Wittvector

Witt vectors are a generalization introduced by the mathematician Ernst Witt of the construction of the (whole) p-adic numbers to any perfect residue class field . In addition to these -typical Witt vectors there are the large Witt vectors, from which the -typical Witt vectors can be reconstructed for anything . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$

p -typical Witt vectors

Be a fixed prime number . For a ring (commutative, with one element) the Witt vectors form a ring that is dependent on . It is particularly interesting for rings of the characteristic , but the construction makes it necessary to allow other rings as well. ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle A}$ ${\displaystyle p}$

motivation

Be an integer. As an approximation to an alternative construction of the -adic numbers , a ring that is isomorphic to the remainder class ring, denoted by , should initially be constructed using addition and multiplication in the body . ${\displaystyle n>1}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle W_{p,n}(\mathbb {F} _{p})}$

The first, naive approach would be to use the mapping that maps the residual class of in to the residual class of in for integers . The bijection ${\displaystyle \sigma :\mathbb {F} _{p}\to \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle 0\leq k<p}$ ${\displaystyle k}$ ${\displaystyle \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle k}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$

${\displaystyle \mathbb {F} _{p}^{n}\to \mathbb {Z} /p^{n}\mathbb {Z} ,\ \ (x_{0},x_{1},\dots ,x_{n-1})\mapsto \sigma (x_{0})+p\sigma (x_{1})+\dots +p^{n-1}\sigma (x_{n-1})}$

corresponds to the representation of whole numbers in the base value system . The addition transferred by is then in the case : ${\displaystyle \{0,1,\dots ,p^{n}-1\}}$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle n=2}$

${\displaystyle (x_{0},x_{1})+(y_{0},y_{1})=(x_{0}+y_{0},x_{1}+y_{1}+c(x_{0},y_{0})),}$

where is the carry. This construction cannot be generalized well to fields other than , also because the definition of makes use of the representative system which is unfavorable from an algebraic point of view . ${\displaystyle c(x_{0},y_{0})}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle \sigma }$ ${\displaystyle \{0,1,\dots ,p-1\}\subset \mathbb {Z} }$

The correct approach is based on the following statement from elementary number theory: The following applies to whole numbers : ${\displaystyle u,v}$

${\displaystyle u\equiv v\mod p\implies u^{p^{n-1}}\equiv v^{p^{n-1}}\mod p^{n}}$

(see congruence (number theory) ). That means: If and is a representative of , then the remainder class of in depends only on , but not on the choice of . We write suggestively for this element of . (This figure is essentially the Teichmüller representative system for the -adic numbers .) More generally, the remainder of the class does not depend on itself either , we disk . ${\displaystyle x\in \mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z} }$ ${\displaystyle u\in \mathbb {Z} }$ ${\displaystyle x}$ ${\displaystyle u^{p^{n-1}}}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle x}$ ${\displaystyle u}$ ${\displaystyle x^{p^{n-1}}}$ ${\displaystyle \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle \mathbb {F} _{p}\to \mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle p^{k}u^{p^{n-1-k}}}$ ${\displaystyle u}$ ${\displaystyle p^{k}x^{p^{n-1-k}}}$

Because , is bijective in each case , we get a bijective map by adding up: ${\displaystyle \mathbb {F} _{p}\to p^{k}\mathbb {Z} /p^{k+1}\mathbb {Z} }$ ${\displaystyle x\mapsto p^{k}x^{p^{n-1-k}}}$

${\displaystyle {\begin{aligned}w_{n-1}{:}\ &\mathbb {F} _{p}^{n}\to \mathbb {Z} /p^{n}\mathbb {Z} ,\\&(x_{0},x_{1},\dots ,x_{n-1})\mapsto x_{0}^{p^{n-1}}+px_{1}^{p^{n-2}}+\dots +p^{k}x_{k}^{p^{n-1-k}}+\dots +p^{n-1}x_{n-1}\end{aligned}}}$

Let be the set together with the addition and multiplication that make an isomorphism. ${\displaystyle W_{p,n}(\mathbb {F} _{p})}$ ${\displaystyle \mathbb {F} _{p}^{n}}$ ${\displaystyle w_{n-1}}$

Now be special and with it . If two vectors and are to be added , then one obtains the equation modulo , thus . So is ${\displaystyle n=2}$ ${\displaystyle w_{1}(x_{0},x_{1})=x_{0}^{p}+px_{1}}$ ${\displaystyle (x_{0},x_{1})}$ ${\displaystyle (y_{0},y_{1})}$ ${\displaystyle (x_{0},x_{1})+(y_{0},y_{1})=(z_{0},z_{1})}$ ${\displaystyle p}$ ${\displaystyle z_{0}^{p}=x_{0}^{p}+y_{0}^{p}}$ ${\displaystyle z_{0}=x_{0}+y_{0}}$

${\displaystyle pz_{1}=px_{1}+py_{1}+x_{0}^{p}+y_{0}^{p}-(x_{0}+y_{0})^{p}}$

The polynomial

${\displaystyle X^{p}+Y^{p}-(X+Y)^{p}\in \mathbb {Z} [X,Y]}$

has divisible coefficients, so it is equal to a polynomial . So is ${\displaystyle p}$ ${\displaystyle p\cdot S_{1}'(X,Y)}$ ${\displaystyle S_{1}'(X,Y)\in \mathbb {Z} [X,Y]}$

${\displaystyle pz_{1}=p(x_{1}+y_{1}+S_{1}'(x_{0},y_{0}))}$

so overall

${\displaystyle (x_{0},x_{1})+(y_{0},y_{1})=(x_{0}+y_{0},x_{1}+y_{1}+S_{1}'(x_{0},y_{0}))}$

The associativity of addition translates into an equation

${\displaystyle S_{1}'(x_{0},y_{0})+S_{1}'(x_{0}+y_{0},z_{0})=S_{1}'(x_{0},y_{0}+z_{0})+S_{1}'(y_{0},z_{0})}$

One can easily convince oneself that this equation is already valid in. That means that you can go for any commutative ring by defining ${\displaystyle \mathbb {Z} [X,Y,Z]}$ ${\displaystyle A}$

${\displaystyle (x_{0},x_{1})+(y_{0},y_{1})=(x_{0}+y_{0},x_{1}+y_{1}+S_{1}'(x_{0},y_{0}))}$

can define the structure of an Abelian group on . The same applies to ${\displaystyle W_{p,2}(A)=A^{2}}$

${\displaystyle (x_{0},x_{1})\cdot (y_{0},y_{1})=(x_{0}y_{0},P_{1}(x_{0},x_{1},y_{0},y_{1}))}$

with so that it becomes a commutative ring with one element . ${\displaystyle P_{1}(X_{0},X_{1},Y_{0},Y_{1})=X_{0}Y_{1}+X_{1}Y_{0}+pX_{1}Y_{1}}$ ${\displaystyle W_{p,2}(A)}$ ${\displaystyle (1,0)}$

definition

Denote the set of nonnegative integers. It is also a fixed prime number. ${\displaystyle \mathbb {N} _{0}}$ ${\displaystyle p}$

There are uniquely definite polynomials for each such that for each commutative ring with one element the following applies: is a ring with addition: ${\displaystyle S_{i},P_{i}\in \mathbb {Z} [X_{0},\dots ,X_{i},Y_{0},\dots ,Y_{i}]}$ ${\displaystyle i\in \mathbb {N} _{0}}$ ${\displaystyle A}$ ${\displaystyle W_{p}(A)=A^{\mathbb {N} _{0}}}$

${\displaystyle x+y=(S_{0}(x_{0},y_{0}),S_{1}(x_{0},x_{1},y_{0},y_{1}),S_{2}(x_{0},x_{1},x_{2},y_{0},y_{1},y_{2}),\dots )}$

and multiplication

${\displaystyle x\cdot y=(P_{0}(x_{0},y_{0}),P_{1}(x_{0},x_{1},y_{0},y_{1}),\dots )}$

and for each is the picture ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle W_{p}(A)\to A,\ \ x\mapsto x_{0}^{p^{n}}+px_{1}^{p^{n-1}}+\dots +p^{k}x_{k}^{p^{n-k}}+\dots +p^{n}x_{n}}$

a ring homomorphism. is called the ring of -typical Witt vectors with entries . If there is only talk of -typical Witt vectors, there is only writing. ${\displaystyle W_{p}(A)}$ ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle W(A)}$

For is with the corresponding truncated addition and multiplication also a commutative ring with one member of the ring of the -typical Witt vectors of length . ${\displaystyle n\in \mathbb {N} _{1}}$ ${\displaystyle W_{p,n}(A)=A^{n}}$ ${\displaystyle p}$ ${\displaystyle n}$

The ring element

${\displaystyle x^{(n)}=x_{0}^{p^{n}}+px_{1}^{p^{n-1}}+\dots +p^{k}x_{k}^{p^{n-k}}+\dots +p^{n}x_{n}\in A}$

is referred to as the -th ghost component or minor component of . With the Witt polynomials ${\displaystyle n}$ ${\displaystyle x\in W_{p}(A)}$

