Ring of Witt vectors

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Definition

The term ring of Witt vectors or Witt ring is a commutative unital ring defined in the context of a prime number p and a commutative unital ring R. There are two version of this:

  • The ring of Witt vectors, by default, refers to infinite sequences, i.e., Witt vectors of infinite length (the detailed definition is given below).
  • The ring of Witt vectors of length d refers to the quotient of this ring obtained by simply looking at the first d coordinates of the vector. The operations are defined in a manner that the first d coordinates of the sum and product depend only on the first d coordinates of the vectors being added or multiplied. Thus, it is legitimate to pass to quotients in this manner.

There is a related notion called the ring of universal Witt vectors that makes sense in the context of any commutative unital ring R and does not require specification of any prime number.

Definition of the ring of Witt vectors of infinite length

We first define the Witt polynomials as the following polynomials in \mathbb{Z}[X_0,X_1,X_2,\dots,] (and hence interpretable in R[X_0,X_1,X_2,\dots,]):

W_n(X) := \sum_{i=0}^n p^iX_i^{p^{n-i}}

The first few Witt polynomials are given below:

n W_n
0 X_0
1 X_0^p + pX_1
2 X_0^{p^2} + pX_1^p + p^2X_2
3 X_0^{p^3} + pX_1^{p^2} + p^2X_2^p + p^3X_3

We now consider the set (X_0,X_1,X_2,\dots,X_n,\dots), X_i \in R of all sequences over R. We call the elements of this set Witt vectors, and we define the addition and multiplication as follows:

Addition and multiplication are defined in the unique manner so as to make the set of Witt vectors a commutative unital ring with the following two properties: (a) each Witt polynomial is a ring homomorphism from that ring to R, and (b) both addition and multiplication are given by polynomials with integer coefficients that depend on p but not on R.

Explicitly, the sum and product of the Witt vectors (X_0,X_1,X_2,\dots,) and (Y_0,Y_1,Y_2,\dots,) are given by the following formulas:

n n^{th} coordinate of sum n^{th} coordinate of product
0 X_0 + Y_0 X_0Y_0
1 X_1 + Y_1 - \sum_{i=1}^{p-1} \frac{(p-1)!}{i!(p-i)!} X_0^iY_0^{p-i}, which can be thought of as X_1 + Y_1 + \frac{(X_0^p + Y_0^p) - (X_0 + Y_0)^p}{p} (although the latter notation does not directly make sense unless p is invertible in R). X_0^pY_1 + Y_0^pX_1 + pX_1Y_1

Some elaboration on why these choices work is provided below:

n W_n(X) W_n(Y) W_n(X) + W_n(Y), which equals W_n of the Witt sum W_n(X)W_n(Y), which equals W_n of the Witt product Comment
0 X_0 Y_0 X_0 + Y_0 X_0Y_0 Zeroth Witt polynomial is projection onto zeroth coordinate; pointwise operations mean this is a homomorphism.
1 X_0^p + pX_1 Y_0^p + pY_1 X_0^p + Y_0^p + p(X_1 + Y_1), which is the same as (X_0 + Y_0)^p + p\left[X_1 + Y_1 + \frac{(X_0^p + Y_0^p) - (X_0 + Y_0)^p}{p}\right] X_0^pY_0^p + pX_0^pY_1 + pY_0^pX_1 + p^2X_1Y_1, which is the same as (X_0Y_0)^p + p(X_0^pY_1 + Y_0^pX_1 + pX_1Y_1) Things work out.

Truncations to finite length

We can also consider the ring of Witt vectors of length d. This is obtained by taking the ring of Witt vectors and projecting it to the first d coordinates X_0,X_1,X_2,\dots,X_{d-1}. The operations all have the property that the formula of the i^{th} coordinate of the sum and product involves only vectors X_j,Y_j, j \le i, so the ring operations descend to the quotient and we get a ring structure on the quotient.

Particular cases

Note that p is the prime with respect to which we are defining the Witt polynomials.

Condition on R Conclusion for ring of Witt vectors (infinite) Conclusion for truncation to length d
p is invertible in R Isomorphic to the external direct product of countably many copies of R (i.e., sequences with pointwise sum and product), i.e., R^{\omega} Isomorphic to R^d
R is the finite prime field \mathbb{F}_p Isomorphic to the ring of p-adic integers, where a Witt vector (X_0,X_1,X_2,\dots) is identified with the p-adic integer \sum_{i=0}^\infty X_ip^i. Isomorphic to the ring of integers mod p^d, i.e., the ring \mathbb{Z}/p^d\mathbb{Z}.
R is a finite field of order p^k, k a natural number Isomorphic to an unramified degree k extension of the ring of p-adic integers. There is only one such extension up to isomorphism. Isomorphic to the Galois ring, which is an unramified extension of \mathbb{Z}/p^d\mathbb{Z} of degree k, and can be constructed using any minimal polynomial for \mathbb{F}_{p^k} over \mathbb{F}_p.

Facts