# 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 $a = (a _{0} , a _{1} , . . . )$, $a _{i} \in A$, which are added and multiplied in accordance with the following rules: $$(a _{0} ,\ a _{1} ,\dots ) \dot{+} (b _{0} ,\ b _{1} ,\dots ) =$$ $$= (S _{0} (a _{0} ,\ b _{0} ),\ S _{1} (a _{0} ,\ a _{1} ; \ b _{0} ,\ b _{1} ) , . . . ),$$ $$(a _{0} ,\ a _{1} , . . . ) \dot \times (b _{0} ,\ b _{1} , . . . ) =$$ $$= (M _{0} (a _{0} ,\ b _{0} ),\ M _{1} (a _{0} ,\ a _{1} ; \ b _{0} ,\ b _{1} ) , . . . ),$$ where $S _{n}$, $M _{n}$ are polynomials in the variables $X _{0} \dots X _{n}$, $Y _{0} \dots Y _{n}$ with integer coefficients, uniquely defined by the conditions $$\Phi _{n} (S _{0} \dots S _{n} ) = \Phi _{n} (X _{0} \dots X _{n} ) + \Phi _{n} (Y _{0} \dots Y _{n} ),$$ $$\Phi _{n} (M _{0} \dots M _{n} ) = \Phi _{n} (X _{0} \dots X _{n} ) \cdot \Phi _{n} (Y _{0} \dots Y _{n} );$$ where $$\Phi _{n} = Z _{0} ^ {p ^ n} + pZ _{1} ^ {p ^ n-1} + \dots + p ^{n} Z _{n}$$ are polynomials, $n \in \mathbf N$ and $p$ is a prime number. In particular, $$S _{0} = X _{0} + Y _{0} ; S _{1} = X _{1} + Y _{1} - \sum _ {i = 1} ^ {p-1} { \frac{1}{p} } \binom{p}{i} X _{0} ^{i} Y _{0} ^{p-i} ;$$ $$M _{0} = X _{0} Y _{0} , M _{1} = X _{0} ^{p} Y _{1} + X _{1} Y _{0} ^{p} + pX _{1} Y _{1} .$$ 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 $a = (a _{0} \dots a _{n-1} )$, $a _{i} \in A$, with the addition and multiplication operations described above. The canonical mappings $$R: \ W _{n+1} (A) \rightarrow W _{n} (A),$$ $$R ((a _{0} \dots a _{n} )) = (a _{0} \dots a _{n-1} ) ,$$ $$T: \ W _{n} (A) \rightarrow W _{n+1} (A),$$ $$T ((a _{0} \dots a _{n-1} )) = (0,\ a _{0} \dots a _{n-1} ),$$ are homomorphisms. The rule $A \mapsto W(A)$( or $A \mapsto W _{n} (A)$) 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 $\mathbf Z [X _{0} \dots X _{n} ,\dots ]$( or $\mathbf Z [X _{0} \dots X _{n-1} ]$) on which the structure of a ring object has been defined. The spectrum $\mathop{\rm Spec}\nolimits \ \mathbf Z [X _{0} \dots X _{n} ,\dots ]$( or $\mathop{\rm Spec}\nolimits \ \mathbf Z [X _{0} \dots X _{n-1} ]$) 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 $$a ^ \tau = (a,\ 0,\ 0 , . . . ) \in W \ (A),$$ called the Teichmüller representative of the element $a$. If $A = k$ is a perfect field of characteristic $p > 0$, $W(k)$ is a complete discrete valuation ring of zero characteristic with field of residues $k$ and maximal ideal $pW(k)$. Each element $\omega \in W(k)$ can be uniquely represented as $$\omega = \omega _{0} ^ \tau + p \omega _{1} ^ \tau + p ^{2} \omega _{2} ^ \tau + \dots ,$$ where $\omega _{i} \in k$. 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 $k \rightarrow W(k)$, splitting the mapping $$W (k) \rightarrow W (k) / p \simeq k.$$ If $k = \mathbf F _{p}$ is the prime field of $p$ elements, $W( \mathbf F _{p} )$ is the ring of integral $p$- adic numbers $\mathbf Z _{p}$.