${\displaystyle w_{p,n}(X)=X_{0}^{p^{n}}+pX_{1}^{p^{n-1}}+\dots +p^{n}X_{n}}$

can and recursively be calculated: ${\displaystyle S_{n}}$ ${\displaystyle P_{n}}$

${\displaystyle {\begin{aligned}S_{n}(X,Y)&={\frac {1}{p^{n}}}\left(w_{p,n}(X)+w_{p,n}(Y)-\sum _{k=0}^{n-1}p^{k}S_{k}^{p^{n-k}}(X,Y)\right)\\P_{n}(X,Y)&={\frac {1}{p^{n}}}\left(w_{p,n}(X)\cdot w_{p,n}(Y)-\sum _{k=0}^{n-1}p^{k}P_{k}^{p^{n-k}}(X,Y)\right)\end{aligned}}}$

Examples:

${\displaystyle {\begin{aligned}S_{0}(X,Y)&=X_{0}+Y_{0}\\S_{1}(X,Y)&=X_{1}+Y_{1}-\sum _{k=1}^{p-1}{\frac {1}{p}}{\binom {p}{k}}X_{0}^{k}Y_{0}^{p-k}\\P_{0}(X,Y)&=X_{0}Y_{0}\\P_{1}(X,Y)&=X_{0}^{p}Y_{1}+X_{1}Y_{0}^{p}+pX_{1}Y_{1}\end{aligned}}}$

The negation in the ring is also given by universal polynomials. For is: ${\displaystyle x\mapsto -x}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle p\neq 2}$

${\displaystyle -(x_{0},x_{1},x_{2},\dots )=(-x_{0},-x_{1},-x_{2},\dots )}$

For is against with ${\displaystyle p=2}$ ${\displaystyle -(x_{0},x_{1},x_{2},\dots )=(I_{0}(X),I_{1}(X),I_{2}(X),\dots )}$

${\displaystyle {\begin{aligned}I_{0}(X)&=-X_{0}\\I_{1}(X)&=-X_{0}^{2}-X_{1}\\I_{2}(X)&=-X_{0}^{4}-X_{0}^{2}X_{1}-X_{1}^{2}-X_{2}\end{aligned}}}$

The figure is multiplicative and is called the Teichmüller representative system (after Oswald Teichmüller ). ${\displaystyle \tau \colon A\to W_{p}(A),\ a\mapsto (a,0,0,\dots )}$

Evidence sketch

The recursive description provides . In order to prove the integer on the one hand, and the ring properties on the other hand, the classical proof approach shows more generally: ${\displaystyle S_{n},P_{n}\in \mathbb {Q} [X,Y]}$

Lemma. If a polynomial is (e.g. ), then there are uniquely certain integer polynomials with ${\displaystyle \phi \in \mathbb {Z} [X,Y]}$ ${\displaystyle \phi (X,Y)=X+Y}$ ${\displaystyle \phi _{n}\in \mathbb {Z} [X_{0},\dots ,X_{n},Y_{0},\dots ,Y_{n}]}$

${\displaystyle w_{p,n}(\phi _{0},\dots ,\phi _{n})=\phi (w_{p,n}(X_{0},\dots ,X_{n}),w_{p,n}(Y_{0},\dots ,Y_{n}))}$

for everyone . Corresponding versions of this statement also apply to instead of or only . ${\displaystyle n}$ ${\displaystyle X,Y,Z}$ ${\displaystyle X,Y}$ ${\displaystyle X}$

Rational uniqueness is clear, the integer proof is based on the properties and as well as the implication mentioned above ${\displaystyle w_{p,n}(X)\equiv w_{p,n-1}(X^{p})\mod p}$ ${\displaystyle f(X^{p})\equiv f(X)^{p}\mod p}$

${\displaystyle u\equiv v\mod p\implies u^{p^{n-1}}\equiv v^{p^{n-1}}\mod p^{n}}$

The ring properties of follow from the uniqueness statement of the lemma: Both and are given by polynomials that are solutions of the following equation: ${\displaystyle W_{p}(A)}$ ${\displaystyle x+(y+z)}$ ${\displaystyle (x+y)+z}$

${\displaystyle w_{p,n}(\phi _{0},\dots ,\phi _{n})=w_{p,n}(X_{0},\dots ,X_{n})+w_{p,n}(Y_{0},\dots ,Y_{n})+w_{p,n}(Z_{0},\dots ,Z_{n})}$

So these polynomials are the same.

Another approach to proof uses the identification of the ring of the large Witt vectors with the ring , see below . ${\displaystyle \Lambda (A)}$

Simple properties

• ${\displaystyle W_{p,1}(A)}$can be identified with, and with the projection . All projections are surjective ring homomorphisms, and ${\displaystyle A}$ ${\displaystyle w_{p,0}:W_{p}(A)\to A}$ ${\displaystyle x\mapsto x_{0}}$ ${\displaystyle W_{p}(A)\to W_{p,n}(A)}$

${\displaystyle W_{p}(A)=\varprojlim _{n}W_{p,n}(A)}$

(see projective Limes )

• ${\displaystyle W_{p}(\mathbb {F} _{p})=\mathbb {Z} _{p}}$ and ${\displaystyle W_{p,n}(\mathbb {F} _{p})=\mathbb {Z} /p^{n}\mathbb {Z} }$
• If in is invertible, then the mapping to the ghost components is a ring isomorphism. ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle w_{p}:W_{p}(A)\to A^{\mathbb {N} _{0}}}$
• Further examples (under both isomorphisms corresponds to the vector ): ${\displaystyle X}$ ${\displaystyle (0,1)}$

${\displaystyle {\begin{aligned}&W_{p,2}(\mathbb {Z} )\cong \mathbb {Z} [X]/(X^{2}-pX)\\&W_{p,2}(\mathbb {Z} /p^{2}\mathbb {Z} )\cong \mathbb {Z} /p^{3}\mathbb {Z} [X]/(X^{2}-pX,pX-p^{2})\end{aligned}}}$

W ( k ) for perfect bodies k

Be a perfect body of characteristic . Then there is a complete discrete evaluation ring of mixed characteristics (i.e. ), the maximum ideal of which is generated by. This property characterizes up to isomorphism. ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle W_{p}(k)}$ ${\displaystyle {\text{char}}(W_{p}(k))=0}$ ${\displaystyle p}$ ${\displaystyle W_{p}(k)}$

Witt vectors play an important role in the structure theory of complete local rings (according to IS Cohen ):

• Teichmüller-Witt theorem: If there is a complete Noetherian local ring with a residue class field , then there is exactly one homomorphism , so that the concatenation with the projection is equal to the projection . There is exactly one multiplicative section of the projection , called the Teichmüller representative system, and the figure is: ${\displaystyle (A,{\mathfrak {m}})}$ ${\displaystyle k}$ ${\displaystyle W_{p}(k)\to A}$ ${\displaystyle A\to k}$ ${\displaystyle W_{p}(k)\to k}$ ${\displaystyle \tau _{A}\colon k\to A}$ ${\displaystyle A\to k}$ ${\displaystyle W_{p}(k)\to A}$

${\displaystyle x\mapsto \sum _{n=0}^{\infty }p^{n}\cdot \tau _{A}(x_{n}^{1/p^{n}})}$

• ${\displaystyle A}$as -Algebra is isomorphic to a quotient of with . ${\displaystyle W_{p}(k)}$ ${\displaystyle W_{p}(k)[[T_{1},\dots ,T_{m}]]}$ ${\displaystyle m=\dim _{k}({\mathfrak {m}}/({\mathfrak {m}}^{2}+pA))}$
• If there is no zero divisor in , then there are elements with , so that the induced homomorphism is injective and is finitely generated as a module. ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle t_{1},\dots ,t_{d-1}}$ ${\displaystyle d=\dim(A)}$ ${\displaystyle W_{p}(k)[[T_{1},\dots ,T_{d-1}]]\to A}$ ${\displaystyle A}$ ${\displaystyle W_{p}(k)[[T_{1},\dots ,T_{d-1}]]}$
• In the special case, this means more precisely: If there is a complete discrete evaluation ring of characteristic 0 with a remainder class field , then a finite extension of is of degree if the normalized evaluation of is, that is, applies. ${\displaystyle d=1}$ ${\displaystyle (A,{\mathfrak {m}})}$ ${\displaystyle k}$ ${\displaystyle A}$ ${\displaystyle W_{p}(k)}$ ${\displaystyle e}$ ${\displaystyle e}$ ${\displaystyle p}$ ${\displaystyle pA={\mathfrak {m}}^{e}}$

For imperfect bodies, Cohen's rings take on the role of . ${\displaystyle W_{p}(k)}$

Frobenius and displacement

In characteristic p

Be a ring of characteristic . The shift is the picture ${\displaystyle A}$ ${\displaystyle p}$

${\displaystyle {\begin{aligned}V{:}\ &W_{p}(A)\to W_{p}(A),\\&(x_{0},x_{1},x_{2},\dots )\mapsto (0,x_{0},x_{1},x_{2},\dots )\end{aligned}}}$

