Witt vector

In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\mathbb{F}_p)} over the finite field of order $p$ is isomorphic to $\mathbb {Z} _{p}$, the ring of $p$-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers.

The main idea behind Witt vectors is instead of using the standard $p$-adic expansion

$a=a_{0}+a_{1}p+a_{2}p^{2}+\cdots$

to represent an element in $\mathbb {Z} _{p}$, we can instead consider an expansion using the Teichmüller character

$\omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}$

which sends each element in the solution set of $x^{p-1}-1$ in $\mathbb {F} _{p}$ to an element in the solution set of $x^{p-1}-1$ in $\mathbb {Z} _{p}$. That is, we expand out elements in $\mathbb {Z} _{p}$ in terms of roots of unity instead of as profinite elements in $\prod \mathbb {F} _{p}$. We can then express a $p$-adic integer as an infinite sum

$\omega (a)=\omega (a_{0})+\omega (a_{1})p+\omega (a_{2})p^{2}+\cdots$

which gives a Witt vector

$(\omega (a_{0}),\omega (a_{1}),\omega (a_{2}),\ldots )$

Then, the non-trivial additive and multiplicative structure in Witt vectors comes from using this map to give $W(\mathbb {F} _{p})$ an additive and multiplicative structure such that $\omega$ induces a commutative ring morphism.

History

In the 19th century, Ernst Eduard Kummer studied cyclic extensions of fields as part of his work on Fermat's Last Theorem. This led to the subject now known as Kummer theory. Let $k$ be a field containing a primitive $n$-th root of unity. Kummer theory classifies degree $n$ cyclic field extensions $K$ of $k$. Such fields are in bijection with order $n$ cyclic groups $\Delta \subseteq k^{\times }/(k^{\times })^{n}$, where $\Delta$ corresponds to $K=k({\sqrt[{n}]{\Delta }})$.

But suppose that $k$ has characteristic $p$. The problem of studying degree $p$ extensions of $k$, or more generally degree $p^{n}$ extensions, may appear superficially similar to Kummer theory. However, in this situation, $k$ cannot contain a primitive $p$-th root of unity. Indeed, if $x$ is a $p$-th root of unity in $k$, then it satisfies $x^{p}=1$. But consider the expression $(x-1)^{p}=0$. By expanding using binomial coefficients we see that the operation of raising to the $p$-th power, known here as the Frobenius homomorphism, introduces the factor $p$ to every coefficient except the first and the last, and so modulo $p$ these equations are the same. Therefore $x=1$. Consequently, Kummer theory is never applicable to extensions whose degree is divisible by the characteristic.

The case where the characteristic divides the degree is today called Artin–Schreier theory because the first progress was made by Artin and Schreier. Their initial motivation was the Artin–Schreier theorem, which characterizes the real closed fields as those whose absolute Galois group has order two. This inspired them to ask what other fields had finite absolute Galois groups. In the midst of proving that no other such fields exist, they proved that degree $p$ extensions of a field $k$ of characteristic $p$ were the same as splitting fields of Artin–Schreier polynomials. These are by definition of the form $x^{p}-x-a.$ By repeating their construction, they described degree $p^{2}$ extensions. Abraham Adrian Albert used this idea to describe degree $p^{n}$ extensions. Each repetition entailed complicated algebraic conditions to ensure that the field extension was normal.

Schmid generalized further to non-commutative cyclic algebras of degree $p^{n}$. In the process of doing so, certain polynomials related to the addition of $p$-adic integers appeared. Witt seized on these polynomials. By using them systematically, he was able to give simple and unified constructions of degree $p^{n}$ field extensions and cyclic algebras. Specifically, he introduced a ring now called $W_{n}(k)$, the ring of $n$-truncated $p$-typical Witt vectors. This ring has $k$ as a quotient, and it comes with an operator $F$ which is called the Frobenius operator because it reduces to the Frobenius operator on $k$. Witt observes that the degree $p^{n}$ analog of Artin–Schreier polynomials is

$F(x)-x-a,$

where $a\in W_{n}(k)$. To complete the analogy with Kummer theory, define $\wp$ to be the operator $x\mapsto F(x)-x.$ Then the degree $p^{n}$ extensions of $k$ are in bijective correspondence with cyclic subgroups $\Delta \subseteq W_{n}(k)/\wp (W_{n}(k))$ of order $p^{n}$, where $\Delta$ corresponds to the field $k(\wp ^{-1}(\Delta ))$.

