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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{ring of Witt vectors} \begin{quote}% This article is about of groups of Witt vectors and rings of Witt vectors; which are often called just Witt groups and Witt rings. However, there is also a different notion of [[Witt group]] and [[Witt ring]]. \end{quote} \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak \noindent\hyperlink{Definituion}{Definition}\dotfill \pageref*{Definituion} \linebreak \noindent\hyperlink{DefinitionInComponents}{In components}\dotfill \pageref*{DefinitionInComponents} \linebreak \noindent\hyperlink{the_ring_structure}{The ring structure}\dotfill \pageref*{the_ring_structure} \linebreak \noindent\hyperlink{the_witt_polynomials_the__the__phantom_components}{The Witt polynomials, the $\Sigma_i$, the $\Pi_i$, phantom components}\dotfill \pageref*{the_witt_polynomials_the__the__phantom_components} \linebreak \noindent\hyperlink{AdditionAndMultiplicationPolynomials}{The addition- and multiplication polynomials}\dotfill \pageref*{AdditionAndMultiplicationPolynomials} \linebreak \noindent\hyperlink{DefinitionUniversalCharacterization}{Universal characterization}\dotfill \pageref*{DefinitionUniversalCharacterization} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{operations_on_the_padic_witt_vectors}{Operations on the p-adic Witt vectors}\dotfill \pageref*{operations_on_the_padic_witt_vectors} \linebreak \noindent\hyperlink{the_shift_map}{The shift map}\dotfill \pageref*{the_shift_map} \linebreak \noindent\hyperlink{the_restriction_map}{The restriction map}\dotfill \pageref*{the_restriction_map} \linebreak \noindent\hyperlink{the_frobenius_morphism}{The Frobenius morphism}\dotfill \pageref*{the_frobenius_morphism} \linebreak \noindent\hyperlink{duality_of_finite_witt_groups}{Duality of finite Witt groups}\dotfill \pageref*{duality_of_finite_witt_groups} \linebreak \noindent\hyperlink{properties_of_the_witt_group}{Properties of the Witt group}\dotfill \pageref*{properties_of_the_witt_group} \linebreak \noindent\hyperlink{properties_of_the_witt_ring}{Properties of the Witt ring}\dotfill \pageref*{properties_of_the_witt_ring} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{basic_examples}{Basic examples}\dotfill \pageref*{basic_examples} \linebreak \noindent\hyperlink{lubintate_ring}{Lubin-Tate ring}\dotfill \pageref*{lubintate_ring} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{original_texts_and_classical_surveys}{Original texts and classical surveys}\dotfill \pageref*{original_texts_and_classical_surveys} \linebreak \noindent\hyperlink{modern_surveys}{Modern surveys}\dotfill \pageref*{modern_surveys} \linebreak \noindent\hyperlink{further_development_of_the_theory}{Further development of the theory}\dotfill \pageref*{further_development_of_the_theory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Rings of \emph{Witt vectors} are the [[co-free functor|co-free]] [[Lambda-rings]]. Depending on whether one defines the latter via Frobenius lifts at a single [[prime number]] $p$ one speaks of \emph{$p$-typical Witt vectors}, or of \emph{big Witt vectors} if all primes are considered at once. In [[arithmetic geometry]] the impact of rings of Witt vectors $W(R)$ of a given ring $R$ is that they are like rings [[formal power series]] on $Spec(R)$, such as rings of [[p-adic numbers]]. For more on this see at \emph{[[arithmetic jet space]]} and at \emph{[[Borger's absolute geometry]]}. In components, a \emph{Witt vector is an infinite sequence of elements of a given [[commutative ring]] $k$. There is a [[ring]] structure on the set $W(k)$ of Witt vectors of $k$ and $W(k)$ is therefore called \emph{the Witt ring of $k$}. The multiplication is defined by means of [[Witt polynomials]] $w_i$ for every [[natural number]] $i$. If the [[characteristic]] of $k$ is $0$ the Witt ring of $k$ is sometimes called \emph{universal Witt ring} to distinguish it from the case where $k$ is of prime characteristic and a similar but different construction is of interest.