Witt vector

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An element of an algebraic construct, first proposed by E. Witt  in 1936 in the context of the description of unramified extensions of $p$-adic number fields. Witt vectors were subsequently utilized in the study of algebraic varieties over a field of positive characteristic , in the theory of commutative algebraic groups , , and in the theory of formal groups . Let $A$ be an associative, commutative ring with unit element. Witt vectors with components in $A$ are infinite sequences $a = (a_0,a_1,\ldots)$, $a_i \in A$, which are added and multiplied in accordance with the following rules: $$(a_0,a_1,\ldots) \oplus (b_0,b_1,\ldots) = (S_0(a_0;b_0), S_1(a_0,a_1;b_0,b_1), \ldots)$$ $$(a_0,a_1,\ldots) \otimes (b_0,b_1,\ldots) = (M_0(a_0;b_0), M_1(a_0,a_1;b_0,b_1), \ldots)$$
where $S_n,M_n$ are polynomials in the variables $X_0,\ldots,X_n$, $Y_0,\ldots,Y_n$ with integer coefficients, uniquely defined by the conditions $$\Phi_n(S_0,\ldots,S_n) = \Phi_n(X_0,\ldots,X_n) + \Phi_n(Y_0,\ldots,Y_n)$$ $$\Phi_n(M_0,\ldots,M_n) = \Phi_n(X_0,\ldots,X_n) \cdot \Phi_n(Y_0,\ldots,Y_n)$$ where $$\Phi_n(Z_0,\ldots,Z_n) = Z_0^{p^n} + p Z_1^{p^{n-1}} + \cdots + p^n Z^n$$ are polynomials, $n \in \mathbf{N}$ and $p$ is a prime number. In particular, $$S_0 = X_0 + Y_0 \ ;\ \ \ S_1 = X_1 + Y_1 - \sum_{i=1}^{p-1} \frac{1}{p} \binom{p}{i} X_0^i Y_0^{p-i}$$ $$M_0 = X_0 \cdot Y_0 \ ;\ \ \ M_1 = X_0^p Y_1 + X_1 Y_0^p + p X_1 Y_1 \ .$$
The Witt vectors with the operations introduced above form a ring, called the ring of Witt vectors and denoted by $W(A)$. For any natural number $n$ there also exists a definition of the ring $W_n(A)$ of truncated Witt vectors of length $n$. The elements of this ring are finite tuples $a = (a_0,\ldots,a_{n-1})$, $a_i \in A$ with the addition and multiplication operations described above. The canonical mappings $$R : W_{n+1}(A) \rightarrow W_n(A)$$ $$R : (a_0,\ldots,a_n) \mapsto (a_0,\ldots,a_{n-1})$$ $$T : W_n(A) \rightarrow W_{n+1}(A)$$ $$T : (a_0,\ldots,a_{n-1}) \mapsto (0,a_0,\ldots,a_{n-1})$$ are homomorphisms. The rule $A \to W(A)$ (or $A \to W_n(A)$) defines a covariant functor from the category of commutative rings with unit element into the category of rings. This functor may be represented by the ring of polynomials $\mathbf{Z}[X_0,X_1,\ldots]$ (or $\mathbf{Z}[X_0,X_1,\ldots,X_n]$) on which the structure of a ring object has been defined. The spectrum $\mathrm{Spec}\mathbf{Z}[X_0,X_1,\ldots]$ (or $\mathrm{Spec}\mathbf{Z}[X_0,X_1,\ldots,X_n]$) is known as a Witt scheme (or a truncated Witt scheme) and is a ring scheme .
Each element $a \in A$ defines a Witt vector $$a^T = (a,0,0,\ldots) \in W(A)$$ called the Teichmüller representative of the element $a$. If $A = k$ is a perfect field of characteristic $p>0$, then $W(k)$ is a complete discrete valuation ring of zero characteristic with field of residues $k$ and maximal ideal $pW(k)$. Each element $w \in W(k)$ can be uniquely represented as $$w = w_0^T + pw_1^T + p^2 w_2^T + \cdots$$ where $w_i \in k$. Conversely, each such ring $A$ with field of residues $k = A/(p)$ is canonically isomorphic to the ring $W(k)$. The Teichmüller representation makes it possible to construct a canonical multiplicative homomorphism $k \to W(k)$, splitting the mapping $W(k) \to W(k)/(p)$.
If $k = \mathbf{F}_p$ is the prime field of $p$ elements, $W(k)$ is the ring of integral $p$-adic numbers $\mathbf{Z}_p$.