Ring of Witt vectors

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:

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:

Some elaboration on why these choices work is provided below:

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.

Facts

 * Linear representation of finite group over finite field of coprime characteristic lifts uniquely to Witt ring
 * Additive group of ring of Witt vectors inherits algebraic group structure