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

YOSHIFUMI TSUCHIMOTO

No.03: ℤ_{p} as a projective limit of {ℤ/p^{k}ℤ}

Definition 3.1. An ordered set Λ is said to be directed if for all i, j ∈ Λ there exists k ∈ Λ such that i ≤ k and j ≤ k.

Definition 3.2. Let Λ be a directed set. Let {X_{λ}}_{λ∈Λ} be a family of topological rings. Assume we are given for each pair of elements (λ, μ) ∈ Λ^{2} such that λ ≥ μ, a continuous homomorphisms

ϕ_{μλ} : X_{λ} → X_{μ}.

We say that such a system ({X_{λ}}, {ϕ_{μλ}}) is a projective system of topological rings if it satisfies the following axioms.

(1) ϕ_{νμ}ϕ_{μλ} = ϕ_{νλ} (∀λ, ∀μ∀ν such that λ ≥ μ ≥ ν).

(2) ϕ_{λλ} = id (∀λ ∈ Λ).

Definition 3.3. Let 𝒳 = ({X_{λ}}, {ϕ_{μλ}}) be a projective system of topological rings. Then we say that a projective limit (X, {ϕ_{λ}}) of 𝒳 is given if

(1) X is a topological ring.

(2) ϕ_{λ} : X → X_{λ} is a continuous homomorphism.

(3) ϕ_{μλ} ∘ ϕ_{λ} = ϕ_{μ} for ∀μ, λ such that λ ≥ μ.)

(4) (X, {ϕ_{λ}}) is a universal object among objects which satisfy (1)- (3).

The “universal” here means the following: If (Y, ψ_{λ}) satisfies

(1) Y is a topological ring.

(2) ψ_{λ} : Y → X_{λ} is a continuous homomorphism.

(3) ϕ_{μλ} ∘ ψ_{λ} = ψ_{μ} for ∀μ, λ such that λ ≥ μ.) Then there exists a unique continuous homomorphism

Φ : Y → X

such that

ψ_{λ} = ϕ_{λ} ∘ Φ(∀λ ∈ Λ).

Proposition 3.4. For any projective system of topological rings, a projective limit of the system exists. It is unique up to a unique isomorphism. (Hence we may call it the projective limit of the system.)

YOSHIFUMI TSUCHIMOTO

Definition 3.5. For any projective system (X, {ϕ_{λ}}) of topological rings, We denote the projective limit of it by

\underleftarrow{lim}_{λ} X_{λ}.

Note: projective limits of systems of topological spaces, rings, groups, modules, and so on, are defined in a similar manner.

Theorem 3.6.

ℤ_{p} ≅ \underleftarrow{lim}_{k→∞}(ℤ/p^{k}ℤ)

as a topological ring.

Corollary 3.7. ℤ_{p} is a compact space.

Note: There are several ways to prove the result of the above corollary. For example, the ring ℤ with the metric d_{p} is easily shown to be totally bounded.

Proposition 3.8. Each element of ℤ_{p} is expressed uniquely as

[0.a_{1}a_{2}a_{3}a_{4} …]_{p} (a_{i} ∈ {0, 1, …, p − 1} (i = 1, 2, 3, …)).

Exercise 3.1. Is −4 = 1 − 5 invertible in ℤ_{5}? (Hint: use formal expansion

(1 − x)^{−1} = 1 + x + x^{2} + …

is it possible to write down a correct proof to see that the result is true?)