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

No.12: The ring of Witt vectors and ℤ_{p}

Definition 12.1. Let A be a ring of characteristic p. We equip W^{(p)} with the ring strucure defined by binary operations (⊞, ⊠). Then we call a ring

𝒲_{n}^{(p)}(A) = 𝒲^{(p)}(A)/V_{p}^{n}(𝒲^{(p)}(A))

the ring of Witt vectors of length n. Its elements are called Witt vectors of lenth n.

Note that

𝒲^{(p)}(A)/V_{p}^{n}(𝒲^{(p)}(A))

may be considered as a set A^{n} with a ring structure (⊞, ⊠) (other than the usual one).

Proposition 12.2.

ℤ_{p} ≅ 𝒲^{(p)}(𝔽_{p}).

Proof. Since 𝒲^{(p)} is a unital commutative ring, there naturally exists a natural ring homomorphism

ι : ℤ → 𝒲^{(p)}(𝔽_{p}).

Let us first fix a positive integer n and examine the kernel K_{n} of a map

π_{n} ∘ ι : ℤ → 𝒲^{(p)}(𝔽_{p})/V_{p}^{n}(𝒲^{(p)}(𝔽_{p}))

where π_{n} is the natural projection. Since

#𝒲^{(p)}(𝔽_{p})/V_{p}^{n}(𝒲^{(p)}(𝔽_{p})) = #(𝔽_{p}^{n}) = p^{n},

we have

#(ℤ/K_{n})|p^{n}.

In other words, K_{n} = p^{s}ℤ for some integer s. On the other hand, we have

π_{n} ∘ ι(p^{s}) = (1 − T)^{p^{s}} = (1 − T^{p^{s}})

thus

π_{n} ∘ ι(p^{s}) ∈ K_{n} ⟺ s ≥ n.

This implies that K_{n} = p^{n}ℤ and therefore we have an inclusion

ℤ/p^{n}ℤ ↪ W_{n}^{(p)}(𝔽_{p})

which turns to be a bijection (∵ #(ℤ/p^{n}ℤ) = #(𝒲_{n}^{(p)}(𝔽_{p}))).

We then take a projective limit of the both hand sides and obtain the resired isomorphism.

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