Witt vectors José Felipe Voloch

We will recall here the definition and basic properties of Witt vectors and prove a few useful lemmas about them.

Fix a prime number p. Let X_{0}, Y_{0}, X_{1}, Y_{1}, … be indeterminates and define the Witt polynomials S_{0}, S_{1}, …, P_{0}, P_{1}, … inductively by S_{0} = X_{0} + Y_{0}, P_{0} = X_{0}Y_{0} and, for n ≥ 1,

∑_{i=0}^{n} p^{i}S_{i}^{p^{n−i}} = ∑_{i=0}^{n} p^{i}X_{i}^{p^{n−i}} + ∑_{i=0}^{n} p^{i}Y_{i}^{p^{n−i}},

∑_{i=0}^{n} p^{i}P_{i}^{p^{n−i}} = (∑_{i=0}^{n} p^{i}X_{i}^{p^{n−i}})(∑_{i=0}^{n} p^{i}Y_{i}^{p^{n−i}}).

The polynomials S_{0}, S_{1}, …, P_{0}, P_{1}, …, which a priori have rational coefficients, actually have integer coefficients, as will be proved below, so we may reduce them modulo p and henceforth we will consider them as polynomials over a field of characteristic p. Notice that S_{n} and P_{n} are polynomials in X_{0}, Y_{0}, X_{1}, Y_{1}, …, X_{n}, Y_{n}.

To prove that S_{0}, S_{1}, … have integer coefficients, we start with Lemma. If F (X_{0}, …, X_{n}) ∈ Z[X_{0}, …, X_{n}] then

F (X_{0}^{p}, …, X_{n}^{p})^{p^{r−1}} ≡ F (X_{0}, …, X_{n})^{p^{r}} (mod p^{r})

for r ≥ 1.

Proof: By induction on r. The case r = 1 is the well-known congruence

F (X_{0}^{p}, …, X_{n}^{p}) ≡ F (X_{0}, …, X_{n})^{p} (mod p).

Assuming the result is true for r we can write

F (X_{0}^{p}, …, X_{n}^{p})^{p^{r−1}} = F (X_{0}, …, X_{n})^{p^{r}} + p^{r}G,

for some G ∈ Z[X_{0}, …, X_{n}]. Raising this equation to the p-th power yields the result for r + 1.

Now, S_{0} = X_{0}+Y_{0} has integer coefficients. Assume by induction that S_{0}, S_{1}, …, S_{n−1} have integer coefficients. Clearly p^{n}S_{n} has integer coefficents. Let W_{n} = ∑_{i=0}^{n} p^{i}X_{i}^{p^{n−i}} ∈ Z[X_{0}, …, X_{n}]. By definition we have:

p^{n}S_{n} + W_{n−1}(S_{0}^{p}, …, S_{n−1}^{p}) = W_{n}(S_{0}, …, S_{n}) = W_{n}(X_{0}, …, X_{n}) + W_{n}(Y_{0}, …, Y_{n}) =

W_{n−1}(X_{0}^{p}, …, X_{n−1}^{p}) + W_{n−1}(Y_{0}^{p}, …, Y_{n−1}^{p}) + p^{n}(X_{n} + Y_{n}) =
W_{n−1}(S_{0}(X_{0}^{p}, …, Y_{0}^{p}, …), …, S_{n−1}(X_{0}^{p}, …, Y_{0}^{p}, …) + p^{n}(X_{n} + Y_{n}).

To show therefore that S_{n} has integer coefficients, it enough to show that

W_{n−1}(S_{0}^{p}, …, S_{n−1}^{p}) ≡ W_{n−1}(S_{0}(X_{0}^{p}, …, Y_{0}^{p}, …), …, S_{n−1}(X_{0}^{p}, …, Y_{0}^{p}, …)) (mod p^{n})

which amounts to show (using the definition of W_{n−1}) that S_{r}(X_{0}^{p}, …, Y_{0}^{p}, …)^{p^{n−1−r}} ≡ S_{r}^{p^{n−r}} (mod p^{n−1−r}) and this follows from the lemma. The proof that P_{n} has integer coefficients is entirely analogous.

Let k be a field of characteristic p. Denote by W_{e}(k) (resp. W (k)), the e-fold (resp. infinite) cartesian product of k with itself and define operations on W_{e}(k) (resp. W (k)) by

(X_{0}, X_{1}, …) + (Y_{0}, Y_{1}, …) = (S_{0}, S_{1}, …),
(X_{0}, X_{1}, …)(Y_{0}, Y_{1}, …) = (P_{0}, P_{1}, …).

Which make these sets into rings. W (k) is a complete discrete valuation ring of characteristic zero and residue field k, such that p generates its maximal ideal and W (k) is uniquely characterized by these properties. W_{e}(k) = W (k)/(p^{e}) and it can also be uniquely characterized as an Artinian local ring with p^{e} = 0 and residue field k, such that p generates its maximal ideal.

The subset T = {(a, 0, …, 0) | a ∈ k} is called the set of Teichmüller representatives of k and is in bijection with k under reduction modulo p. Moreover, reduction modulo p establishes an isomorphism between the groups T \ {0} and k^{∗} under multiplication.

S_{n} satisfies the following homogeneity property:

S_{n}(λX_{0}, λ^{p}X_{1}, …, λ^{p^{n}}X_{n}) = λ^{p^{n}}S_{n}(X_{0}, X_{1}, …, X_{n}).

This can be proved by multiplying by (λ, 0, …, 0) the equation

(X_{0}, X_{1}, …) + (Y_{0}, Y_{1}, …) = (S_{0}, S_{1}, …)

and using (λ, 0, …, 0)(X_{0}, X_{1}, …) = (λX_{0}, λ^{p}X_{1}, …), which can be easily checked. Hence the monomials occuring in S_{n} are of the form X_{0}^{i_{0}}Y_{0}^{j_{0}} … X_{n}^{i_{n}}Y_{n}^{j_{n}}, where i_{0} + j_{0} + … + p^{n}(i_{n} + j_{n}) = p^{n}. Likewise,

P_{n}(λX_{0}, λ^{p}X_{1}, …, λ^{p^{n}}X_{n}) = λ^{2p^{n}}P_{n}(X_{0}, X_{1}, …, X_{n}).

Also, (X_{0}, X_{1}, …)^{−1} = (I_{0}X_{0}^{−1}, I_{1}X_{0}^{−2p}, I_{2}X_{0}^{−3p^{2}}, …), where I_{0}, I_{1}, … are polynomials. Computing the inverse of (λ, 0, …, 0)(X_{0}, X_{1}, …) in two ways gives:

I_{n}(λX_{0}, λ^{p}X_{1}, …, λ^{p^{n}}X_{n}) = λ^{np^{n}}I_{n}(X_{0}, X_{1}, …, X_{n}).

Again, the monomials occuring in F_{n} are of the form X_{0}^{i_{0}} … X_{n}^{i_{n}}, where i_{0} + … + p^{n}i_{n} = np^{n}.