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

No.09: Witt algebras (3) The ring of p-adic Witt vectors

Definition 9.1. Let A be any commutative ring. Let n be a positive integer. Let us define additive operators V_{n}, F_{n} on 𝒲_{1}(A) by the following formula.

V_{n}(f(T)) = f(T^{n}).

F_{n}(f(T)) = ∏_{ζ∈μ_{n}} f(ζT^{1/n})

(The latter definition is a formal one. That means, F_{n} is actually defined to be the unique continuous additive map which satisfies

F_{n}(1 − aT^{l}) = (1 − a^{m/l}T^{m/n})^{ln/m} (m = l.c.m(n, l)).

)

Lemma 9.2. Let p be a prime number. Let A be acommutative ring of characteristic p. Then:

(1) We have

F_{p}(f(T)) = (f(T^{1/p}))^{p} (∀f ∈ 𝒲_{1}(A)).

in partucular, F_{p} is an algebra endomorphism of 𝒲_{1}(A) in this case.

(2)

V_{p}(F_{p}(f) = F_{p}(V_{p}(f)) = (f(T))^{p} = p ⊡ f(T)(= \overbrace{f(T) ⊞ ⋯ ⊞ f(T)}^{p}).

Definition 9.3. Let A be any commutative ring. Let p be a prime number. We denote by

𝒲^{(p)}(A) = A^{ℕ}.

and define

π_{p} : 𝒲_{1}(A) → 𝒲^{(p)}(A)

by

π_{p} ( \begin{matrix}∞ \\ ∑ & ⊞ \\ j=1 \end{matrix}(1 − x_{j}T^{j})) = (x_{1}, x_{p}, x_{p^{2}}, x_{p^{3}} …).

Lemma 9.4. Let us define polynomials α_{j}(X, Y) ∈ ℤ[X, Y] as follows.

(1 − xT)(1 − yT) = ∏_{j=1}^{∞}(1 − α_{j}(x, y)T^{j}).

Then we have the following rule for “carry operation”:

(1 − xT^{n}) ⊞ (1 − yT^{n}) = \begin{matrix}∞ \\ ∑ & ⊞ \\ j=1 \end{matrix}(1 − α_{j}(x, y)T^{jn}).

Proposition 9.5. There exist unique binary operators ⊞ and ⊠ on 𝒲^{(p)}(A) such that the following diagrams commute.

𝒲_{1}(A) × 𝒲_{1}(A) ⟶^{⊞} 𝒲_{1}(A)
π_{p}↓  ^{π_{p}}↓
𝒲^{(p)}(A) × 𝒲^{(p)}(A) ⟶^{⊞} 𝒲^{(p)}(A)
𝒲_{1}(A) × 𝒲_{1}(A) ⟶^{⊠} 𝒲_{1}(A)
π_{p}↓  ^{π_{p}}↓
𝒲^{(p)}(A) × 𝒲^{(p)}(A) ⟶^{⊠} 𝒲^{(p)}(A)

Proof. Using the rule as in the previous lemma, we see that addition descends to an addition of 𝒲^{(p)}(A). It is easier to see that the multiplication also descends.

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Definition 9.6. For any commutative ring A, elements of W^{(p)}(A) are called p-adic Witt vectors over A. The ring (W^{(p)}(A), ⊞, ⊠) is called the ring of p-adic Witt vectors over A.