} A [[Witt vector|p-adic Witt vector]] is an infinite sequence of elements af a commutative ring of [[prime]] [[characteristic]] $p$. There exists a ring structure whose construction parallels that in characteristic $0$ except that only [[Witt polynomials]] $w_{p^l}$ whose index is a power of $p$ are taken. More abstractly, the ring of Witt vectors carries the structure of a [[Lambda-ring]] and the construction $W \colon k\mapsto W(k)$ of the Witt ring $W(k)$ on a [[commutative ring]] $k$ is [[right adjoint]] to the [[forgetful functor]] from [[Lambda-rings]] to [[commutative rings]]. Hence rings of Witt vectors are the \emph{[[co-free functor|co-free]] [[Lambda-rings]]}. Moreover $W(-)$ is [[representable functor|representable]] by [[symmetric function|Symm]], the ring of [[symmetric functions]] which is a [[Hopf algebra]] and consequently $W$ is a [[group scheme]]. This is explained at [[Lambda-ring]]. The construction of Witt vectors gives a functorial way to lift a commutative ring $A$ of [[prime]] [[characteristic]] $p$ to a commutative ring $W(A)$ of [[characteristic]] 0. Since this construction is functorial, it can be applied to the [[structure sheaf]] of an [[algebraic variety]]. In interesting special cases the resulting ring $W(A)$ has even more desirable properties: If $A$ is a [[perfect field]] then $W(A)$ is a [[discrete-valuation ring|discrete valuation]]. This is partly due to the fact that the construction of $W(A)$ involves a ring of [[power series]] and a ring of power series over a field is always a discrete valuation ring. There is a generalization, $W_G$, to any profinite group, $G$, due to Dress and Siebeneicher (\hyperlink{DS88}{DS88}), known as [[Witt-Burnside functors|Witt-Burnside functor]]. There is a generalization to [[non-commutative Witt vectors]], however these only carry a group- but no ring structure. The [[Lubin-Tate ring]] in [[Lubin-Tate theory]] is a [[polynomial ring]] on a ring of Witt vectors and this way Witt vectors control much of [[chromatic homotopy theory]]. \hypertarget{motivation}{}\subsection*{{Motivation}}\label{motivation} In an \emph{expansion} of a $p$-adic number $a=\Sigma a_i p^i$ the $a^i$ are called \emph{digits}. Usually these digits are defined to be taken elements of the set $\{0,1,\dots,p-1\}$. Equivalently the digits can be defined to be taken from the set $T_p:=\{x|x^{p-1}=1\}\cup \{0\}$. Elements from this set are called \emph{Teichm\"u{}ller digits} or \emph{Teichm\"u{}ller representatives}. The set $T$ is in bijection with the [[finite field]] $F_p$. The set $W(F_p)$ of (countably) infinite sequences of elements in $F_p$ hence is in bijection to the set $\mathbb{Z}_p$ of $p$-adic integers. There is a ring structure on $W(F_p)$ called \emph{Witt ring structure} such that all `'truncated expansion polynomials'` $\Phi_n=X^{p^n}+pX^{p^{n-1}}+p^2X^{p^{n-2}}+\dots +p^n X$ called \emph{Witt polynomials} are morphisms \begin{displaymath} \Phi_n:W(F_p)\to \mathbb{Z}_p \end{displaymath} of groups. \hypertarget{Definituion}{}\subsection*{{Definition}}\label{Definituion} We first give the \begin{itemize}% \item \hyperlink{DefinitionInComponents}{Explicit definition in components} \end{itemize} and then discuss the \begin{itemize}% \item \href{DefinitionUniversalCharacterization}{Universal characterization} \end{itemize} \hypertarget{DefinitionInComponents}{}\subsubsection*{{In components}}\label{DefinitionInComponents} \hypertarget{the_ring_structure}{}\paragraph*{{The ring structure}}\label{the_ring_structure} \begin{defn} \label{}\hypertarget{}{} Let $k$ be a [[commutative ring]]. If the [[characteristic]] of $k$ is $0$ then the Witt ring $W(k)$ of $k$ is defined defined by the addition \begin{displaymath} (a_1, a_2, \ldots, )+(b_1, b_2, \ldots) :=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots) \end{displaymath} and the multiplication \begin{displaymath} (a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots ) :=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots ) \end{displaymath} If $k$ is of prime characteristic $p$ we index the defining formulas by $p^1,p^2,p^3,\dots$ instead of $1,2,3,\dots$. Here the $\Sigma_i$ are called \emph{addition polynomials} and the $\Pi_i$ are called \emph{multiplication polynomials}, these are described \hyperlink{AdditionAndMultiplicationPolynomials}{below}. \end{defn} \hypertarget{the_witt_polynomials_the__the__phantom_components}{}\paragraph*{{The Witt polynomials, the $\Sigma_i$, the $\Pi_i$, phantom components}}\label{the_witt_polynomials_the__the__phantom_components} let $x_1, x_2, \ldots$ be a collection of [[variables]]. We can define an infinite collection of [[polynomial]]s in $\mathbb{Z}[x_1, x_2, \ldots ]$ using the following formulas: $w_1(X)=x_1$ $w_2(X)=x_1^2+2x_2$ $w_3(X)=x_1^3+3x_3$ $w_4(X)=x_1^4+2x_2^2+4x_4$ and in general $\displaystyle w_n(X)=\sum_{d|n} dx_d^{n/d}$. The value $w_n(w)$ of the $n$-th Witt polynomial in some element $w\in W(k)$ of the Witt ring of $k$ is sometimes called the \emph{$n$-th phantom component of $w$} or \emph{the $n$-th ghost component of $w$.} Now let $\phi(z_1, z_2)\in\mathbb{Z}[z_1, z_2]$. This just an arbitrary two variable polynomial with coefficients in $\mathbb{Z}$. We can define new polynomials $\Phi_i(x_1, \ldots x_i, y_1, \ldots y_i)$ such that the following condition is met $\phi(w_n(x_1, \ldots ,x_n), w_n(y_1, \ldots , y_n))=w_n(\Phi_1(x_1, y_1), \ldots , \Phi_n(x_1, \ldots x_n, y_1, \ldots , y_n))$. In short we'll notate this $\phi(w_n(X),w_n(Y))=w_n(\Phi(X,Y))$. The first thing we need to do is make sure that such polynomials exist. Now it isn't hard to check that the $x_i$ can be written as a $\mathbb{Q}$-linear combination of the $w_n$ just by some linear algebra. $x_1=w_1$, and $x_2=\frac{1}{2}w_2+\frac{1}{2}w_1^2$, etc. so we can plug these in to get the existence of such polynomials with coefficients in $\mathbb{Q}$. It is a fairly tedious lemma to prove that the coefficients $\Phi_i$ are actually in $\mathbb{Z}$, so we won't detract from the construction right now to prove it. \hypertarget{AdditionAndMultiplicationPolynomials}{}\paragraph*{{The addition- and multiplication polynomials}}\label{AdditionAndMultiplicationPolynomials} Define yet another set of polynomials $\Sigma_i$, $\Pi_i$ and $\iota_i$ by the following properties: $w_n(\Sigma)=w_n(X)+w_n(Y)$, $w_n(\Pi)=w_n(X)w_n(Y)$ and $w_n(\iota)=-w_n(X)$. We now can construct $W(A)$, the ring of generalized Witt vectors over $A$. Define $W(A)$ to be the set of all infinite sequences $(a_1, a_2, \ldots)$ with entries in $A$. Then we define addition and multiplication by $(a_1, a_2, \ldots, )+(b_1, b_2, \ldots)=(\Sigma_1(a_1,b_1), \Sigma_2(a_1,a_2,b_1,b_2), \ldots)$ and $(a_1, a_2, \ldots )\cdot (b_1, b_2, \ldots )=(\Pi_1(a_1,b_1), \Pi_2(a_1,a_2,b_1,b_2), \ldots )$. \hypertarget{DefinitionUniversalCharacterization}{}\subsubsection*{{Universal characterization}}\label{DefinitionUniversalCharacterization} \begin{theorem} \label{}\hypertarget{}{} The assignment \begin{displaymath} W \;\colon\; k\mapsto W(k) \end{displaymath} is a [[functor]] \begin{displaymath} W \;\colon\; CRing \longrightarrow \Lambda Ring \end{displaymath} from the [[category]] of [[commutative rings]] to that of [[Lambda-rings]]. Composed with the [[forgetful functor]] \begin{displaymath} U \;\colon\; \Lambda Ring \longrightarrow CRing \end{displaymath} this is the unique [[endofunctor]] $W \;\colon\; CRing \longrightarrow CRing$ such that all [[Witt polynomials]] \begin{displaymath} w_n : \begin{cases} W(A)\to A \\ a\mapsto w_n(a) \end{cases} \end{displaymath} are [[homomorphisms]] of [[rings]]. \end{theorem} \begin{proof} There is a nice trick to prove that $W(A)$ is a ring when $A$ is a $\mathbb{Q}$-algebra. Just define $\psi: W(A)\to A^\mathbb{N}$ by $(a_1, a_2, \ldots) \mapsto (w_1(a), w_2(a), \ldots)$. This is a [[bijection]] and the addition and multiplication is taken to component-wise addition and multiplication, so since this is the standard ring structure we know $W(A)$ is a ring. Also, $w(0,0,\ldots)=(0,0,\ldots)$, so $(0,0,\ldots)$ is the additive identity, $W(1,0,0,\ldots)=(1,1,1,\ldots)$ which shows $(1,0,0,\ldots)$ is the multiplicative identity, and $w(\iota_1(a), \iota_2(a), \ldots)=(-a_1, -a_2, \ldots)$, so we see $(\iota_1(a), \iota_2(a), \ldots)$ is the additive inverse. We can actually get this idea to work for any characteristic $0$ ring by considering the embedding $A\to A\otimes\mathbb{Q}$. We have an induced injective map $W(A)\to W(A\otimes\mathbb{Q})$. The addition and multiplication is defined by polynomials over $\mathbb{Z}$, so these operations are preserved upon tensoring with $\mathbb{Q}$. We just proved above that $W(A\otimes\mathbb{Q})$ is a ring, so since $(0,0,\ldots)\mapsto (0,0,\ldots)$ and $(1,0,0,\ldots)\mapsto (1,0,0,\ldots)$ and the map preserves inverses we get that the image of the embedding $W(A)\to W(A\otimes \mathbb{Q})$ is a subring and hence $W(A)$ is a ring. Lastly, we need to prove this for positive [[characteristic]] rings. Choose a characteristic $0$ ring that surjects onto $A$, say $B\to A$. Then since the induced map again preserves everything and $W(B)\to W(A)$ is [[surjective]], the image is a ring and hence $W(A)$ is a ring. \end{proof} \begin{prop} \label{}\hypertarget{}{} The construction of the ring of Witt vectors $W(k)$ on a given [[commutative ring]] $k$ is the [[right adjoint]] to the [[forgetful functor]] $U$ from [[Lambda-rings]] to [[commutative rings]] \begin{displaymath} (U \dashv W) \;\colon\; CRing \stackrel{\overset{U}{\leftarrow}}{\underset{W}{\longrightarrow}} \Lambda Ring \,. \end{displaymath} Hence rings of Witt-vectors are the \emph{[[co-free functors|co-free]] [[Lambda-rings]]}. \end{prop} This statement appears in (\hyperlink{Hazewinkel08}{Hazewinkel 08, p. 87, p. 97}). \begin{remark} \label{}\hypertarget{}{} On the other hand, the [[free construction|free]] [[Lambda-ring]] (on one generator) (hence the [[left adjoint]] construction to the [[forgetful functor]]) is the ring of [[symmetric functions]]. \end{remark} This statement appears in (\hyperlink{Hazewinkel08}{Hazewinkel 08, p. 98}). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{operations_on_the_padic_witt_vectors}{}\subsubsection*{{Operations on the p-adic Witt vectors}}\label{operations_on_the_padic_witt_vectors} On untruncated $p$-adic Witt vectors there are two operations, the [[Frobenius morphism]] and the [[Verschiebung morphism]] satisfying relations (\hyperlink{FVrelations}{Lemma 1}) being constitutive for the definition of the [[Dieudonné ring]]: In fact the Dieudonn\'e{} ring is generated by two objects satisfying these relations. Also the $n$-truncations of a Witt ring are rings since by definition the ring operations (addition and multiplication) of the first $n$ components only involve the first $n$ components. We have $W\simeq lim_n W_n$ and the projection map $W(A)\to W_n(A)$ is a [[ring homomorphism]]. We also have operations on the truncated Witt rings. \hypertarget{the_shift_map}{}\paragraph*{{The shift map}}\label{the_shift_map} For $W(k)$ as for every $k$-scheme we have the [[Verschiebung morphism]]. It is defined to be the adjoint operation to the [[Frobenius morphism]]. For $W(k)$ the Verschiebung morphism coincides with the shift $(a_0, a_1,\dots)\mapsto (0, a_0, a_1,\dots)$ For the truncated Witt rings and the shift operation $V:W_n(k)\to W_{n+1}(k)$ the Verschiebung morphism equals the $VR=RV$ where $R$ is the restriction map. \hypertarget{the_restriction_map}{}\paragraph*{{The restriction map}}\label{the_restriction_map} The restriction map $R: W_{n+1}(A)\to W_n(A)$ is given by $(a_0, \ldots, a_n)\mapsto (a_0, \ldots, a_{n-1})$. \hypertarget{the_frobenius_morphism}{}\paragraph*{{The Frobenius morphism}}\label{the_frobenius_morphism} The Frobenius endomorphism $F: W_n(A)\to W_n(A)$ is given by $(a_0, \ldots , a_{n-1})\mapsto (a_0^p, \ldots, a_{n-1}^p)$. This is also a ring map, but only because of our necessary assumption that $A$ is of characteristic $p$. Just by brute force checking on elements we see a few relations between these operations, namely that $V(x)y=V(x F(R(y)))$ and $RVF=FRV=RFV=p$ the multiplication by $p$ map. \hypertarget{duality_of_finite_witt_groups}{}\subsubsection*{{Duality of finite Witt groups}}\label{duality_of_finite_witt_groups} For a $k$-ring $R$ let $W^\prime(R)$ denote the [[ideal]] in $W(R)$ consisting of sequences $x=(x_n)_n$ of nilpotent elements in $W(R)$ such that $x_n=0$ for large $n$. Let $E$ denote the [[Artin-Hasse exponential]]. Then we have $E(x,1)$ is a polynomial for $x\in W^\prime(R)$ and \begin{displaymath} E(-,1):\begin{cases}W^\prime\to \mu_k \\ w\mapsto E(w,1)\end{cases} \end{displaymath} is a morphism of [[group schemes]] to the [[multiplicative group scheme]] $\mu_k$. \begin{prop} \label{}\hypertarget{}{} a) $W^\prime (R)$ is an ideal in $W(R)$. b) $E(-,1): W^\prime\to \mu_k$ is an morphism of [[group schemes]]. c) The morphism \begin{displaymath} \begin{cases} W\times W^\prime\to \mu_k \\ (x,y)\mapsto E(xy,1) \end{cases} \end{displaymath} is bilinear and gives an isomorphism of [[group schemes]] \begin{displaymath} W^\prime\to D(W) \end{displaymath} where $D$ denotes the [[Cartier dual]] of $W$ (maybe it is equivalently the [[Pontryagin dual]] of the underlying group of the (plain) ring $W$). That this map is a morphism of group schemes follows from the definition of the [[Cartier dual]]. d) For $x\in W(R)$ and $y\in W^\prime (R)$ we have $E(xy,1)\in R^\times$ and \begin{displaymath} E(V^n x y,1)=E (T^n(x F^n y),1)=E(x F^n y,1) \end{displaymath} \end{prop} \begin{defn} \label{}\hypertarget{}{} Let $ker(F_n^m):=ker (F^m:W_{nk}\to W_{nk})$ denote the [[kernel]] of $m$-times iterated [[Frobenius endomorphism]] of the $n$-truncated Witt ring.Let \begin{displaymath} \sigma_n:\begin{cases} W_{nk}\to W_k \\ (\alpha_0,\dots,\alpha_{n-1})\mapsto(\alpha_0,\dots,\alpha_{n-1},0,\dots) \end{cases} \end{displaymath} be the section of the restriction $R_n:W_k\to W_{nk}$. $\sigma_n$ sends $ker(F_n^m)$ in $W^\prime$. Note that $\sigma_n$ is not a morphism of groups. We define the bilinear map \begin{displaymath} \lt-,-\gt:\begin{cases} ker(F^m_n)\times ker(F^n_m)\to R^\times \\ \lt x,y\gt\mapsto E(\sigma_n(x)\sigma_m(y),1) \end{cases} \end{displaymath} then $\lt x,y\gt$ is bilinear and gives an isomorphism \begin{displaymath} ker(F^m_n)\simeq D(ker(F^n_m) \end{displaymath} and satisfies \begin{displaymath} \lt x,t y\gt=\lt f x,y\gt \end{displaymath} \begin{displaymath} \lt x,r y\gt=\lt i x,y\gt \end{displaymath} where the morphisms are \begin{displaymath} \itexarray{ ker(F^m_n) &\stackrel{t}{\to}& ker(F^m_{n+1}) \\ \downarrow^f&&\downarrow^r \\ ker(F^{m-1}_n) &\stackrel{i}{\hookrightarrow}& ker(F^m_n) } \end{displaymath} where $i$ is the canonical inclusion, and $r,f,t$ are induced by $R,F,T$ where $F$ is [[Frobenius morphism|Frobenius]], $T$ is [[Witt ring|Verschiebung]] and $R:W\to W_n$ is [[Witt ring|restriction]]. $i$ and $t$ are monomorphisms, $f$ and $r$ are epimorphisms, and for $ker(F^n_m)$ we have $F=if$ and $V=rt$. \end{defn} References: \hyperlink{Pink}{Pink, \S{}25}, \hyperlink{Demazure}{Demazure, III.4} \hypertarget{properties_of_the_witt_group}{}\subsection*{{Properties of the Witt group}}\label{properties_of_the_witt_group} The group of universal (i.e. not $p$-adic) Witt vectors equals $W(k)= 1+k [ [ X] ]$ i.e. the multiplicative group of power series in one variable $X$ with constant term $1$. \hypertarget{properties_of_the_witt_ring}{}\subsection*{{Properties of the Witt ring}}\label{properties_of_the_witt_ring} \begin{theorem} \label{}\hypertarget{}{} Let $k$ be a perfect field of prime characteristic $p$. Then $W(k)$ is a discrete valuation ring with maximal ideal generated by $p$. From the above we see that $pW(k)=(0, a_0^p, a_1^p, \ldots )$. This clearly gives $W(k)/pW(k)\simeq k$. Also, $W(k)/p^nW(k)\simeq W_n(k)$. Thus the completion of $W(k)$ with respect to the maximal ideal is just $lim W_n(k)\simeq W(k)$ which shows that $W(k)$ is a complete discrete valuation ring. \end{theorem} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{basic_examples}{}\subsubsection*{{Basic examples}}\label{basic_examples} \begin{itemize}% \item $W_{p^\infty}(\mathbb{F}_p)\simeq \mathbb{Z}_p$ the $p$-adic integers. \item $W_{p^\infty}(\mathbb{F}_{p^n})$ is the unique [[unramified extension]] of $\mathbb{Z}_p$ of degree $n$. \end{itemize} \hypertarget{lubintate_ring}{}\subsubsection*{{Lubin-Tate ring}}\label{lubintate_ring} The [[Lubin-Tate ring]] in [[Lubin-Tate theory]] is a [[power series]] ring over a [[Witt ring]] and this way Witt rings govern much of [[chromatic homotopy theory]]. As an Abelian group $W(A)$ is isomorphic to the group of curves in the one-dimensional [[multiplicative group|multiplicative formal group]]. 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. (\hyperlink{eom}{eom}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Witt Cohomology]] \item [[Lambda-ring]] \item [[Hochschild-Witt complex]] and Kaledin's non-commutative Witt vectors \item [[Lubin-Tate ring]] \item [[Borger's absolute geometry]] \item [[de Rham-Witt complex]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{original_texts_and_classical_surveys}{}\subsubsection*{{Original texts and classical surveys}}\label{original_texts_and_classical_surveys} witt vectors were introduced in \begin{itemize}% \item [[Ernst Witt]], \emph{Zyklische K\"o{}rper und Algebren der