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

No.08: ring of Witt vectors (2) The ring of universal Witt vectors

In the following, we use infinite sums and infinite products of elements of 𝒲_{1}(A) = 1 + T A[[T]]. They are defined as limits of sums and products with respect to the filtration topology defined in the usual way.

Lemma 8.1. Let A be any commutative ring. Then every element of 1 + T A[[T]] is written uniquely as

∏_{j=1}^{∞}(1 − x_{j}T^{j}) (x_{j} ∈ A).

Proof. We may use an expansion

∏_{j=1}^{∞}(1 − x_{j}T^{j}) ≡ −x_{n}T^{n} + poly(x_{1}, …, x_{n−1}, T) (mod T^{n+1})

to inductively determine x_{j}. More precisely, for each n ∈ ℤ_{>0}, let us define a polynomial f_{n}(X_{1}, X_{2}, …, X_{n−1}) in the following way:

f_{n}(X_{1}, …, X_{n−1}) = coeff(∏_{j=1}^{n−1}(1 − X_{j}T^{j}), T^{n})

Then for any element 1 + ∑_{j=1}^{∞} y_{j}T^{j} ∈ 1 + T A[[T]], we define

x_{1} = −y_{1}, x_{n} = −y_{n} + f_{n}(x_{1}, …, x_{n−1}) (∀n > 1).

Then it is easy to verify that an equation

1 + ∑_{j=1}^{∞} y_{j}T^{j} = ∏_{j=1}^{∞}(1 − x_{j}T^{j})

holds. □

Corollary 8.2. 𝒲_{1}(A) = 1 + T A[[T]] is topologically generated by

{(1 − x_{j}T^{j}); x_{j} ∈ A, j = 1, 2, 3, … }.

Lemma 8.3. Let d, e be positive integers. Let m be the least common multiple of d, e. Then we have

(1−x_{d}T^{d})∗_{ℒ} (1−x_{e}T^{e}) = (1−x_{d}^{m/d}x_{e}^{m/e}T^{m})^{de/m}(= (\frac{de}{m})·_{ℒ} (1 − x_{d}^{m/d}x_{e}^{m/e}T^{m})).

(Note that m ≥ d, e.)

Proof. let d, e be positive integers. Let m be the least common multiple of d, e. We have,

ℒ(1 − x_{d}T^{d}) ∗ ℒ(1 − x_{e}T^{e}) = \frac{dx_{d}T^{d}}{1 − x_{d}T^{d}} ∗ \frac{ex_{e}T^{e}}{1 − x_{e}T^{e}} = de(∑_{i=1}^{∞}(x_{d}T^{d})^{i} ∗ ∑_{j=1}^{∞}(x_{e}T^{e})^{j})

=de ∑_{u=1}^{∞} x_{d}^{mu/d}x_{e}^{mu/e}T^{mu} = \frac{dex_{d}^{m/d}x_{e}^{m/e}T^{m}}{1 − x_{d}^{m/d}x_{e}^{m/e}T^{m}} = −\frac{de}{m} \frac{d}{dT} log(1 − x_{d}^{m/d}x_{e}^{m/e}T^{m})
=ℒ((1 − x_{d}^{m/d}x_{e}^{m/e}T^{m})^{de/m}).
□

Definition 8.4. Let A be any commutative ring. Then we define an addition ⊞ and a multiplication ⊠ on 𝒲_{1}(A) who satisfy the following requirements:

(1) f ⊞ g = fg.

(2) For any positive integer d, e, Let m be the least common multiple of d, e. Then we have

(1 − x_{d}T^{d}) ⊠ (1 − x_{e}T^{e}) = (1 − x_{d}^{m/d}x_{e}^{m/e}T^{m})^{\frac{de}{m}} .

(3) for general f, g , the multiplication f ⊠ g is defined by first expressing f, g as a formal ⊞-sum as in Lemma 8.3 and then applying the rule 2 formally to each “⊞-summand”.

