####
Definition 4.1.1.

We started our description of the \(p\)-typical Witt vector functor with the fact that the underlying functor to sets is represented by the ring \(A = \ZZ\{y\}\text{,}\) but crucially we *already* had produced a functor valued in rings (and even in \(\delta\)-rings). If we had needed to construct from scratch a functor valued in rings, we would have needed structures on \(A\) giving rise to the addition and multiplication maps. These structures are:

A ring \(A\) equipped with these structures represents a functor from rings to sets equipped with two binary operations \(+, \times\text{.}\) A biring is a ring equipped with coaddition and comultiplication operators which are further subject to the axioms that correspond to the ring axioms on \(+, \times\text{.}\) Namely, the coaddition map is cocommutative, coassociative, and admits a counit and an antipode (giving rise to additive inverses); the comultiplication map is cocommutative, coassociative, codistributive over coaddition, and admits a counit.

A shorter way to say this is that a biring is a commutative ring object in the category of affine schemes. (Remember that the functor \(\Spec\colon \Ring \to \Sch\) is contravariant!)

####
Proposition 4.1.2.

There is a unique functor \(\WW\) from \(\Ring\) to \(\Ring\) characterized by the following conditions.

The underlying functor to sets is

\begin{equation*}
\WW(A) = A \times A \times \cdots.
\end{equation*}

There is a natural transformation \(w\) from \(\WW\) to the ordinary product \(A \mapsto A^{\NN}\) given by the ghost map:

\begin{equation*}
(x_n)_{n=1}^\infty \mapsto (w_n)_{n=1}^\infty, \qquad w_n = \sum_{d|n} d x_d^{n/d}.
\end{equation*}

(Again, the individual factors of this map are called ghost components.)

The ring \(\WW(A)\) is called the ring of big Witt vectors over \(A\text{.}\)

It suffices to produce a unique biring structure on \(\ZZ[x_1, x_2, \dots]\) representing the desired functor. To begin with, since the ghost map is an isomorphism whenever \(A\) is a \(\QQ\)-algebra, we obtain a biring structure on \(\QQ[w_0,w_1,\dots] = \QQ[x_0, x_1,\dots]\text{;}\) this already implies uniqueness. For existence, it suffices to check that for each prime \(p\text{,}\) this biring structure descends to \(\ZZ_{(p)}[x_1, x_2, \dots]\text{;}\) this will imply that the coaddition and comultiplication maps act on \(\bigcap_p \ZZ_{(p)}[x_1,x_2,\dots] = \ZZ[x_1,x_2,\dots]\text{.}\)

Define the family of elements \(y_n\) of \(\QQ[x_1, x_2, \dots]\) as follows: for each positive integer \(m\) coprime to \(p\) and each nonnegative integer \(i\text{,}\)

\begin{equation*}
w_{mp^i} = \sum_{j=0}^i p^j y_{mp^j}^{p^{i-j}}.
\end{equation*}

By a calculation which we omit (see

Exercise 4.3.1), we see that

\(\ZZ_{(p)}[y_1,y_2,\dots] = \ZZ_{(p)}[x_1,x_2,\dots]\text{.}\) In the

\(y\)-coordinates,

\(\WW(A)\) splits into a collection of copies of

\(W(A)\) indexed by positive integers coprime to

\(p\text{;}\) hence we obtain a biring structure on

\(\ZZ_{(p)}[y_1,y_2,\dots] = \ZZ_{(p)}[x_1,x_2,\dots]\) as needed.