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