Witt vector

An element of an algebraic construct, first proposed by E. Witt [1] 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 [3], in the theory of commutative algebraic groups [4], [5], and in the theory of formal groups [6]. 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 [3].

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 $\mathbf Z _{p}$.

References

[1]	E. Witt, "Zyklische Körper und Algebren der characteristik vom Grad . Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik " J. Reine Angew. Math. , 176 (1936) pp. 126–140 Zbl 0016.05101
[2]	S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[3]	D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[4]	J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191
[5]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503
[6]	J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique VII" Math. Ann. , 134 (1957) pp. 114–133

Comments

There is a generalization of the construction above which works for all primes $p$ simultaneously, [a3]: a functor $W : \ \mathbf{Ring} \rightarrow \mathbf{Ring}$ called the big Witt vector. Here, $\mathbf{Ring}$ is the category of commutative, associative rings with unit element. The functor described above, of Witt vectors of infinite length associated to the prime $p$, is a quotient of $W$ which can be conveniently denoted by $W _ {p ^ \infty}$.

For each $n \in \{ 1,\ 2,\dots \}$, let $w _{n} (X)$ be the polynomial $$w _{n} (X) = \sum _{d\mid n} dX _{d} ^{n/d} .$$ Then there is the following characterization theorem for the Witt vectors. There is a unique functor $W : \ \mathbf{Ring} \rightarrow \mathbf{Ring}$ satisfying the following properties: 1) as a functor $W: \ \mathbf{Ring} \rightarrow \mathop{\rm Set}\nolimits$, $W (A) = \{ {(a _{1} ,\ a _{2} , \dots )} : {a _{i} \in A} \}$ and $W ( \phi ) (a _{1} ,\ a _{2} , . . ) = ( \phi (a _{1} ) ,\ \phi (a _{2} ) ,\dots )$ for any ring homomorphism $\phi : \ A \rightarrow B$; 2) $w _ {n , A} : \ W(A) \rightarrow A$, $( a _{1} ,\ a _{2} ,\dots ) \mapsto w _{n} (a _{1} ,\ a _{2} ,\dots )$ is a functorial homomorphism of rings for every $A$ and $n \in \{ 1,\ 2,\dots \}$.

The functor $W$ admits functorial ring endomorphisms $\mathbf f _{n} : \ W \rightarrow W$, for every $n \in \{ 1,\ 2,\dots \}$, that are uniquely characterized by $w _{n} \mathbf f _{m} = w _{nm}$ for all $n,\ m \in \{ 1,\ 2,\dots \}$. Finally, there is a functorial homomorphism $\Delta : \ W(-) \rightarrow W(W(-))$ that is uniquely characterized by the property $w _ {n, W(A)} \Delta _{A} = \mathbf f _ {n, A}$ for all $n$, $A$.

To construct $W(A)$, define polynomials $\Sigma _{1} \dots \Sigma _{n} ,\dots$; $\Pi _{1} \dots \Pi _{n} ,\dots$; $r _{1} \dots r _{n} ,\dots$ by the requirements $$w _{n} ( \Sigma _{1} \dots \Sigma _{n} ) = w _{n} (X) + w _{n} (Y),$$ $$w _{n} ( \Pi _{1} \dots \Pi _{n} ) = w _{n} (X) w _{n} (Y),$$ $$w _{n} ( r _{1} \dots r _{n} ) = - w _{n} ( X) .$$ The $\Sigma _{n}$ and $\Pi _{n}$ are polynomials in $X _{1} \dots X _{n}$; $Y _{1} \dots Y _{n}$ and the $r _{n}$ are polynomials in the $X _{1} \dots X _{n}$ and they all have integer coefficients. $W(A)$ is now defined as the set $W(A) = \{ {\mathbf a = (a _{1} ,\ a _{2} ,\dots )} : {a _{i} \in A} \}$ with addition, multiplication and "minus" : $$(a _{1} ,\ a _{2} ,\dots ) + (b _{1} ,\ b _{2} ,\dots ) = ( \Sigma _{1} ( \mathbf a ) ,\ \Sigma _{2} ( \mathbf a ) ,\dots )$$ $$(a _{1} ,\ a _{2} ,\dots ) (b _{1} ,\ b _{2} ,\dots ) = ( \Pi _{1} ( \mathbf a ) ,\ \Pi _{2} ( \mathbf a ) ,\dots ) -$$ $$- (a _{1} ,\ a _{2} ,\dots ) = ( r _{1} ( \mathbf a ) ,\ r _{2} ( \mathbf a ) ,\dots ) .$$ The zero of $W(A)$ is $( 0,\ 0 ,\dots )$ and the unit element is $( 1,\ 0 ,\ 0 ,\dots )$. The Frobenius endomorphisms $\mathbf f _{n}$ and the Artin–Hasse exponential $\Delta$ are constructed by means of similar considerations, i.e. they are also given by certain universal polynomials. In addition there are the Verschiebung morphisms $\mathbf V _{n} : \ W(-) \rightarrow W(-)$, which are characterized by $$w _{m} \mathbf V _{n} = \left \{ \begin{array}{ll} 0 & \textrm{ if } n \textrm{ does not divide } m, \\ nw _{m/n} & \textrm{ if } n \textrm{ divides } m. \\ \end{array} \right .$$ The $\mathbf V _{m}$ are group endomorphisms of $W(-)$ but not ring endomorphisms.

