Difference between revisions of "Witt vector"
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− | An element of an algebraic construct, first proposed by E. Witt [[#References|[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 [[#References|[3]]], in the theory of commutative algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[6]]]. Let $A$ be an associative, commutative ring with unit element. Witt vectors with components in $A$ are infinite sequences $a = ( | + | {{TEX|done}} |
+ | |||
+ | An element of an algebraic construct, first proposed by E. Witt [[#References|[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 [[#References|[3]]], in the theory of commutative algebraic groups [[#References|[4]]], [[#References|[5]]], and in the theory of formal groups [[#References|[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} $, | |
− | where $ | + | $ 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 [[#References|[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==== | ||
+ | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Witt, "Zyklische Körper und Algebren der characteristik <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810058.png" /> vom Grad <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810059.png" />. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassen-körper der Charakteristik <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810060.png" />" ''J. Reine Angew. Math.'' , '''176''' (1936) pp. 126–140 {{MR|}} {{ZBL|0016.05101}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S. Lang, "Algebra" , Addison-Wesley (1974) {{MR|0783636}} {{ZBL|0712.00001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) {{MR|0103191}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w09810061.png" /> VII" ''Math. Ann.'' , '''134''' (1957) pp. 114–133 {{MR|}} {{ZBL|}} </TD></TR></table> | ||
+ | |||
+ | |||
+ | |||
+ | ====Comments==== | ||
+ | There is a generalization of the construction above which works for all primes $ p $ | ||
+ | simultaneously, [[#References|[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 | + | $$ |
+ | (a _{1} ,\ a _{2} ,\dots ) (b _{1} ,\ b _{2} ,\dots ) | ||
+ | = ( \Pi _{1} ( \mathbf a ) ,\ \Pi _{2} ( \mathbf a ) ,\dots ) - | ||
$$ | $$ | ||
− | + | $$ | |
+ | - | ||
+ | (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 .$$ | |
− | + | The $ \mathbf V _{m} $ | |
− | + | are group endomorphisms of $ W(-) $ | |
− | + | but not ring endomorphisms. | |
− | |||
− | |||
− | |||
− | |||
− | = | + | The ideals $ I _{n} = \{ ( 0 \dots 0,\ a _{n+1} ,\ a _{n+2} ,\dots ) \} \subset W(A) $ |
− | + | define a topology on $ W(A) $ | |
+ | making $ W(A) $ | ||
+ | a separated complete topological ring. | ||
− | For each $ | + | For each $ A \in \mathbf{Ring} $, |
+ | let $ \Lambda (A) $ | ||
+ | be the Abelian group $ 1 + t A [[t]] $ | ||
+ | under multiplication of power series; $$ | ||
+ | \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)) | ||
+ | $$ | ||
+ | making $ W(A) $ | ||
+ | a functorial special [[Lambda-ring| $ \lambda $- | ||
+ | ring]]. The Artin–Hasse exponential $ \Delta : \ W \rightarrow W \circ W $ | ||
+ | defines a cotriple structure on $ W $ | ||
+ | and the co-algebras for this co-triple are precisely the special $ \lambda $- | ||
+ | rings (cf. also [[Category|Category]] and [[Triple|Triple]]). | ||
− | + | On $ \Lambda (A) $ | |
+ | the Frobenius and Verschiebung endomorphisms satisfy $$ | ||
+ | \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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[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|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, [[#References|[a3]]], [[#References|[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. [[#References|[a5]]]. | |
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====References==== | ====References==== | ||
− | <table> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100202.png" />-ten Potenzreste im Körper der <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/w/w098/w098100/w098100203.png" />-ten Einheitswurzeln" ''Abh. Math. Sem. Univ. Hamburg'' , '''6''' (1928) pp. 146–162 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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 {{MR|73645}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> 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 {{MR|0561822}} {{ZBL|0437.13014}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) {{MR|506881}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR></table> |
− | <TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Artin, H. Hasse, "Die beide Ergänzungssätze zum Reciprozitätsgesetz der | ||
− | <TR><TD valign="top">[a2]</TD> <TD valign="top"> 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 {{MR|73645}} {{ZBL|}} </TD></TR> | ||
− | <TR><TD valign="top">[a3]</TD> <TD valign="top"> 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 {{MR|0561822}} {{ZBL|0437.13014}} </TD></TR> | ||
− | <TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "Formal group laws and applications" , Acad. Press (1978) {{MR|506881}} {{ZBL|}} </TD></TR> | ||
− | <TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR> | ||
− | </table> | ||
− | |||
− |
Latest revision as of 17:25, 22 December 2019
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. $$ If $ k = \mathbf F _{p} $ is the prime field of $ p $ elements, $ W( \mathbf F _{p} ) $ is the ring of integral $ p $- adic numbers $ \mathbf Z _{p} $.
References
[1] | E. Witt, "Zyklische Körper und Algebren der characteristik ![]() ![]() ![]() |
[2] | S. Lang, "Algebra" , Addison-Wesley (1974) MR0783636 Zbl 0712.00001 |
[3] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |
[4] | J.-P. Serre, "Groupes algébrique et corps des classes" , Hermann (1959) MR0103191 |
[5] | M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503 |
[6] | J. Dieudonné, "Groupes de Lie et hyperalgèbres de Lie sur un corps de charactéristique ![]() |
Comments
There is a generalization of the construction above which works for all primes $ p $ simultaneously, [a3]: a functor $ W : \ \mathbf{Ring} \rightarrow \mathbf{Ring} $ called the big Witt vector. Here, $ \mathbf{Ring} $ is the category of commutative, associative rings with unit element. The functor described above, of Witt vectors of infinite length associated to the prime $ p $, is a quotient of $ W $ which can be conveniently denoted by $ W _ {p ^ \infty} $.
For each $ n \in \{ 1,\ 2,\dots \} $,
let $ w _{n} (X) $
be the polynomial $$
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 ![]() ![]() |
[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 |
Witt vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Witt_vector&oldid=37670