ℤ_{p}, ℚ_{p}, AND THE RING OF WITT VECTORS

No.11: The ring of Witt vectors(4) The ring of p-adic Witt vectors revisited

Lemma 11.1. Let A be a commutative ring. Then:

(1) For any a, b ∈ A, we have

[a] ⊠ [b] = [ab]

(2) If a ∈ A satisfies a^{q} = a, then

[a]^{⊠q} = [a].

(3) Let q be a positive integer. If b ∈ A satisfies

∀n ∈ ℤ_{>0}∃b_{n} ∈ A such that b_{n}^{q^{n}} = b,

then we have

∀n ∈ ℤ_{>0}∃c_{n} ∈ 𝒲_{1}(A) such that c_{n}^{q^{n}} = [b].
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Recall that the ring of p-adic Witt vectors is a quotient of the ring of universal Witt vectors. We have therefore a projection ϖ : 𝒲_{1}(A) → 𝒲^{(p)}(A). But in the following we intentionally omit to write ϖ.

Proposition 11.2. Let p be a prime number. Let A be a ring of characteristic. Then:

(1) Every element of 𝒲^{(p)}(A) is written uniquely as

\begin{matrix}∞ \\ ∑ & ⊞ \\ j=0 \end{matrix} V_{p}^{j}([x_{j}]) (x_{j} ∈ A).

(2) For any x, y ∈ A, we have

V_{p}^{n}([x]) ⊠ V_{p}^{m}([y]) = V^{n+m}([x^{p^{m}}y^{p^{n}}]).

(3) A map

φ : 𝒲^{(p)}(A) ∋ \begin{matrix}∞ \\ ∑ & ⊞ \\ j=0 \end{matrix} V_{p}^{n}([x_{j}]) ↦ x_{0} ∈ A

is a ring homomorphism from (𝒲^{(p)}, ⊞, ⊠) to (A, +, ×).

(4) Ker(φ) = Image(V_{p}).

(5) An element x ∈ 𝒲^{(p)} is invertible in 𝒲^{(p)} if and only if φ(x) is invertible in A.

□

Corollary 11.3. If k is a field of characteristic p ≠ 0, then 𝒲^{(p)} is a local ring with the residue field k. If furthermore the field k is perfect (that means, every element of k has a p-th root in k), then every non-zero element of 𝒲^{(p)} may be writen as

p^{k} ⊡ x (k ∈ ℕ, x ∈ (𝒲^{(p)})^{⊠} (i.e. x:invertible))

Since any integral domain can be embedded into a perfect field, we deduce the following

ℤ_{P}, ℚ_{P}, AND THE RING OF WITT VECTORS

Corollary 11.4. Let A be an integral domain of characteristic p ≠ 0. Then 𝒲^{(p)}(A) is an integral domain of characteristic 0.

Proof. 𝒲^{(p)}(ι) is always an injection when ι is. □