Characteristik $p$ vom Grad $p^n$. Struktur diskret bewerteter perfekter K\"o{}rper mit vollkommenem Restklassenk\"o{}rper der Charakteristik $p^n$}, J. Reine Angew. Math. , 176 (1936) pp. 126--140, (\href{http://www.digizeitschriften.de/main/dms/img/?IDDOC=504725}{web}) \end{itemize} In the context of [[formal group laws]] they were used in \begin{itemize}% \item [[Jean Dieudonné]], \emph{Groupes de Lie et hyperalg\`e{}bres de Lie sur un corps de charact\'e{}ristique $p \gt 0$ VII`` Math. Ann. , 134 (1957) pp. 114--133} \end{itemize} See also \begin{itemize}% \item [[Michiel Hazewinkel]], \emph{Twisted Lubin-Tate formal group laws, ramified Witt vectors and (ramified) Artin-Hasse exponentials}, Transactions of the AMS (1980) \end{itemize} Surveys incluce \begin{itemize}% \item [[eom]], \emph{\href{http://www.encyclopediaofmath.org/index.php/Witt_vector}{Witt vector}} \end{itemize} \begin{itemize}% \item Michel Demazure, \emph{[[Demazure, lectures on p-divisible groups|Lectures on p-divisible groups]]} (\href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}) \end{itemize} \hypertarget{modern_surveys}{}\subsubsection*{{Modern surveys}}\label{modern_surveys} \begin{itemize}% \item [[Michiel Hazewinkel]], \emph{[[Hazewinkel, Witt vectors|Witt vectors]]}, (\href{http://arxiv.org/abs/0804.3888}{arXiv}) \item [[Michiel Hazewinkel]], \emph{Formal Groups and Applications}, review in \href{http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183548600}{projecteuclid} \item Joseph Rabinoff, \emph{The theory of Witt vectors} (\href{http://math.stanford.edu/~rabinoff/misc/witt.pdf}{pdf}) \item [[Richard Pink]], finite group schemes, \href{http://www.math.ethz.ch/~pink/ftp/FGS/CompleteNotes.pdf}{pdf} \end{itemize} Review in the context of the [[Kummer-Artin-Schreier-Witt exact sequence]] is in \begin{itemize}% \item Ariane M\'e{}zard, Matthieu Romagny, Dajano Tossici, section 2 of \emph{Sekiguchi-Suwa theory revisited} (\href{http://arxiv.org/abs/1104.2222}{arXiv:1104.2222}) \end{itemize} \hypertarget{further_development_of_the_theory}{}\subsubsection*{{Further development of the theory}}\label{further_development_of_the_theory} \begin{itemize}% \item [[Dmitri Kaledin]], [[non-commutative Witt vectors]] \item [[Dmitri Kaledin]], universal Witt vectors and the `'Japanese cocycle'', \href{http://imperium.lenin.ru/~kaledin/math/jap.pdf}{pdf} \item [[Lars Hesselholt]], [[Ib Madsen]], \emph{on the de Rham-Witt comples in [[mixed characteristic]]}, \href{http://www.math.uiuc.edu/K-theory/0551/paper.pdf}{pdf} \item [[Lars Hesselholt]], Witt vectors of non-commutative rings and topological cyclic homology, \href{http://www.math.uiuc.edu/K-theory/0135/derived.pdf}{pdf} \item A. Dress, C. Siebeneicher, \emph{The Burnside ring of profinite groups and the Witt vector construction}, Adv. Math., 70, (1988), 87--132. \end{itemize} In the context of [[Borger's absolute geometry]]: \begin{itemize}% \item [[James Borger]], \emph{The basic geometry of Witt vectors, I: The affine case} (\href{http://arxiv.org/abs/0801.1691}{arXiv:0801.1691}) \item [[James Borger]], \emph{The basic geometry of Witt vectors, II: Spaces} (\href{http://arxiv.org/abs/1006.0092}{arXiv:1006.0092}) \end{itemize} [[!redirects rings of Witt vectors]] [[!redirects rings of Witt-vectors]] [[!redirects Witt ring]] [[!redirects Witt rings]] [[!redirects Witt polynomial]] [[!redirects Witt polynomials]] [[!redirects group of Witt vectors]] [[!redirects Witt vector]] [[!redirects Witt vectors]] [[!redirects ring of Witt vectors]] [[!redirects big-Witt-vectors functor]] \end{document}