(Note that Lemma 8.1 guarantees the existence and the uniqueness of such multiplication ⊠.)

Theorem 8.5. Let A be any commutative ring. Then:

(1) Any element of 𝒲_{1}(A) is written uniquely as

\begin{matrix}∞ \\ ∑ & ⊞ \\ j=1 \end{matrix}(1 − x_{j}T^{j}).

(2) 𝒲_{1}(A) forms a commutative ring under the binary operations ⊞ and ⊠. More precisely,

(a) (𝒲_{1}(A), ⊞) is an additive group with the zero element 1.

(b) The multiplication ⊠ is an associative commutative product on 𝒲_{1}(A) with the unit element 1 − T.

(c) The distributive law holds.

(3) When A ⊃ ℚ, the ring (𝒲_{1}(A), ⊞, ⊠) is isomorphic to (𝒲_{0}(A), +, ∗) via the map ℒ_{A} = −T \frac{d}{dT} log(•).

Proof. When A ⊃ ℚ, the statements trivially hold. This implies in particular that rules such as distributivity and associativity hold for universal cases (that means, for formal power series with indeterminate coefficients). Thus we conclude by specialzation arguments that the rule also hold for any ring A.

□

Definition 8.6. For any commutative ring A, elements of 𝒲_{1}(A) are called universal Witt vectors over A. The ring (𝒲_{1}(A), ⊞, ⊠) is called the ring of universal Witt vectors over A.

Proposition 8.7. (𝒲_{1}(•), ⊞, ⊠) is uniquely determined by the following properties.

(1) f ⊞ g = fg (∀f, g ∈ 𝒲_{1}(A).

(2) The multiplication ⊠ is ⊞-biadditive. (That means, 𝒲_{1}(A), ⊞, ⊠) obeys the distributive law.)

(3) (1 − xT) ⊠ (1 − yT) = (1 − (xy)T) (∀x, y ∈ A).

(4) ⊠ is continuous.

(5) ⊠ is functorial.

Proof. We only need to prove the requirement (2) of Definition 8.4. With the help of distributive law, the requirement is satisfied if an equation

(#) (1−xT^{a})⊠(1−yT^{b}) = (1−x^{m/a}y^{m/b}T^{m})^{ab/m} (m = l.c.m(a, b))

holds for each (a, b) ∈ (ℤ_{>0})^{2}.

To that aim, we first deal with a special case where x = α^{a}, y = β^{b}, A = ℂ[α, β], α, β algebraically independent over ℂ. In that case we may easily decompose the polynomials (1 − xT^{a}) and (1 − yT^{b}) and then we use the distributive law to see that the requirement actually holds. Indeed, let us put

ζ_{k} = exp(2π\sqrt{−1}/k)

and compute as follows.

(1 − xT^{a}) ⊠ (1 − yT^{b})
= \begin{matrix}∑ & ⊞ \\ j,l \end{matrix}(1 − ζ_{a}^{j}(α)T) ⊠ (1 − ζ_{b}^{l}(β)T)

= \begin{matrix}∑ & ⊞ \\ j,l \end{matrix}(1 − ζ_{a}^{j}ζ_{b}^{l}αβT)
= ∏_{l}(1 − ζ_{b}^{al}α^{a}β^{a}T^{a})
= ∏_{l′}(1 − xβ^{a}T^{a}ζ_{b/d}^{l′})^{d} (d = g.c.d(a, b))
=(1 − x^{a/d}y^{b/d}T^{ab/d})^{d}.

We second deal with a case where A = ℤ[x, y], x, y algebraically independent over ℂ. In that case we take a look at an inclusion

ι : ℤ[x, y] ↪ ℂ[a, b].

and consider 𝒲_{1}(ι). It is easy to see that 𝒲_{1}(ι) is injection so that the equation (#) is also true in this case. The general case now follows from specialization argument. □