It is a homomorphism of the additive groups. Truncation gives induced homomorphisms

${\displaystyle {\begin{aligned}&W_{p,n}(A)\to W_{p,n+1}(A),\\&(x_{0},x_{1},\dots ,x_{n-1})\mapsto (0,x_{0},x_{1},\dots ,x_{n-1})\end{aligned}}}$

The Frobenius homomorphism (based on the Frobenius homomorphism of bodies of the characteristic ) is the illustration ${\displaystyle p}$

${\displaystyle {\begin{aligned}F{:}\ &W_{p}(A)\to W_{p}(A),\\&(x_{0},x_{1},x_{2},\dots )\mapsto (x_{0}^{p},x_{1}^{p},x_{2}^{p},\dots )\end{aligned}}}$

It is a ring homomorphism that is restricted to ring homomorphisms . Be the multiplication with on . Then ${\displaystyle W_{p,n}(A)\to W_{p,n}(A)}$ ${\displaystyle [p]}$ ${\displaystyle p}$ ${\displaystyle W_{p}(A)}$

${\displaystyle F\circ V=V\circ F=[p]}$

Consequently

${\displaystyle [p](x_{0},x_{1},x_{2},\dots )=(0,x_{0}^{p},x_{1}^{p},\dots )}$

in particular

${\displaystyle p\cdot 1_{W_{p}(A)}=(0,1,0,0,0,\dots )}$

Also is

${\displaystyle V(a\cdot F(b))=V(a)\cdot b}$

Frobenius and displacement are special cases of a more general construction, see Frobenius homomorphism # displacement .

Let be the quotient field of . Then the (arithmetic) Frobenius automorphism is for the body expansion . ${\displaystyle K}$ ${\displaystyle W_{p}(\mathbb {F} _{q})}$ ${\displaystyle F}$ ${\displaystyle K/\mathbb {Q} _{p}}$

Dieudonné ring

Be a perfect body of characteristic . If you write and for the module , where the module structure is given by , then you get module homomorphisms ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle W=W_{p}(k)}$ ${\displaystyle W^{\sigma }}$ ${\displaystyle W}$ ${\displaystyle W}$ ${\displaystyle F}$

${\displaystyle F\colon W\to W^{\sigma },\ \ V\colon W^{\sigma }\to W}$

in analogy to Frobenius and shift for algebraic groups in characteristic . If more general is a module together with two module homomorphisms and , this structure can be summarized as a module for the Dieudonné ring (after Jean Dieudonné ), the non-commutative ring that is generated by and two symbols with the relations ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle W}$ ${\displaystyle F\colon M\to M^{\sigma }}$ ${\displaystyle V\colon M^{\sigma }\to M}$ ${\displaystyle D_{k}}$ ${\displaystyle W_{p}(k)}$ ${\displaystyle \mathbf {F} ,\mathbf {V} }$

${\displaystyle {\begin{aligned}&\mathbf {V} \cdot \mathbf {F} =\mathbf {F} \cdot \mathbf {V} =p\\&\mathbf {V} \cdot F(a)=a\cdot \mathbf {V} \\&\mathbf {F} \cdot a=F(a)\cdot \mathbf {F} \end{aligned}}}$

The classical Dieudonné theory is an equivalence of categories between commutative unipotent algebraic groups and certain modules. See also below . ${\displaystyle D_{k}}$

General

For any rings , the definition of the Frobenius homomorphism has to be modified: it is characterized by the equation . In particular, is the 0 th component . The Frobenius homomorphism is also a ring homomorphism in the general case. It applies ${\displaystyle A}$ ${\displaystyle w_{p,n+1}=w_{p,n}\circ F}$ ${\displaystyle F_{0}(x)=x_{0}^{p}+px_{1}}$

${\displaystyle Fx\equiv x^{p}\mod pW(A)}$

By cutting off, ring homomorphisms are obtained

${\displaystyle W_{p,n}(A)\to W_{p,n-1}(A)}$

(So ​​no longer with a goal as in the case of the characteristic ). Generally still applies ${\displaystyle W_{p,n}(A)}$ ${\displaystyle p}$

${\displaystyle FV=[p]}$

and

${\displaystyle V(a\cdot F(b))=V(a)\cdot b}$

Frobenius lifts and comonade structure

Be one - torsion ring. A Frobenius lift is a ring homomorphism with . According to Dieudonné-Cartier, there is a clearly defined continuation for a Frobenius lift that applies to all . You met . Since the Frobenius lift is available, a natural transformation is first obtained for -torsion-free rings and through universal formulas for any rings , which is characterized by . It is also called the Artin-Hasse exponential function, see also below , and defines a comonad . ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle \sigma \colon A\to A}$ ${\displaystyle \sigma (a)\equiv a^{p}\mod pA}$ ${\displaystyle s\colon A\to W_{p}(A)}$ ${\displaystyle w_{p,n}\circ s=\sigma ^{n}}$ ${\displaystyle n}$ ${\displaystyle F\circ s=s\circ \sigma }$ ${\displaystyle W_{p}(A)}$ ${\displaystyle F}$ ${\displaystyle p}$ ${\displaystyle \mu \colon W_{p}\to W_{p}\circ W_{p}}$ ${\displaystyle w_{p,n}\circ \mu =F^{n}}$ ${\displaystyle (W_{p},\mu ,w_{0})}$

The restriction to -torsion-free rings can be eliminated by switching to -Derivations : For a ring , a -derivation is a mapping , for which the mapping ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle \delta \colon A\to A}$

${\displaystyle s_{2,\delta }\colon A\to W_{p,2}(A),\ a\mapsto (a,\delta (a))}$

is a ring homomorphism. Specifically, this means that the following equations are met: ${\displaystyle \delta }$

${\displaystyle {\begin{aligned}&\delta (1)=0\\&\delta (a+b)=S_{1}(a,\delta (a),b,\delta (b))=\delta (a)+\delta (b)+\sum _{k=1}^{p-1}{\frac {1}{p}}{\binom {p}{k}}a^{k}b^{p-k}\\&\delta (ab)=P_{1}(a,\delta (a),b,\delta (b))=a^{p}\delta (b)+\delta (a)b^{p}+p\cdot \delta (a)\delta (b)\end{aligned}}}$

A derivation defined by a Frobenius lift . Is torsion-free, is obtained from a reversed Frobeniuslift a -Derivation ${\displaystyle p}$ ${\displaystyle \delta }$ ${\displaystyle w_{p,1}\circ s_{2,\delta }\colon a\mapsto a^{p}+p\cdot \delta (a)}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle \sigma }$ ${\displaystyle p}$

${\displaystyle \delta \colon A\to A,\ \delta (a)={\frac {\sigma (a)-a^{p}}{p}}}$

A ring together with a derivation is called a δ ring. ${\displaystyle p}$

The situation is analogous to ordinary derivations insofar as these can be characterized by the fact that there is a ring homomorphism. ${\displaystyle d\colon A\to A}$ ${\displaystyle A\to A[T]/T^{2},\ a\mapsto (a,d(a))}$

The coalgebras for the comonad defined above can be identified with the δ-rings. In particular, it is right adjoint to the forgetting functor from the category of δ-rings to the category of rings. There is also a dual description based on the "Plethorie" that as Endofunktor the category of rings represents . ${\displaystyle W_{p}}$ ${\displaystyle \Lambda _{p}}$ ${\displaystyle W_{p}}$

Further properties in characteristics p

Be a ring with . ${\displaystyle A}$ ${\displaystyle pA=0}$

• If there is an area of integrity , then there is, and it holds . ${\displaystyle A}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle {\text{char}}(W_{p}(A))=0}$
• The units of are exactly the elements with . ${\displaystyle W_{p}(A)}$ ${\displaystyle x}$ ${\displaystyle x_{0}\in A^{*}}$
• If there is a body, then there is a local ring with maximum ideal . In addition, if and noetherian if is perfect. ${\displaystyle A}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle V(W_{p}(A))}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle A}$
• If is surjective, then is and thus . ${\displaystyle A\to A,\ a\mapsto a^{p}}$ ${\displaystyle V(W_{p}(A))=pW_{p}(A)}$ ${\displaystyle W_{p}(A)/pW_{p}(A)\cong A}$
• Is perfect, i.e. H. bijective, then a Witt vector with the Teichmüller map can be written as -adic convergent series: ${\displaystyle A}$ ${\displaystyle a\mapsto a^{p}}$ ${\displaystyle x=(x_{0},x_{1},x_{2},\dots )}$ ${\displaystyle \tau \colon A\to W_{p}(A)}$ ${\displaystyle p}$

${\displaystyle x=\sum _{n=0}^{\infty }V^{n}\tau (x_{n})=\sum _{n=0}^{\infty }p^{n}\cdot \tau (x_{n}^{1/p^{n}})}$

• If an integrity domain is, and all prime numbers are invertible (e.g. if a field is), then one can describe the unit groups and ( formal power series or Laurent series ) as well as by , see below . ${\displaystyle A}$ ${\displaystyle \neq p}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle A[[T]]^{*}}$ ${\displaystyle A((T))^{*}}$ ${\displaystyle (A[T]/T^{n}A[T])^{*}}$ ${\displaystyle W_{p}(A)}$

