Definition 2

Let $A$ be a ring and let $n \geq 1$ be an integer. We denote by $W_n(A)$ the quotient ring $W(A)/V_n(A)$.

Given elements $a_0, ..., a_{n-1}$ of $A$, we denote by $[a_0, ..., a_{n-1}]$ or $[a_i]_{0 \leq i < n}$ the class modulo $V_n(A)$ of the element $(a_0, ..., a_{n-1}, 0, 0, ...)$ of $W(A)$. According to Lemma 4 of No. 6, the mapping $(a_0, ..., a_{n-1}) \mapsto [a_0, ..., a_{n-1}]$ from $A^n$ into $W_n(A)$ is a bijection. For this reason, we say that the elements of $W_n(A)$ are the Witt vectors of length $n$; by analogy, one sometimes qualifies the elements of $W(A)$ as Witt vectors of infinite length.

We denote by $\pi_n$ the canonical homomorphism from $W(A)$ into $W_n(A)$. According to Lemma 4 of No. 6, we have

for every $a = (a_n)_{n \in \mathbf{N}}$ in $W(A)$.

According to the definition of the operations in $W(A)$, we have the following description of the operations in $W_n(A)$:

Moreover, the identity element of the addition in $W_n(A)$ is $[0, ..., 0]$ and that of the multiplication is $[1, 0, ..., 0]$.

Let $i$ be an integer such that $0 \leq i \leq n$. By passing to the quotient, the homomorphism $\Phi_i$ from $W(A)$ into $A$ defines a homomorphism $\Phi_i$ from $W_n(A)$ into $A$. This associates with the Witt vector $[a_0, ..., a_{n-1}]$ the element $\Phi_i(a_0, ..., a_i)$ of $A$ (also called the ghost component of index $i$ of $[a_0, ..., a_{n-1}]$).

Let $\rho : B \to A$ be a ring homomorphism. By passing to the quotients, the homomorphism $W(\rho)$ from $W(B)$ into $W(A)$ defines a homomorphism $W_n(\rho)$ from $W_n(B)$ into $W_n(A)$. It is described by the formula

for every $[b_0, ..., b_{n-1}]$ in $W_n(B)$.

Let $m$ and $n$ be two integers such that $1 \leq n \leq m$. We have $V_n(A) \supset V_m(A)$, whence a canonical homomorphism from $W_m(A) = W(A)/V_m(A)$ onto $W_n(A) = W(A)/V_n(A)$; this homomorphism will be denoted by $\pi_{n,m}$. Explicitly, we have

for $[a_0, ..., a_{m-1}]$ in $W_m(A)$. The family $(W_n(A), \pi_{n,m})$ is a projective system of rings and the mapping $\pi : a \mapsto (\pi_n(a))_{n \geq 1}$ is a ring homomorphism from $W(A)$ into $\lim_{\leftarrow} W_n(A)$, called canonical. Since $W(A)$ is separated and complete for the filtration $(V_n(A))_{n \in \mathbf{Z}}$ (cf. No. 6), the canonical homomorphism $\pi$ is an isomorphism of topological rings, when $W_n(A)$ is endowed with the discrete topology for every integer $n \geq 1$ (III, § 2, No. 6).

Henceforth, the homomorphisms $\pi_n$ and $\pi_{n,m}$ will be called projection homomorphisms from $W(A)$ into $W_n(A)$, and from $W_m(A)$ into $W_n(A)$ respectively.