring of Witt vectors was a ring, but in the proof we noted that {W} actually was a functor {\mathrm{Ring}\rightarrow\mathrm{Ring}}. In fact, since it exists and is the unique functor that has the three properties we listed, we could have just defined the ring of Witt vectors over {A} to be {W(A)}.

We also said that {W} was representable, and this is just because {W(A)=Hom(\mathbb{Z}[x_1, x_2, \ldots ], A)}. We can use our {\Sigma_i} to define a (co)commutative Hopf algebra structure on {\mathbb{Z}[x_1, x_2, \ldots]}.

For instance, define the comultiplication {\mathbb{Z}[x_1, x_2, \ldots ]\rightarrow \mathbb{Z}[x_1,x_2,\ldots]\otimes \mathbb{Z}[x_1,x_2,\ldots]} by {x_i\mapsto \Sigma_i(x_1\otimes 1, \ldots , x_i\otimes 1, 1\otimes x_1, \ldots 1\otimes x_i)}.

Since this is a Hopf algebra we get that {W=\mathrm{Spec}(\mathbb{Z}[x_1,x_2,\ldots])} is an affine group scheme. The {A}-valued points on this group scheme are by construction the elements of {W(A)}. In some sense we have this “universal” group scheme keeping track of all of the rings of Witt vectors.

Another thing we could notice is that {\Sigma_1(X,Y)}, {\Sigma_2(X,Y)}, {\ldots} are polynomials and hence power series. If we go through the tedious (yet straightfoward since it is just Witt addition) details of checking, we will find that they satisfy all the axioms of being an infinite-dimensional formal group law. We will write this formal group law as {\widehat{W}(X,Y)} and {\widehat{W}} as the associated formal group.

Next time we’ll start thinking about the length {n} formal group law of Witt vectors (truncated Witt vectors).


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