Other uses

• Artin-Schreier-Witt theory : If a field has the characteristic , Abelian extensions of the exponent of can be classified with the help of the Witt vectors . ${\displaystyle K}$ ${\displaystyle p}$ ${\displaystyle p^{n}}$ ${\displaystyle K}$ ${\displaystyle W_{p,n}}$
• If a scheme is over a body of the characteristic , then there is not always a flat scheme over with . The existence of a lift to plays a role in evidence of the degeneration of the Hodge-de-Rham spectral sequence by Pierre Deligne and Luc Illusie . ${\displaystyle X}$ ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle W_{p}(k)}$ ${\displaystyle {\tilde {X}}\times _{{\text{Spec }}W_{p}(k)}{\text{Spec }}k\cong X}$ ${\displaystyle W_{p,2}(k)}$
• Is smooth, lifts exist locally. If the local lifts are equipped with a PD structure , which has the effect that an analogue of the Poincaré lemma applies, the crystalline cohomology is obtained . The crystalline cohomology groups are modules. If you tensor with the quotient field, you get a Weil cohomology , the l-adic cohomology for complementary. ${\displaystyle X/k}$ ${\displaystyle W(k)}$ ${\displaystyle l\neq p}$
• If a scheme is over , the topological space with the sheaf is again a scheme . The De-Rham-Witt complex is a suitable quotient of . For smooth the crystalline cohomology is isomorphic to the hypercohomology of . ${\displaystyle X}$ ${\displaystyle \mathbb {F} _{p}}$ ${\displaystyle X}$ ${\displaystyle W_{n}{\mathcal {O}}_{X}}$ ${\displaystyle W_{n}(X)}$ ${\displaystyle W_{n}\Omega _{X}^{\cdot }}$ ${\displaystyle \Omega _{W_{n}{\mathcal {O}}_{X}}^{\cdot }}$ ${\displaystyle X}$ ${\displaystyle H^{*}(X/W_{n})}$ ${\displaystyle W_{n}\Omega _{X}^{\cdot }}$
• There are approaches to apply Witt vectors to the analysis of the encryption method NTRUEncrypt .

Witt vectors as an algebraic group

Be a perfect body of characteristic . The Witt vectors of length form a commutative algebraic group over , as variety isomorphic to the affine space is. is a unipotent group : This follows from the filtration with subquotients or the Artin-Hasse embedding . ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle W_{n}}$ ${\displaystyle k}$ ${\displaystyle \mathbb {A} _{k}^{n}}$ ${\displaystyle W_{n}}$ ${\displaystyle V^{k}W_{n}}$ ${\displaystyle \cong \mathbb {G} _{a}}$ ${\displaystyle W_{n}(A)\to (A[T]/(T^{p^{n}+1}))^{*}}$

In characteristic 0 every commutative unipotent group is isomorphic to . In a positive way, the theory is much more complex: There are nontrivial extensions, and besides there are the possible compositional factors and (the core of the Frobenius morphism on , explicit ). ${\displaystyle \mathbb {G} _{a}^{d}}$ ${\displaystyle \mathbb {G} _{a}}$ ${\displaystyle {\underline {\mathbb {Z} /p\mathbb {Z} }}}$ ${\displaystyle \alpha _{p}}$ ${\displaystyle \mathbb {G} _{a}}$ ${\displaystyle \alpha _{p}(A)=\{a\in A:a^{p}=0\}}$

Every commutative unipotent group above is isogenic to a product of Witt vector groups. The main clause of the classical Dieudonné theory says: The functor ${\displaystyle k}$

${\displaystyle M(G)=\varinjlim _{n}{\text{Hom}}(G,W_{n})}$

defines an equivalence between the category of the commutative unipotent algebraic groups and the category of the finitely generated modules on which nilpotent acts. With the help of the Cartier duality or with Witt co-vectors one can construct an analogous equivalence for finite groups as well as for p-divisible groups . ${\displaystyle k}$ ${\displaystyle D_{k}}$ ${\displaystyle V}$ ${\displaystyle p}$

For an Abelian variety there is a canonical isomorphism of -modules . It is the core of the multiplication with on and the algebraic De Rham cohomology of . The Dieudonné module of the -divisible group of is isomorphic to the crystalline cohomology . ${\displaystyle X/k}$ ${\displaystyle D_{k}}$ ${\displaystyle M({}_{p}X)\cong H_{\text{DR}}^{1}(X)}$ ${\displaystyle {}_{p}X}$ ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle H_{\text{DR}}^{1}(X)}$ ${\displaystyle X}$ ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle H_{\text{cris}}^{1}(X/W)}$

Witt co-vectors

Just as Witt vectors are a generalization of the -adic numbers , so Witt covectors are a generalization of the group of examiners . The functor allows a uniform representation of the Dieudonné theory for finite commutative groups and divisible groups over a perfect field. ${\displaystyle p}$ ${\displaystyle W_{p}(\mathbb {F} _{p})=\mathbb {Z} _{p}=\varprojlim _{n}\mathbb {Z} /p^{n}\mathbb {Z} }$ ${\displaystyle CW(\mathbb {F} _{p})=\mathbb {Q} _{p}/\mathbb {Z} _{p}=\varinjlim _{n}{\tfrac {1}{p^{n}}}\mathbb {Z} /\mathbb {Z} }$ ${\displaystyle M(G)={\text{Hom}}(G,CW)}$ ${\displaystyle p}$ ${\displaystyle p}$

For a ring , the direct Limes is from ${\displaystyle A}$ ${\displaystyle CW^{u}(A)}$

${\displaystyle W_{p,1}(A){\stackrel {V}{\longrightarrow }}W_{p,2}(A){\stackrel {V}{\longrightarrow }}W_{p,3}(A){\stackrel {V}{\longrightarrow }}\dots }$

This becomes an Ind group scheme . The symbol is also used in older literature . is the name of the group of unipotent Witt co-vectors. ${\displaystyle CW^{u}}$ ${\displaystyle \mathop {W} _{\to }}$ ${\displaystyle CW^{u}(A)}$

The construction of the topological group of all Witt covectors is more complicated: elements in can be identified with sequences that are zero from an index. The same universal formulas can be used to explain an addition for sequences that have values ​​in a fixed nilpotent ideal from a fixed index . Equip these groups with the product topology of with discrete factors and set . The unipotent covectors form a dense subgroup of . ${\displaystyle CW(A)}$ ${\displaystyle CW^{u}(A)}$ ${\displaystyle (x_{0},x_{-1},x_{-2},\dots )}$ ${\displaystyle r}$ ${\displaystyle {\mathfrak {n}}}$ ${\displaystyle CW(A,{\mathfrak {n}},r)}$ ${\displaystyle A^{r}\times {\mathfrak {n}}^{\mathbb {N} }}$ ${\displaystyle CW(A)=\varinjlim _{{\mathfrak {n}},r}CW(A,{\mathfrak {n}},r)}$ ${\displaystyle CW^{u}(A)}$ ${\displaystyle CW(A)}$

Be a perfect ring of characteristic and an algebra. The image ${\displaystyle A}$ ${\displaystyle p}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle A}$

${\displaystyle W_{p}(A)\times W_{p,n}(B)\to W_{p,n}(B),\ \ (a,b)\mapsto W(f)(F^{1-n}a)\cdot b}$

makes a module (deviating from the module structure defined above), and with the Frobenius and the shift it becomes a module. The shift is -linear, and one obtains a -module structure on and . ${\displaystyle W_{p,n}(B)}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle W_{p,n}(B)}$ ${\displaystyle D_{A}}$ ${\displaystyle W_{p,n}(B)\to W_{p,n+1}(B)}$ ${\displaystyle D_{A}}$ ${\displaystyle D_{A}}$ ${\displaystyle CW^{u}(B)}$ ${\displaystyle CW(B)}$

Branched Witt vectors

Let be a complete discrete evaluation ring of characteristic 0 with uniformizer , whose remainder class field is a finite field with elements. Then there is exactly one functional algebra structure on for algebras such that ${\displaystyle A}$ ${\displaystyle \pi }$ ${\displaystyle q}$ ${\displaystyle A}$ ${\displaystyle W_{\pi }^{A}(B)=B^{\mathbb {N} _{0}}}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle w_{\pi ,n}:W_{\pi }^{A}(B)\to B,\ \ (x_{0},x_{1},\dots )\mapsto \sum _{k=0}^{n}\pi ^{k}x_{k}^{q^{n-k}}}$

for each is a homomorphism of -algebras. There are Frobenius and displacement operators that go through ${\displaystyle n\in \mathbb {N} _{0}}$ ${\displaystyle A}$ ${\displaystyle F_{\pi },V_{\pi }}$