The ideals $I _{n} = \{ ( 0 \dots 0,\ a _{n+1} ,\ a _{n+2} ,\dots ) \} \subset W(A)$ define a topology on $W(A)$ making $W(A)$ a separated complete topological ring.

For each $A \in \mathbf{Ring}$, let $\Lambda (A)$ be the Abelian group $1 + t A [[t]]$ under multiplication of power series; $$\overline{E}\; : \ W(A) \rightarrow \Lambda (A),$$ $$( a _{1} ,\ a _{2} ,\dots ) \mapsto \prod _{i=1} ^ \infty (1- a _{i} t ^{i} ) ,$$ defines a functional isomorphism of Abelian groups, and using the isomorphism $\overline{E}\;$ there is a commutative ring structure on $\Lambda (A)$. Using $\overline{E}\;$ the Artin–Hasse exponential $\Delta$ defines a functorial homomorphism of rings $$W(A) \rightarrow \Lambda (W(A))$$ making $W(A)$ a functorial special $\lambda$- ring. The Artin–Hasse exponential $\Delta : \ W \rightarrow W \circ W$ defines a cotriple structure on $W$ and the co-algebras for this co-triple are precisely the special $\lambda$- rings (cf. also Category and Triple).

On $\Lambda (A)$ the Frobenius and Verschiebung endomorphisms satisfy $$\mathbf f _{n} (1-at) = (1-a ^{n} t) ,$$ $$\mathbf V _{n} f(t) = f(t ^{n} ) ,$$ and are completely determined by this (plus functoriality and additivity in the case of $\mathbf f _{n}$).

For each supernatural number $\mathbf n = \prod _{p} p ^ {\alpha _ p}$, $\alpha _{p} \in \{ 0,\ 1,\ 2,\dots \} \cup \{ \infty \}$, one defines $N ( \mathbf n ) = \{ {n \in \{ 1,\ 2,\dots \}} : {v _{p} (n) \leq \alpha _{p } \textrm{ for all "prime" numbers } p} \}$, where $v _{p}$ is the $p$- adic valuation of $n$, i.e. the number of prime factors $p$ in $n$. Let $$\mathfrak a _ {\mathbf n} (A) =$$ $$= \{ {(a _{1} ,\ a _{2} ,\dots )} : { a _{d} = 0 \textrm{ for all } d \in N ( \mathbf n )} \} .$$ Then $\mathfrak a _ {\mathbf n} (A)$ is an ideal in $W(A)$ and for each supernatural $\mathbf n$ a corresponding ring of Witt vectors is defined by $$W _ {\mathbf n} (A) = W(A) / \mathfrak a _ {\mathbf n} (A) .$$ In particular, one thus finds $W _ {p ^ \infty} (A)$, the ring of infinite-length Witt vectors for the prime $p$, discussed in the main article above, as a quotient of the ring of big Witt vectors $W(A)$.

The Artin–Hasse exponential $\Delta : \ W \rightarrow W \circ W$ is compatible in a certain sense with the formation of these quotients, and using also the isomorphism $\overline{E}\;$ one thus finds a mapping $$\mathbf Z _{p} = W _ {p ^ \infty} ( \mathbf F _{p} ) \rightarrow \Lambda (W _ {p ^ \infty} ( \mathbf F _{p} ) ) = \Lambda ( \mathbf Z _{p} ) ,$$ where $\mathbf Z _{p}$ denotes the $p$- adic integers and $\mathbf F _{p}$ the field of $p$ elements, which can be identified with the classical morphism defined by Artin and Hasse [a1], [a2], [a3].

