Zp , Qp , AND THE RING OF WITT VECTORS
No.05: ring of Witt vectors (1) Preparations
From here on, we make use of several notions of category theory. Readers
who are unfamiliar with the subject is advised to see a book such as [?] forbasic definitions and first properties.

Let p be a prime number. For any commutative ring k of characteristic p 6= 0, we want to construct a ring W (k) of characteristic 0 insuch a way that:(1) W (Fp ) = Zp .
(2) W (•) is a functor. That means,(a) For any ring homomorphism ϕ : k1 → k2 between rings ofcharacterisic p, there is given a unique ring homomorphism
W (ϕ) : W (k1) → W (k2 ).
(b) W (•) should furthermore commutes with compositions ofhomomorphisms.
Recent days, it gets easier for us on the net to i find some good articles concerning the ring of Witt vectors. The treatment here borrows
some ideas from them. See for example the “comments” section inhttps://www.encyclopediaofmath.org/index.php/Witt_vector
5.1. Λ(A).
Definition 5.1. For any commutative ring A,(1) we defineΛ(A) = (1 + T A[[T ]])(as a set)

For any f ∈ (1+T A[[T ]]), we denote by (f )W the corresponding
element in Λ(A).
(2) For any (f )W , (g)W ∈ Λ(A), we define their sum by(f )W + (g)W = (f g)W
It is easy to see that Λ(A) is an additive group. It also carries the“T -addic topology” so that Λ(A) is a topological additive group.
The next task is to define multiplicative structure on Λ(A). To that
end, we do something somewhat different to others.
Definition 5.2. For any commutative ring A, we define E(A) =Endadditive (Λ(A)). It has the usual structure of a ring. For any a ∈ A,
we define its “Teichmüler” lift [a] as(f (T ))W 7→ (f (aT ))W .
The basic idea is to define E0 (A) as the subalgebra of E(A) topologicalalgebraically generated by all the Teichmüller lifts {[a]; a ∈ A} andidentify E0 (A) with Λ(A). To avoid some difficulties doing so, we first
do this when A is a very good one:Proposition 5.3. Assume A = Ω, an algebraically closed field.
Then:(1) Λ(A) is generated by {(1−aT )W |a ∈ A} as a topological additive
group.

Zp , Qp , AND THE RING OF WITT VECTORS(2) The subring E0 (A) of E(A) generated by {[a]|a ∈ A} as a topological ring is equal to {x ∈ E(A); x commutes with all Teichmüller lifts}.
(3) (1 − T )W is a generating separating vector of Λ(A) over E0 (A).
Thus we have a module isomorphism
E0 (A) ∋ ϕ → ϕ((1 − T )W ) ∈ Λ(A).
(Note that This isomorphism sends [a] to (1 − aT )W .
We may thus identify E0 (A) and Λ(A) via this isomorphism and employ
a ring structure on Λ(A).
Here after, for any algebraically closed field A, we employ the ringstructure of Λ(A) defined as the above proposition. In this language
we have:(1 − aT )W · (1 − bT )W = (1 − abT )W(a, b ∈ A)

More generally, for any f (T ) ∈ 1 + T A[[T ]] , we have a formula for
multiplication by degree-1-object (1 − aT )W :(1 − aT )W · (f (T ))W = (f (aT )W )(a ∈ A)We may extend this formula to any polynomial g(T ) ∈ 1 + T A[T ] withQ
constant term=1. Indeed, we factorize g as g(T ) = kj=1 (1 − αj T ) andY(g(T ))W · (f (T ))W =f (αj T )jExercise 5.1. Compute (1+aT +bT 2 )W (1+pT +qT 2 )W . Notice thatthe result of the computation only needs polynomials with coefficentsin Z[a, b, p, q] rather than some extension of the ring.