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

No.10: The ring of Witt vectors when A is a ring of characteristic p ≠ 0.

Definition 10.1. Let A be a commutative ring. For any a ∈ A, we denote by [a] the element of 𝒲_{1}(A) defined as follows:

[a] = (1 − aT)

We call [a] the Teichmüller lift” of a

Lemma 10.2. Let A be a commutative ring. Then:

(1) (𝒲_{1}(A), ⊞, ⊠) is a commutative ring with the zero element [0] and the unity [1].

(2) For any a, b ∈ A, we have

[a] ⊠ [b] = [ab]
□

Proposition 10.3. Let p be a prime number. Let A be a ring of characteristic p. Then:

(1) If n is a positive integer which is not divisible by p, then n is invertible in 𝒲_{1}(A). To be more precise,

\frac{1}{n} ⊡ [1] = (1 − T)^{\frac{1}{n}} = 1 + ∑_{j=1}^{∞} \binom{\frac{1}{n}}{j}(−T)^{j}.

(2) p⊡ : 𝒲_{1}(A) → 𝒲_{1}(A) is an injection.

(3) For any positive integer n which is not divisible by p, we define

e_{n} = \frac{1}{n} ⊡ (1 − T^{n}).

as an element of 𝒲_{1}(A).

(4) For any positive integer n, e_{n} is an idempotent. (That means, e_{n}^{⊠2} = e_{n}.)

(5) If n|m, then e_{n} ⪰ e_{m} in the order of idempotents. That means, e_{n} ⊠ e_{m} = e_{m}.

Proof. (1) follows from the next lemma. □

Lemma 10.4. Let n be a positive integer. Let k be a non negative integer. Then we have always

\binom{\frac{1}{n}}{k} ∈ ℤ [\frac{1}{n}] .

Proof.

\binom{\frac{1}{n}}{k} ∈ ℤ [\frac{1}{n}]

= \frac{\frac{1}{n}(\frac{1}{n} − 1) ⋯ (\frac{1}{n} − (k − 1))}{k!}
=\frac{1}{n^{k}} \frac{(1(1 − n)(1 − 2n) … (1 − (k − 1)n)}{k!}

So the result follows from the next sublemma. □

Sublemma 10.5. Let n be a positive integer. Let k be a non negative integer. Let {a_{j}}_{j=1}^{k} ⊂ ℤ be an arithmetic progression of common difference n. Then:

(1) For any positive integer m which is relatively prime to n, we have

#\{j; m|a_{j} \} ≥ ⌊\frac{k}{m}⌋

(2) For any prime p which does not divide n, let us define

c_{k,p} = ∑_{i=1}^{∞}⌊\frac{k}{p^{i}} ⌋

(which is evidently a finite sum in practice.) Then

p^{c_{k,p}}| ∏_{j=1}^{k} a_{j}

(3)

p^{c_{k,p}}|k!, p^{c_{k,p}+1} |̸k!

(4)

\frac{∏_{j=1}^{k} a_{j}}{k!} ∈ ℤ_{(p)}

Proof. (1) Let us put t = ⌊\frac{k}{m}⌋. Then we divide the set of first kt-terms of the sequence {a_{j}} into disjoint sets in the following way.

S_{0} = {a_{1}, a_{2}, …, a_{m}},
S_{1} = {a_{m+1}, a_{m+2}, a_{m+m}},
S_{2} = {a_{2m+1}, a_{2m+2}, a_{2m+m}},
…
S_{t−1} = {a_{(t−1)m+1}, a_{(t−1)m+2}, …, a_{(t−1)m+m}}

Since m is coprime to n, we see that each of the S_{u} gives a complete representative of ℤ/nℤ.

(2): Apply (1) to the cases where m = p, p^{2}, p^{3}, … and count the powers of p which appear in ∏ a_{j}.

(3): Easy. (4) is a direct consequence of (2),(3). □

Proposition 10.6. Let p be a prime. Let A be an integral domain of characteristic p. Let us define an idempotent f of 𝒲_{1}(A) as follows.

f = ⋁_{\begin{matrix}n>1 \\ p|̸n \end{matrix}} e_{n}(= [1] ⊟ ^{∏⊠} ; \begin{matrix}p|̸n \\ n>1 \end{matrix}([1] ⊟ e_{n}))

Then f defines a direct product decomposition

𝒲_{1}(A) ≅ (f ⊠ 𝒲_{1}(A)) × ((1 ⊟ f) ⊠ 𝒲_{1}(A)) .

Furthermore, the factor algebra (1 ⊟ f) ⊠ 𝒲_{1}(A) is isomorphic to the ring 𝒲^{(p)}(A) of p-adic Witt vectors.

The following proposition tells us the importance of the ring of p-adic Witt vectors.

Proposition 10.7. Let p be a prime. Let A be a commutative ring of characteristic p. For each positive integer k which is not divisible by p, let us define an idempotent f_{k} of 𝒲_{1}(A) as follows.

f_{k} = ⋁_{\begin{matrix}p|̸n \\ n>1 \end{matrix}} e_{kn}(= e_{k} ⊟ ^{∏⊠} ; \begin{matrix}p|̸n \\ n>1 \end{matrix}(e_{k} ⊟ e_{kn}))

Then f_{k} defines a direct product decomposition

e_{k}𝒲_{1}(A) ≅ (f_{k} ⊠ 𝒲_{1}(A)) × ((1 ⊟ f_{k}) ⊠ 𝒲_{1}(A)) .

Furthermore, the factor algebra (1 ⊟ f_{k}) ⊠ 𝒲_{1}(A) is isomorphic to the ring 𝒲^{(p)}(A) of p-adic Witt vectors. Thus we have a direct product decomposition

𝒲_{1}(A) ≅ 𝒲^{(p)}(A)^{ℕ}.

To understand the mechanism which appears in the proposition above, it would be better to prove the following

Lemma 10.8. Let p be a prime number. Let A be a ring of characteristic p. Then for any n which is not divisible by p, a map

\frac{1}{n} ⊡ V_{n} : (𝒲_{1}(A), ⊞, ⊠) → (𝒲_{1}(A), ⊞, ⊠)

is a ring homomorphism. Its image is equal to the range of the idempotent e_{n}. That means,

Image(\frac{1}{n} ⊡ V_{n}) = e_{n} ⊠ 𝒲_{1}(A) = \{ \begin{matrix}∑ & ⊞ \\ j \end{matrix}(1 − y_{j}T^{nj}); y_{j} ∈ A\}.

Proof. V_{n} is already shown to be additive. The following calculation shows that \frac{1}{n} · V_{n} preserves the ⊠-multiplication.

(\frac{1}{n} ⊡ V_{n}(1 − xT^{a})) ⊠ (\frac{1}{n} ⊡ V_{n}(1 − yT^{b}))

=(\frac{1}{n} ⊡ (1 − xT^{an})) ⊠ (\frac{1}{n} ⊡ (1 − yT^{bn}))
=\frac{1}{n^{2}} ⊡ (1 − x^{m/a}y^{m/b}T^{nm})^{d}
=\frac{1}{n} ⊡ ((1 − xT^{a}) ⊠ (1 − yT^{b}))
□

In preparing from No.7 to No.10 of this lecture, the following reference (especially its appendix) has been useful:

http://www.math.upenn.edu/~chai/course_notes/cartier_12_2004.pdf