As an Abelian group $W(A)$ is isomorphic to the group of curves ${\mathcal C} ( \mathbf G _{m} ; \ A)$ of curves in the one-dimensional multiplicative formal group $\mathbf G _{m}$. In this way there is a Witt-vector-like Abelian-group-valued functor associated to every one-dimensional formal group. For special cases, such as the Lubin–Tate formal groups, this gives rise to ring-valued functors called ramified Witt vectors, [a3], [a4].

Let $r _{n} (X,\ Y)$ be the sequence of polynomials with coefficients in $\mathbf Z$ defined by $$X ^{n} + Y ^{n} = \sum _{d\mid n} d r _{d} (X,\ Y) ^{n/d} .$$ The Cartier ring $\mathop{\rm Cart}\nolimits (A)$ is the ring of all formal expressions $$\tag{*} \sum _ {i,j \in \{ 1, 2,\dots \}} \mathbf V _{i} \langle a _{ij} \rangle \mathbf f _{j}$$ with the calculation rules $$\langle a><b\rangle = \langle ab\rangle , \langle 1\rangle = \mathbf f _{1} = \mathbf V _{1} = \textrm{ unit element } 1 ,$$ $$\mathbf V _{n} \mathbf V _{m} = \mathbf V _{nm} , \mathbf f _{n} \mathbf f _{m} = \mathbf f _{nm} ,$$ $$\langle a\rangle \mathbf V _{m} = \mathbf V _{m} \langle a ^{m} \rangle , \mathbf f _{m} \langle a\rangle = \langle a ^{m} \rangle \mathbf f _{m} ,$$ $$\mathbf V _{m} \mathbf f _{n} = \mathbf f _{n} \mathbf V _{m} \textrm{ if } (n,\ m) = 1 ,$$ $$\mathbf f _{n} \mathbf V _{n} = 1 + \dots + 1 ( n \textrm{ summands } ) ,$$ $$\langle a+b\rangle = \sum _{n=1} ^ \infty \mathbf V _{n} \langle r _{n} ( a,\ b) \rangle \mathbf f _{n} .$$ Commutative formal groups over $A$ are classified by certain modules over $\mathop{\rm Cart}\nolimits (A)$. In case $A$ is a $\mathbf Z _{(p)}$- algebra, a simpler ring $\mathop{\rm Cart}\nolimits _{p} (A)$ can be used for this purpose. It consists of all expressions (*) where now the $i,\ j$ only run over the powers $p ^{0} ,\ p ^{1} ,\ p ^{2} , . . .$ of the prime $p$. The calculation rules are the analogous ones. In case $k$ is a perfect field of characteristic $p > 0$ and $\sigma$ denotes the Frobenius endomorphism of $W(k)$( which in this case is given by $\sigma ( a _{1} ,\ a _{2} , . . . ) = ( a _{1} ^{p} ,\ a _{2} ^{p} , . . )$), then $\mathop{\rm Cart}\nolimits _{p} (k)$ can be described as the ring of all expressions $$x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} + \sum _{j=1} ^ \infty y _{j} \mathbf f ^{i} ,$$ in two symbols $\mathbf f$ and $\mathbf V$ and with coefficients in $W _ {p ^ \infty} (k)$, with the extra condition $\mathop{\rm lim}\nolimits _ {i \rightarrow \infty} \ y _{i} = 0$ and the calculation rules $$\mathbf f x = \sigma (x) \mathbf f , \mathbf V x = \sigma ^{-1} (x) \mathbf V ,$$ $$\mathbf f \mathbf V = \mathbf V \mathbf f = p .$$ This ring, and also its subring of all expressions $$x _{0} + \sum _{i=1} ^ \infty x _{i} \mathbf V ^{i} + \sum _{j=1} ^ {< \infty} y _{j} \mathbf f ^{j} ,$$ is known as the Dieudonné ring $D(k)$ and certain modules (called Dieudonné modules) over it classify unipotent commutative affine group schemes over $k$, cf. [a5].

References

[a1]	E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der -ten Potenzreste im Körper der -ten Einheitswurzeln" Abh. Math. Sem. Univ. Hamburg , 6 (1928) pp. 146–162
[a2]	G. Whaples, "Generalized local class field theory III: Second form of the existence theorem, structure of analytic groups" Duke Math. J. , 21 (1954) pp. 575–581 MR73645
[a3]	M. Hazewinkel, "Twisted Lubin–Tate formal group laws, ramified Witt vectors and (ramified) Artin–Hasse exponentials" Trans. Amer. Math. Soc. , 259 (1980) pp. 47–63 MR0561822 Zbl 0437.13014
[a4]	M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) MR506881
[a5]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503