${\displaystyle w_{\pi ,n}\circ V_{\pi }=\pi \cdot w_{\pi ,n-1},\ \ w_{\pi ,n}\circ F_{\pi }=w_{\pi ,n+1}}$

are characterized. For a finite extension of the remainder class field of is an unbranched extension of from degree . Branched Witt vectors take on the role of ordinary Witt vectors in transferring the Cartier theory to formal modules. ${\displaystyle \lambda /\mathbb {F} _{q}}$ ${\displaystyle A}$ ${\displaystyle W_{\pi }^{A}(\lambda )}$ ${\displaystyle A}$ ${\displaystyle [\lambda :\mathbb {F} _{q}]}$ ${\displaystyle A}$

Great Witt vectors

definition

Denote the set of positive integers. ${\displaystyle \mathbb {N} _{1}}$

There are clearly definite polynomials such that for every commutative ring with one element the following applies: is a ring with addition ${\displaystyle S_{i},P_{i}\in \mathbb {Z} [X_{1},\dots ,X_{i},Y_{1},\dots ,Y_{i}]}$ ${\displaystyle A}$ ${\displaystyle W(A)=A^{\mathbb {N} _{1}}}$

${\displaystyle x+y=(S_{1}(x_{1},y_{1}),S_{2}(x_{1},x_{2},y_{1},y_{2}),S_{3}(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}),\dots )}$

and multiplication

${\displaystyle x\cdot y=(P_{1}(x_{1},y_{1}),P_{2}(x_{1},x_{2},y_{1},y_{2}),\dots )}$

and for each is the picture ${\displaystyle n\in \mathbb {N} _{1}}$

${\displaystyle W(A)\to A,\ \ x\mapsto \sum _{d|n}dx_{d}^{n/d}}$

a ring homomorphism. The additive inverse is also given by universal polynomials. is called the ring of large or universal Witt vectors with entries . ${\displaystyle -x}$ ${\displaystyle W(A)}$ ${\displaystyle A}$

If it is a subset, so that there is also every divisor of in , then with the correspondingly truncated addition and multiplication of is also a commutative ring with one element. For one obtains the ring of the large Witt vectors of length , for with a prime number one obtains the ring of the -typical Witt vectors except for re-indexing , see below . ${\displaystyle S\subseteq \mathbb {N} _{1}}$ ${\displaystyle n\in S}$ ${\displaystyle n}$ ${\displaystyle S}$ ${\displaystyle W_{S}(A)=A^{S}}$ ${\displaystyle W(A)}$ ${\displaystyle S=\{1,\dots ,n\}}$ ${\displaystyle W_{[n]}(A)}$ ${\displaystyle n}$ ${\displaystyle S=\{1,p,p^{2},\dots \}}$ ${\displaystyle p}$ ${\displaystyle p}$

The ring element

${\displaystyle x^{(n)}=\sum _{d|n}dx_{d}^{n/d}}$

is referred to as the -th ghost component or minor component of . With the Witt polynomials ${\displaystyle n}$ ${\displaystyle x\in W(A)}$

${\displaystyle w_{n}(X)=\sum _{d|n}dX_{d}^{n/d}}$

can and recursively be calculated: ${\displaystyle S_{n}}$ ${\displaystyle P_{n}}$

${\displaystyle {\begin{aligned}S_{n}(X,Y)&={\frac {1}{n}}\left(w_{n}(X)+w_{n}(Y)-\sum _{d|n,\ d<n}dS_{d}^{n/d}(X,Y)\right)\\P_{n}(X,Y)&={\frac {1}{n}}\left(w_{n}(X)\cdot w_{n}(Y)-\sum _{d|n,\ d<n}dP_{d}^{n/d}(X,Y)\right)\end{aligned}}}$

The figure is multiplicative and is called the Teichmüller representative system. ${\displaystyle \tau \colon A\to W(A),\ a\mapsto (a,0,0,\dots )}$

${\displaystyle W}$can be represented as a set-valued functor by a polynomial ring in countably infinite indeterminates. In practice one uses concretely the ring of symmetrical polynomials , and structures of to transfer. ${\displaystyle W}$

Alternative definition with power series

Let be the multiplicative group of the formal power series with constant term 1. The mapping ${\displaystyle \Lambda (A)=1+T\cdot A[[T]]\subseteq A[[T]]^{*}}$

${\displaystyle {\begin{aligned}&W(A)\to \Lambda (A)\\&x\mapsto \prod _{n\geq 1}(1-x_{n}(-T)^{n})\end{aligned}}}$

is an isomorphism of groups . For has ${\displaystyle (W(A),{+})\cong (\Lambda (A),{\cdot })}$ ${\displaystyle x\in W(A)}$

${\displaystyle -T\cdot {\frac {\partial }{\partial T}}\log \prod _{n\geq 1}(1-x_{n}(-T)^{n})=\sum _{n\geq 1}x^{(n)}(-T)^{n}}$

as coefficients the ghost components of . ${\displaystyle x}$

Under the isomorphism, the product of two Witt vectors is mapped to: ${\displaystyle x,y\in W(A)}$

${\displaystyle \prod _{d,e\geq 1}(1-x_{d}^{m/d}y_{e}^{m/e}(-T)^{m})^{de/m}}$

where each . Write down the corresponding linkage for the multiplication in , so that there is an isomorphism of rings. A special case of the multiplication formula results ${\displaystyle m={\text{kgV}}(d,e)}$ ${\displaystyle *}$ ${\displaystyle W(A)}$ ${\displaystyle \Lambda (A)}$ ${\displaystyle (W(A),{+},{\cdot },0,1)\cong (\Lambda (A),{\cdot },{*},1,1+T)}$

${\displaystyle (1+x_{1}T)*(1+y_{1}T)=1+x_{1}y_{1}T}$

Frobenius and displacement

For every natural number there are operators and . Their effect on the ghost components is: ${\displaystyle n\in \mathbb {N} _{1}}$ ${\displaystyle F_{n}}$ ${\displaystyle V_{n}}$

${\displaystyle {\begin{aligned}&w_{m}(V_{n}(x))={\begin{cases}n\cdot w_{m/n}(x)&{\text{falls }}n|m\\0&{\text{sonst}}\end{cases}}\\&w_{m}(F_{n}(x))=w_{mn}(x)\end{aligned}}}$

In is ${\displaystyle \Lambda (A)}$

${\displaystyle {\begin{aligned}&V_{n}(f(T))=f((-1)^{n+1}T^{n})\\&F_{n}(f)((-1)^{n+1}T^{n})=\prod _{k=1}^{n}f(\zeta ^{k}T)=N_{A[[T]]/A[[T^{n}]]}(f(T))\end{aligned}}}$

A formal, primitive -th root of unity is the norm . In particular, ${\displaystyle \zeta }$ ${\displaystyle n}$ ${\displaystyle N}$

${\displaystyle F_{n}(1+aT)=1+a^{n}T}$

For is the multiplication with on , so ${\displaystyle n\in \mathbb {Z} }$ ${\displaystyle [n]}$ ${\displaystyle n}$ ${\displaystyle W(A)}$

${\displaystyle {\begin{aligned}&w_{m}([n](x))=n\cdot w_{m}(x)\\{}&[n](f(T))=f(T)^{n}\end{aligned}}}$

If an algebra is (in particular ) then there is an operator for each : ${\displaystyle A}$ ${\displaystyle R}$ ${\displaystyle R=\mathbb {Z} }$ ${\displaystyle r\in R}$ ${\displaystyle \langle r\rangle \colon W(A)\to W(A)}$

${\displaystyle {\begin{aligned}&w_{m}(\langle r\rangle (x))=r^{m}\cdot w_{m}(x)\\&\langle r\rangle (f(T))=f(rT)\end{aligned}}}$

It applies to , : ${\displaystyle m,n\in \mathbb {N} _{1}}$ ${\displaystyle r,q\in R}$

${\displaystyle {\begin{aligned}V_{m}V_{n}&=V_{mn}\\F_{m}F_{n}&=F_{mn}\\V_{m}F_{n}&=F_{n}V_{m}{\text{ falls ggT}}(m,n)=1\\F_{n}V_{n}&=[n]\\\langle r\rangle \langle s\rangle &=\langle rs\rangle \\\langle r\rangle V_{n}&=V_{n}\langle r^{n}\rangle \\F_{n}\langle r\rangle &=\langle r^{n}\rangle F_{n}\\\langle r\rangle +\langle q\rangle &=\sum _{n=1}^{\infty }V_{n}\langle s_{n}(r,q)\rangle F_{n}\end{aligned}}}$

In the last formula stands for the -th component of . ${\displaystyle s_{n}(r,q)}$ ${\displaystyle n}$ ${\displaystyle \tau (r)+\tau (q)\in W(R)}$

Relationship to the p -type Witt vectors, Artin-Hasse exponential function

Be a prime number. The figure , is a surjective homomorphism. The -typical Witt polynomials under this re-indexing are the same as the large Witt polynomials , the same applies to the ghost components . ${\displaystyle p}$ ${\displaystyle W(A)\to W_{p}(A)}$ ${\displaystyle (x_{n})_{n\in \mathbb {N} _{1}}\mapsto (x_{p^{n}})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle p}$ ${\displaystyle w_{p,n}}$ ${\displaystyle w_{p^{n}}}$

