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

No.07: ring of Witt vectors (1) Preparations

From here on, we make use of several notions of category theory. Readers who are unfamiliar with the subject is advised to see a book such as [1] for basic definitions and first properties.

Let p be a prime number. For any commutative ring k of characteristic p ≠ 0, we want to construct a ring W (k) of characteristic 0 in such a way that:

(1) W (𝔽_{p}) = ℤ_{p}.

(2) W (•) is a functor. That means,

(a) For any ring homomorphism φ : k_{1} → k_{2} between rings of characterisic p, there is given a unique ring homomorphism W (φ) : W (k_{1}) → W (k_{2}).

(b) W (•) should furthermore commutes with compositions of

homomorphisms.

To construct W (k), we construct a new addition and multiplication on a k-module ∏_{j=1}^{∞} k. The ring W (k) will then be called the ring of Witt vectors. The treatment here essentially follows the treatment which appears in [2, VI,Ex.46-49], with a slight modification (which may or may not be good–it may even be wrong) by the author.

We first introduce a nice idea of Witt.

Definition 7.1. Let A be a ring (of any characteristic). Let T be an indeterminate. We define the following copies of A^{ℤ_{>0}}.

𝒲_{0}(A) = T A[[T]] = {∑_{j=1}^{∞} x^{(j)}T^{j} ; x^{(n)} ∈ A(∀n)}

𝒲_{1}(A) = 1 + T A[[T]] = {1 + ∑_{j=1}^{∞} y_{j}T^{j} ; x_{n} ∈ A(∀n)}

Lemma 7.2. 𝒲_{0} and 𝒲_{1} are functors from the category of rings to the category of sets. They are represented by “polynomial rings in infinite indeterminates”

A_{𝒲_{0}} = ℤ[X^{(1)}, X^{(2)}, X^{(3)}, …]

and

A_{𝒲_{1}} = ℤ[Y_{1}, Y_{2}, Y_{3}, …].

That means, there are functorial bijections

Hom_{ring}(A_{𝒲_{0}} , A) ≅ 𝒲_{0}(A)

and

Hom_{ring}(A_{𝒲_{1}} , A) ≅ 𝒲_{1}(A).

Definition 7.3. We define the following “universal elements”.

v_{0} = ∑_{j=1}^{∞} X^{(j)}T^{j} ∈ 𝒲_{0}(A_{𝒲_{0}} ),

v_{1} = 1 + ∑_{j=1}^{∞} Y_{j}T^{j} ∈ 𝒲_{1}(A_{𝒲_{1}} ).

Lemma 7.4. There is an well-defined map

ℒ_{A} = −T \frac{d}{dT} log(•) : 1 + T A[[T]] → T A[[T]].

If A contains an copy of ℚ, then the map is a bijection. The inverse is given by

T g(T) ↦ exp (− ∫_{0}^{T} g(s)ds) .

Proof. To see that ℒ is well defined (that is, “defined over ℤ”), we compute as follows.

−T \frac{d}{dT} log(1+T f_{1}) = −T (f′_{1}+f_{1})(1+T f_{1})^{−1} = −T (f′_{1}+f_{1}) ∑_{j=1}^{∞}(−T f_{1})^{j}

The rest should be obvious.

Note: the condition A ⊃ ℚ is required to guarantee exictence of exponential

exp(•) = ∑_{j=0}^{∞} \frac{1}{j!}•^{j}

and existence of the integration ∫_{0}^{T} g(s)ds. □

Definition 7.5. We equip 𝒲_{0}(A) = T A[[T]] with the usual addition and the following (unusual) multiplication:

(∑_{j=1}^{∞} a^{(j)}T^{j}) ∗ (∑_{j=1}^{∞} b^{(j)}T^{j}) = ∑_{j=1}^{∞}(a^{(j)}b^{(j)})T^{j}

It is easy to see that 𝒲_{0}(A) forms a (unital associative) commutative ring with these binary operations.

Definition 7.6. Let A be a ring which contains a copy of ℚ. Then we define ring structure on 𝒲_{1}(A) by putting

f +_{ℒ} g = ℒ_{A}^{−1}(ℒ_{A}(f) + ℒ_{A}(g)), f ∗_{ℒ} g = ℒ_{A}^{−1}(ℒ_{A}(f) ∗ ℒ_{A}(g)).

Lemma 7.7. Let A be a ring which contains a copy of ℚ. For any f, g ∈ 𝒲_{1}(A), we have

f +_{ℒ} g = fg.

In particular, addition +_{ℒ} is defined over ℤ.

Proof. easy □

We may thus extend the definition +_{ℒ} on 𝒲_{1}(A) to cases where the condition A ⊃ ℚ is no longer satisfied.

References

[1] S. S. Mac Lane, Categories for the working mathematicians, Springer Verlag, 1971.

[2] S. Lang, Algebra (graduate texts in mathematics), Springer Verlag, 2002.