The subset is not a subring of . In certain cases, however, you can in embedding. ${\displaystyle \{x\in W(A):x_{k}\neq 0\Rightarrow k=p^{n}\}}$ ${\displaystyle W(A)}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle W(A)}$

The Artin-Hasse exponential function

${\displaystyle E_{0}(X)=\exp \left(\sum _{m=0}^{\infty }{\frac {X^{p^{m}}}{p^{m}}}\right)=\prod _{p\nmid m}(1-X^{m})^{-\mu (m)/m}}$

can be understood as an element of (i.e. the coefficients have denominators that are not divisible by denominators, see localization ; is the Möbius function ). ${\displaystyle \mathbb {Z} _{(p)}[[X]]}$ ${\displaystyle p}$ ${\displaystyle \mu }$

Is an algebra, i. H. if all prime numbers are invertible, then for a Witt vector is the element ${\displaystyle A}$ ${\displaystyle \mathbb {Z} _{(p)}}$ ${\displaystyle \neq p}$ ${\displaystyle A}$ ${\displaystyle x\in W_{p}(A)}$

${\displaystyle E(x)=\prod _{n=0}^{\infty }E_{0}(x_{n}T^{p^{n}})=\exp \left(\sum _{n}w_{p,n}(x){\frac {T^{p^{n}}}{p^{n}}}\right)\in \Lambda (A)}$

well defined. is an idempotent in , and induces a ring isomorphism . Designate the corresponding subgroup of with . Then: ${\displaystyle e=E(1)=E_{0}(T)}$ ${\displaystyle \Lambda (A)}$ ${\displaystyle E:W_{p}(A)\to \Lambda (A)\cong W(A)}$ ${\displaystyle W_{p}(A)\to e*\Lambda (A)}$ ${\displaystyle e*\Lambda (A)}$ ${\displaystyle W(A)}$ ${\displaystyle W_{(p)}(A)}$

${\displaystyle W_{(p)}(A)=\{x\in W(A):p\nmid n\implies F_{n}x=0\}}$

The ring disintegrates as a direct product of the for . For any rings is when the ratio of indicated that you do not by projection onto the components through gets separable index. ${\displaystyle W(A)}$ ${\displaystyle {\tfrac {1}{m}}V_{m}W_{(p)}(A)}$ ${\displaystyle p\nmid m}$ ${\displaystyle A}$ ${\displaystyle W(A)\cong W^{p}(W_{p}(A))}$ ${\displaystyle W^{p}(R)}$ ${\displaystyle W(A)}$ ${\displaystyle p}$

Frobenius and displacement are restricted to operators on and agree there with the operators or declared on . ${\displaystyle F_{p}}$ ${\displaystyle V_{p}}$ ${\displaystyle W_{(p)}}$ ${\displaystyle W_{p}}$ ${\displaystyle F}$ ${\displaystyle V}$

For a body of characteristic is the one-unit group of , and we obtain the isomorphism ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle \Lambda (k)}$ ${\displaystyle k((T))}$

${\displaystyle k((T))^{*}\cong k^{*}\times \mathbb {Z} \times W_{p}(k)^{I(p)}}$ With ${\displaystyle I(p)=\{n\in \mathbb {N} :p\nmid n\}}$

For every algebra is ${\displaystyle k}$ ${\displaystyle A}$

${\displaystyle W_{[n]}(A)\cong \Lambda _{[n]}(A)=\{1+a_{1}T+\dots +a_{n}T^{n}\in (A[T]/T^{n+1}A[T])^{*}\}}$

Truncating at reduces the factor to , where the smallest integer is with . So we get an isomorphism of algebraic groups (over ) ${\displaystyle n}$ ${\displaystyle {\tfrac {1}{m}}V_{m}W_{(p)}(A)}$ ${\displaystyle {\tfrac {1}{m}}V_{m}W_{p,r_{m}}(A)}$ ${\displaystyle r_{m}}$ ${\displaystyle m\cdot p^{r_{m}}\geq n+1}$ ${\displaystyle k}$

${\displaystyle \Lambda _{[n]}\cong \prod _{m\leq n,\ p\nmid m}W_{p,r_{m}}}$

The Artin-Hasse exponential mapping is also related to the comonade structure : For a perfect body of the characteristic , the concatenation of ${\displaystyle \mu \colon W_{p}\to W_{p}\circ W_{p}}$ ${\displaystyle k}$ ${\displaystyle p}$

${\displaystyle W_{p}(k)\to \Lambda (W_{p}(k)),\ \ x\mapsto \prod _{i=0}^{\infty }E_{0}(\tau (x_{i}^{p^{-i}})\cdot T)^{p^{i}}}$

same with the projection . ${\displaystyle \Lambda (W_{p}(k))\cong W(W_{p}(k))\to W_{p}(W_{p}(k))}$ ${\displaystyle \mu }$

λ rings

There is a canonical ring homomorphism that satisfies. If the Abelian group is torsion-free, this condition is uniquely determined, and for other rings it is characterized by the fact that the equation holds for a surjection with a torsion-free ring . Together with becomes a comonade . If one transfers the coalgebras to this comonad , one obtains the so-called λ-rings. ${\displaystyle \mu _{A}\colon W(A)\to W(W(A))}$ ${\displaystyle w_{n}\circ \mu =F_{n}}$ ${\displaystyle A}$ ${\displaystyle \mu }$ ${\displaystyle \mu }$ ${\displaystyle f\colon A'\to A}$ ${\displaystyle A'}$ ${\displaystyle \mu _{A}\circ W(f)=W(W(f))\circ \mu _{A'}}$ ${\displaystyle w_{1}\colon W(A)\to A}$ ${\displaystyle W}$ ${\displaystyle \Lambda (A)}$

The first ghost component corresponds in the first coefficient: ${\displaystyle w_{1}\colon W(A)\to A}$ ${\displaystyle \Lambda (A)}$

${\displaystyle w_{1}\colon \Lambda (A)\to A,\ 1+a_{1}T+a_{2}T^{2}+\dots \mapsto a_{1}}$

A pre-λ-ring is a ring together with a group homomorphism with . This condition is compatibility with the Koeins of the Comonade. If one denotes the coefficients of with , so ${\displaystyle A}$ ${\displaystyle \lambda \colon A\to \Lambda (A)}$ ${\displaystyle w_{1}\circ \lambda ={\text{id}}_{A}}$ ${\displaystyle \lambda (a)}$ ${\displaystyle \lambda ^{n}(a)}$

${\displaystyle \lambda (a)=\sum _{n=0}^{\infty }\lambda ^{n}(a)T^{n}}$

then a pre-λ structure is equivalent to specifying mappings for which satisfy the following equations: ${\displaystyle \lambda ^{n}\colon A\to A}$ ${\displaystyle n\in \mathbb {N} _{0}}$

${\displaystyle {\begin{aligned}&\lambda ^{0}(a)=1\\&\lambda ^{1}(a)=a\\&\lambda ^{n}(a+b)=\sum _{k=0}^{n}\lambda ^{k}(a)\lambda ^{n-k}(b)\end{aligned}}}$

A λ homomorphism is a ring homomorphism with , i. H. the following diagram commutates : ${\displaystyle (A,\lambda _{A})\to (B,\lambda _{B})}$ ${\displaystyle f\colon A\to B}$ ${\displaystyle \Lambda (f)\circ \lambda _{A}=\lambda _{B}\circ f}$

${\displaystyle {\begin{matrix}A&\rightarrow &B\\\downarrow &&\downarrow \\\Lambda (A)&\rightarrow &\Lambda (B)\end{matrix}}}$

The ring has as discussed above for each ring a canonical pre-λ structure. A λ-ring is a pre-λ-ring for which there is a λ-homomorphism. The above diagram is for precisely the compatibility with the multiplication of the comonade. Translated into the these are additional terms of the following form: ${\displaystyle \Lambda (A)}$ ${\displaystyle A}$ ${\displaystyle \lambda \colon A\to \Lambda (A)}$ ${\displaystyle B=\Lambda (A)}$ ${\displaystyle \lambda ^{n}}$

${\displaystyle {\begin{aligned}&\lambda ^{n}(1)=0{\text{ falls }}n>1\\&\lambda ^{n}(ab)=P_{n}(\lambda ^{1}(a),\dots ,\lambda ^{n}(a),\lambda ^{1}(b),\dots ,\lambda ^{n}(b))\\&\lambda ^{m}(\lambda ^{n}(a))=Q_{n,m}(\lambda ^{1}(a),\dots ,\lambda ^{mn}(a))\end{aligned}}}$

The (universal) polynomials describe the multiplication and, like the polynomials, have a description using elementary symmetric polynomials . ${\displaystyle P_{n}}$ ${\displaystyle *}$ ${\displaystyle \Lambda (A)}$ ${\displaystyle Q_{n,m}}$

The coassociativity of the comonad means that it is itself a λ-ring. The functor is right adjoint to the forgetting functor from the category of λ-rings to the category of rings. ${\displaystyle \Lambda }$ ${\displaystyle \Lambda (A)}$ ${\displaystyle \Lambda }$

If a λ-ring, then the ring is homomorphism ${\displaystyle (A,\lambda )}$

${\displaystyle A{\stackrel {\lambda }{\longrightarrow }}\Lambda (A)\cong W(A){\stackrel {w_{n}}{\longrightarrow }}A}$

the -th Adams operation on . It applies . For a prime number is , therefore , d. H. is a Frobenius lift. Is any ring, then the Adams operation on the λ-ring is Frobenius . ${\displaystyle n}$ ${\displaystyle \psi _{n}}$ ${\displaystyle A}$ ${\displaystyle \psi _{n}\circ \psi _{m}=\psi _{mn}}$ ${\displaystyle p}$ ${\displaystyle w_{p}=X_{1}^{p}+pX_{p}}$ ${\displaystyle \psi _{p}(a)\equiv a^{p}\mod pA}$ ${\displaystyle \psi _{p}}$ ${\displaystyle A}$ ${\displaystyle \psi _{n}}$ ${\displaystyle \Lambda (A)}$ ${\displaystyle F_{n}}$

Cartier theory

The Cartier theory (after Pierre Cartier ) is an equivalence of categories between an appropriate category of commutative formal groups above a ring and a sub-category of modules above the Cartier ring . ${\displaystyle k}$ ${\displaystyle {\text{Cart}}(k)}$

Let be the category of commutative algebras without a unit element, which consist only of nilpotent elements. For the purposes of theory, commutative formal groups are identified with functors . If there is a formal group law, the corresponding functor assigns the set with the group structure to an algebra . The formal group is the formal affine line . The functors can naturally be extended to the category of algebras, which are filtering projective limits of algebras in . ${\displaystyle {\text{Nil}}(k)}$ ${\displaystyle k}$ ${\displaystyle {\text{Nil}}(k)\to {\text{Ab}}}$ ${\displaystyle F\in k[[X,Y]]}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle (a,b)\mapsto F(a,b)}$ ${\displaystyle A\mapsto (A,{+})}$ ${\displaystyle {\widehat {\mathbb {A} }}^{1}}$ ${\displaystyle {\text{Nil}}(k)}$

The formal group of Witt vectors is the functor that assigns to an algebra the subgroup of Witt vectors in which only have a finite number of components different from 0. The corresponding subgroup consists of the elements that are a polynomial with respect to . The ring is labeled with and called the Cartier ring. The operators are restricted to endomorphisms of and thus define elements . The names of and are swapped so that the above relationships apply again because of the swapped order of multiplication. The figure , is an injective ring homomorphism. ${\displaystyle {\widehat {W}}}$ ${\displaystyle A}$ ${\displaystyle W(A)}$ ${\displaystyle {\widehat {\Lambda }}(A)\subseteq \Lambda (A)}$ ${\displaystyle T}$ ${\displaystyle {\text{End}}({\widehat {W}})^{\text{op}}}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle F_{n},V_{n},\langle r\rangle }$ ${\displaystyle {\widehat {W}}}$ ${\displaystyle V_{n},F_{n},\langle r\rangle \in {\text{Cart}}(k)}$ ${\displaystyle F_{n}}$ ${\displaystyle V_{n}}$ ${\displaystyle W(k)\to {\text{Cart}}(k)}$ ${\displaystyle \textstyle a\mapsto \sum _{n\geq 1}V_{n}\langle a_{n}\rangle F_{n}}$

Be a formal group. The following groups are of course isomorphic: ${\displaystyle G}$

• ${\displaystyle G(X\ k[[X]])=\varprojlim _{n}G(X\ k[X]/X^{n}\ k[X])}$
• the group of morphisms (not group homomorphisms, i.e. natural transformations only as set-valued functors). The group structure is induced by the group structure. ${\displaystyle {\widehat {\mathbb {A} }}^{1}\to G}$ ${\displaystyle G}$
• the group of homomorphisms ${\displaystyle {\widehat {W}}\to G}$

Its elements are called curves in , the group with . A canonical left module structure emerges from the last description . ${\displaystyle G}$ ${\displaystyle C(G)}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle C(G)}$

The power series group can be identified with. The Witt polynomials correspond to the group homomorphism that is induced by the logarithmic derivative on . ${\displaystyle \Lambda (k)}$ ${\displaystyle C({\widehat {\mathbb {G} }}_{m})}$ ${\displaystyle C({\widehat {\mathbb {G} }}_{m})\to C({\widehat {\mathbb {G} }}_{a})}$ ${\displaystyle X\ k[[X]]}$

In the operation of of inducing the operation of of . For consider again a formal -th root of unity and add to the sum of the curves obtained by for . For a curve is determined by the image of the coordinate. If one identifies with the corresponding element in , the effects of agree with those defined above (without interchanging and ). ${\displaystyle C(G)=G(X\ k[[X]])}$ ${\displaystyle V_{n}\in {\text{Cart}}(k)}$ ${\displaystyle X\mapsto X^{n}}$ ${\displaystyle \langle r\rangle }$ ${\displaystyle X\mapsto rX}$ ${\displaystyle F_{n}}$ ${\displaystyle n}$ ${\displaystyle \zeta }$ ${\displaystyle G}$ ${\displaystyle X\mapsto \zeta ^{i}X^{1/n}}$ ${\displaystyle i=0,1,\dots ,n-1}$ ${\displaystyle G={\widehat {\mathbb {G} }}_{m}}$ ${\displaystyle f\in X\ k[[X]]}$ ${\displaystyle 1+f}$ ${\displaystyle \Lambda (k)}$ ${\displaystyle F_{n},V_{n},\langle r\rangle }$ ${\displaystyle F_{n}}$ ${\displaystyle V_{n}}$

Both and have natural topologies. The main theorem of Cartier theory is that an equivalence between a category of formal groups over and a category of topological -modules induces. The inverse functor assigns a suitably constructed tensor product to a module . ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle C(G)}$ ${\displaystyle C}$ ${\displaystyle k}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle M}$ ${\displaystyle {\widehat {W}}\otimes _{{\text{Cart}}(k)}M}$

Let be a prime number and an algebra, i.e. H. every prime number is invertible in. Then an idempotent is in , bet . For a module is the subgroup of elements with for all . Such elements are called typical. ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle \mathbb {Z} _{(p)}}$ ${\displaystyle \neq p}$ ${\displaystyle k}$ ${\displaystyle \textstyle \epsilon _{p}=\sum _{p\nmid n}{\frac {\mu (n)}{n}}V_{n}F_{n}}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle {\text{Cart}}_{p}(k)=\epsilon _{p}{\text{Cart}}(k)\epsilon _{p}}$ ${\displaystyle {\text{Cart}}(k)}$ ${\displaystyle M}$ ${\displaystyle \epsilon _{p}M\subseteq M}$ ${\displaystyle m\in M}$ ${\displaystyle F_{n}m=0}$ ${\displaystyle p\nmid n}$ ${\displaystyle p}$

For a formal group, let the group of the -typical curves ( the formal group for the -typical Witt vectors , analogous to ). Then induces an equivalence between the category of formal groups via as above and a category of topological modules. As before, the inverse functor is a tensor product . ${\displaystyle G}$ ${\displaystyle C_{p}(G)=\epsilon _{p}C(G)\cong {\text{Hom}}({\widehat {W}}_{p},G)}$ ${\displaystyle p}$ ${\displaystyle {\widehat {W}}_{p}}$ ${\displaystyle p}$ ${\displaystyle W_{p}}$ ${\displaystyle {\widehat {W}}}$ ${\displaystyle C_{p}}$ ${\displaystyle k}$ ${\displaystyle {\text{Cart}}_{p}(k)}$ ${\displaystyle {\widehat {W}}_{p}\otimes _{{\text{Cart}}_{p}(k)}M}$

For a perfect body of characteristic , the dieudonné ring can be identified with a dense sub-ring in . Under appropriate conditions of Dieudonné module is dual to the module -typical curves . ${\displaystyle k}$ ${\displaystyle p}$ ${\displaystyle D_{k}}$ ${\displaystyle {\text{Cart}}_{p}(k)}$ ${\displaystyle p}$ ${\displaystyle {\text{Hom}}_{W_{p}(k)}(M(G),W_{p}(k))\cong C_{p}(G)}$

Generalizations

• Colette Schoeller extended parts of the typical theory, namely the construction of the Cohen ring and the classification of the unipotent groups, to imperfect bodies. ${\displaystyle p}$
• Andreas Dress and Christian Siebeneicher have specified the construction of a ring from a pro-end group and a ring , so that it is isomorphic to the completed Burnside ring of . For results , for results . ${\displaystyle W_{G}(A)}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle W_{G}(\mathbb {Z} )}$ ${\displaystyle G}$ ${\displaystyle G=\mathbb {Z} _{p}}$ ${\displaystyle W_{p}(A)}$ ${\displaystyle G={\widehat {\mathbb {Z} }}}$ ${\displaystyle W(A)}$

literature

Textbooks and review articles

• George Mark Bergman: Ring Schemes: the Witt Scheme . In: David Mumford, Lectures on Curves on an Algebraic Surface . Princeton University Press, Princeton 1966, ISBN 978-0-691-07993-6 .
• Nicolas Bourbaki : Eléments de mathématique. Algèbre commutative. Chapitres 8 and 9 . Springer, Berlin 2006, ISBN 978-3-540-33942-7 .
• Michiel Hazewinkel : Witt Vectors. Part 1 . In: Michiel Hazewinkel (Ed.): Handbook of Algebra, Volume 6 . Elsevier, Amsterdam 2009, ISBN 978-0-444-53257-2 , pp. 319-472 , arxiv : 0804.3888 .
• Jean-Pierre Serre : Local Fields . Springer, Berlin 1979, ISBN 3-540-90424-7 .

Further topics

• Pierre Berthelot : Exposé V. Généralités sur les λ-anneaux . In: Pierre Berthelot, Alexandre Grothendieck, Luc Illusie (eds.): Séminaire de Géométrie Algébrique du Bois Marie . 1966-67 - Théorie des intersections et théorème de Riemann-Roch (SGA 6) (=  Lecture notes in mathematics ). tape 225 . Springer, Berlin 1971, ISBN 978-3-540-05647-8 , pp. 297-364 .
• James Borger: The basic geometry of Witt vectors, I: The affine case . In: Algebra and Number Theory . tape 5 , no. 2 , 2011, p. 231-285 , doi : 10.2140 / ant.2011.5.231 , arxiv : 0801.1691 ( maths.anu.edu.au ).
• James Borger, Ben Wieland: Plethystic Algebra . In: Advances in Mathematics . tape 194 , no. 2 , 2005, p. 246-283 , arxiv : math.AC/0407227 ( maths.anu.edu.au ).
• Michel Demazure : Lectures on p-Divisible Groups (=  Lecture notes in mathematics . Volume 302 ). Springer-Verlag, Berlin 1972, ISBN 3-540-06092-8 .
• Michel Demazure, Pierre Gabriel : Groupes algébriques. Tome 1 . North-Holland, Amsterdam 1970, ISBN 978-0-7204-2034-0 .
• Michiel Hazewinkel: Formal Groups and Applications . Academic Press, New York 1978, ISBN 0-12-335150-2 .
• Luc Illusie : Complexe de De Rham-Witt et cohomologie cristalline . In: Annales scientifiques de l'École Normale Supérieure, Sér. 4 . tape 12 , no. 4 , 1979, p. 501-661 ( numdam.org ).
• Jean-Pierre Serre: Algebraic Groups and Class Fields . Springer, Berlin 1988, ISBN 3-540-96648-X .

Web links

• Witt vector . In: IV Dolgachev (originator): Encyclopedia of Mathematics .

Individual evidence

1. ↑ Original work: Ernst Witt: Cyclical fields and algebras of the characteristic p of degree p ⁿ . Structure of discretely valued perfect bodies with a perfect residue class field of characteristic p . In: J. Reine Angew. Math. Band 176 , 1936, pp. 126-140 .
2. ↑ James Borger has put forward arguments for preferring the numbering for , see Borger 2011, 2.5 ${\displaystyle W_{p,n}(A)=A^{n+1}}$ ${\displaystyle n=0,1,2,\dots }$
3. ^ Hazewinkel 2009, Theorem 5.2. Bourbaki instead uses a characterization of the image of and obtains the universal polynomials by specializing in polynomial rings . ${\displaystyle w:W_{p}(A)\to A^{\mathbb {N} _{0}}}$ ${\displaystyle A}$
4. ↑ Demazure-Gabriel, V §4, 2.1
5. ↑ Bourbaki, IX §1 Proposition 3. Hazewinkel 2009, 5.30
6. ↑ Illusie 1979, p. 508. Bourbaki, IX §1 Ex. 14, 15
7. ↑ Alexandru Buium : Arithmetic Differential Equations . AMS, Providence 2005, ISBN 0-8218-3862-8 .
8. ^ André Joyal : δ-anneaux et vecteurs de Witt . In: CR Math. Acad. Sci., Soc. R. Can. tape 7 , 1985, pp. 177-182 .
9. ↑ Borger-Wieland 2005
10. ^ Bourbaki, IX §1 Ex. 9
11. ^ Pierre Deligne, Luc Illusie: Relèvements modulo p² et décomposition du complexe de Rham . In: Inv. Math. Band 89 , 1987, pp. 247-270 .
12. ↑ Illusie 1979, chap. II. The following work by Spencer Bloch can be seen as a forerunner of this construction , in which he considers a complex of curves in the sense of the Cartier theory in the K-theory : Spencer Bloch: Algebraic K-Theory and Crystalline Cohomology . In: Publ. Math. de l'IHÉ.S. tape 47 , 1977, pp. 187-268 . A systematic consideration of thickenings of the type and an adjoint construction can be found in: James Borger: The basic geometry of Witt vectors. II: Spaces . In: Mathematical Annals . tape ${\displaystyle W_{n}(X)}$ 351 , no. 4 , 2011, p. 877-933 , doi : 10.1007 / s00208-010-0608-1 , arxiv : 1006.0092 .
13. ^ JH Silverman , Nigel Smart , F. Vercauteren: An algebraic approach to NTRU via Witt vectors and overdetermined systems of nonlinear equations . In: Carlo Blundo, Stelvio Cimato (Ed.): Security in Communication Networks: 4th International Conference, SCN 2004, Amalfi, Italy, September 8-10, 2004, Lecture Notes in Computer Science 3352 . Springer-Verlag, 2005, p. 278-293 .
14. ↑ Demazure-Gabriel, V §3 6.11. Serre 1988, VII §2 10
15. ↑ Demazure-Gabriel, V §1 4
16. ↑ Demazure 1972, III §6-8
17. ^ Corollary 5.11 in: Tadao Oda : The first de Rham cohomology group and Dieudonné modules . In: Annales scientifiques de l'École Normale Supérieure, Sér. 4 . tape 2 , no. 1 , 1969, p. 63-135 ( online ).
18. ↑ Luc Illusie: Crystalline Cohomology . In: Uwe Jannsen et al. (Ed.): Motives (=  Proceedings of Symposia in Pure Mathematics ). tape 55 , no. 1 . American Mathematical Society, Providence 1994, pp. 43-70 .
19. ^ Jean-Marc Fontaine : Groupes p-divisibles sur les corps locaux . In: Astérisque . tape 47-48 , 1977. Pierre Berthelot: Théorie de Dieudonné sur un anneau de valuation parfait . In: Annales scientifiques de l'École Normale Supérieure, Sér. 4 . tape 13 , no. 2 , 1980, p. 225-268 ( online ).
20. ↑ Bourbaki, IX §1 Ex. 23
21. ↑ Bourbaki, IX §1 Ex. 24
22. ↑ Bourbaki, IX §1 Ex. 25
23. ↑ Hazewinkel 1978, 25.3 and 25.6.4. Further generalizations there and in Borger 2011.
24. ↑ Hazewinkel 2009 chap. 10. Borger-Wieland 2005
25. ↑ The choice of sign is inconsistent, three variants can be found in Bourbaki, Hazewinkel and Bergman. With the choice made here (as in Bourbaki or Berthelot 1971) the connection with λ-rings becomes easier.
26. ↑ Bourbaki, IX §1 Ex. 47. Cf. Hazewinkel 2009, (9.15) and (9.27)
27. ↑ Bourbaki, IX §1 Ex. 47. Cf. Hazewinkel 2009, chap. 13
28. ↑ Bourbaki, IX §1 Ex. 40. Demazure-Gabriel, V §5, 3.4
29. ^ Roland Auer: A functorial property of nested Witt vectors . In: Journal of Algebra . tape 252 , no. 2 , 2002, p. 293-299 .
30. ↑ Serre 1988, V §3 Proposition 9
31. ↑ Hazewinkel 1978 May 17
32. ↑ Bourbaki, IX §1 Ex. 41. Hazewinkel 2009 16:59
33. ↑ Bourbaki, IX §1 Ex. 48. Hazewinkel 2009, 16.22
34. ↑ Review article for the entire section: Lawrence Breen: Rapport sur la Théorie de Dieudonné . In: Astérisque . tape 63 , 1979, pp. 39-66 . See also: Thomas Zink : Cartier theory of commutative formal groups . Teubner, 1984, ISBN 3-322-00647-6 . Michel Lazard: Commutative Formal Groups . Springer, Berlin 1975, ISBN 3-540-07145-8 . Ching-Li Chai: Notes on Cartier Theory . ( math.upenn.edu [PDF]).
35. ^ Colette Schoeller: Groupes affines, commutatifs, unipotents sur un corps non parfait . In: Bulletin de la SMF Band 100 , 1972, p. 241-300 ( online ). ; Bourbaki, IX §2 Ex. 10
36. ^ Andreas Dress, Christian Siebeneicher: The Burnside ring of profinite groups and the Witt vector construction . In: Adv. Math. Band 70 , 1988, pp. 87-132 .