Motivation

Any $p$-adic integer (an element of $\mathbb {Z} _{p}$, not to be confused with $\mathbb {Z} /p\mathbb {Z} =\mathbb {F} _{p}$) can be written as a power series $a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots$, where the $a_{i}$ are usually taken from the integer interval $[0,p-1]=\{0,1,2,\ldots ,p-1\}$. It is hard to provide an algebraic expression for addition and multiplication using this representation, as one faces the problem of carrying between digits. However, taking representative coefficients $a_{i}\in [0,p-1]$ is only one of many choices, and Hensel himself (the creator of $p$-adic numbers) suggested the roots of unity in the field as representatives. These representatives are therefore the number $0$ together with the $(p-1)^{\text{th}}$ roots of unity; that is, the solutions of $x^{p}-x=0$ in $\mathbb {Z} _{p}$, so that $a_{i}=a_{i}^{p}$. This choice extends naturally to ring extensions of $\mathbb {Z} _{p}$ in which the residue field is enlarged to $\mathbb {F} _{q}$ with $q=p^{f}$, some power of $p$. Indeed, it is these fields (the fields of fractions of the rings) that motivated Hensel's choice. Now the representatives are the $q$ solutions in the field to $x^{q}-x=0$. Call the field $\mathbb {Z} _{p}(\eta )$, with $\eta$ an appropriate primitive $(q-1)^{\text{th}}$ root of unity (over $\mathbb {Z} _{p}$). The representatives are then $0$ and $\eta ^{i}$ for $0\leq i\leq q-2$. Since these representatives form a multiplicative set they can be thought of as characters. Some thirty years after Hensel's works Teichmüller studied these characters, which now bear his name, and this led him to a characterisation of the structure of the whole field in terms of the residue field. These Teichmüller representatives can be identified with the elements of the finite field $\mathbb {F} _{q}$ of order $q$ by taking residues modulo $p$ in $\mathbb {Z} _{p}(\eta )$, and elements of $\mathbb {F} _{q}^{\times }$ are taken to their representatives by the Teichmüller character $\omega :\mathbb {F} _{q}^{\times }\to \mathbb {Z} _{p}(\eta )^{\times }$. This operation identifies the set of integers in $\mathbb {Z} _{p}(\eta )$ with infinite sequences of elements of $\omega (\mathbb {F} _{q}^{\times })\cup \{0\}$.

Taking those representatives the expressions for addition and multiplication can be written in closed form. We now have the following problem (stated for the simplest case: $q=p$): given two infinite sequences of elements of $\omega (\mathbb {F} _{p}^{\times })\cup \{0\},$ describe their sum and product as $p$-adic integers explicitly. This problem was solved by Witt using Witt vectors.

Detailed motivational sketch

We derive the ring of $p$-adic integers $\mathbb {Z} _{p}$ from the finite field $\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z}$ using a construction which naturally generalizes to the Witt vector construction.

The ring $\mathbb {Z} _{p}$ of $p$-adic integers can be understood as the inverse limit of the rings $\mathbb {Z} /p^{i}\mathbb {Z}$ taken along the obvious projections. Specifically, it consists of the sequences $(n_{0},n_{1},\ldots )$ with $n_{i}\in \mathbb {Z} /p^{i+1}\mathbb {Z} ,$ such that $n_{j}\equiv n_{i}{\bmod {p}}^{i+1}$ for $j\geq i.$ That is, each successive element of the sequence is equal to the previous elements modulo a lower power of p; this is the inverse limit of the projections $\mathbb {Z} /p^{i+1}\mathbb {Z} \to \mathbb {Z} /p^{i}\mathbb {Z} .$

The elements of $\mathbb {Z} _{p}$ can be expanded as (formal) power series in $p$

$a_{0}+a_{1}p^{1}+a_{2}p^{2}+\cdots ,$

where the coefficients $a_{i}$ are taken from the integer interval $[0,p-1]=\{0,1,\ldots ,p-1\}.$ Of course, this power series usually will not converge in $\mathbb {R}$ using the standard metric on the reals, but it will converge in $\mathbb {Z} _{p},$ with the $p$-adic metric. We will sketch a method of defining ring operations for such power series.

Letting $a+b$ be denoted by $c$, one might consider the following definition for addition:

${\begin{aligned}c_{0}&\equiv a_{0}+b_{0}&&{\bmod {p}}\\c_{0}+c_{1}p&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p&&{\bmod {p}}^{2}\\c_{0}+c_{1}p+c_{2}p^{2}&\equiv (a_{0}+b_{0})+(a_{1}+b_{1})p+(a_{2}+b_{2})p^{2}&&{\bmod {p}}^{3}\end{aligned}}$

and one could make a similar definition for multiplication. However, this is not a closed formula, since the new coefficients are not in the allowed set $[0,p-1].$

Representing elements in F_p as elements in the ring of Witt vectors W(F_p)

There is a better coefficient subset of $\mathbb {Z} _{p}$ which does yield closed formulas, the Teichmüller representatives: zero together with the $(p-1)^{\text{th}}$ roots of unity. They can be explicitly calculated (in terms of the original coefficient representatives $[0,p-1]$) as roots of $x^{p-1}-1=0$ through Hensel lifting, the $p$-adic version of Newton's method. For example, in $\mathbb {Z} _{5},$ to calculate the representative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2,} one starts by finding the unique solution of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4-1=0} in $\mathbb {Z} /25\mathbb {Z}$ with $x\equiv 2{\bmod {5}}$; one gets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7.} Repeating this in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z /125\Z,} with the conditions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^4-1=0} and $x\equiv 7{\bmod {2}}5$, gives Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 57,} and so on; the resulting Teichmüller representative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} , denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(2)} , is the sequence

$\omega (2)=(2,7,57,\ldots )\in W(\mathbb {F} _{5}).$

The existence of a lift in each step is guaranteed by the greatest common divisor $(x^{p-1}-1,(p-1)x^{p-2})=1$ in every $\mathbb {Z} /p^{n}\mathbb {Z} .$

This algorithm shows that for every $j\in [0,p-1]$, there is exactly one Teichmüller representative with $a_{0}=j$, which we denote $\omega (j).$ Indeed, this defines the Teichmüller character $\omega :\mathbb {F} _{p}^{*}\to \mathbb {Z} _{p}^{*}$ as a (multiplicative) group homomorphism, which moreover satisfies $m\circ \omega =\mathrm {id} _{\mathbb {F} _{p}}$ if we let $m:\mathbb {Z} _{p}\to \mathbb {Z} _{p}/p\mathbb {Z} _{p}\cong \mathbb {F} _{p}$ denote the canonical projection. Note however that $\omega$ is not additive, as the sum need not be a representative. Despite this, if $\omega (k)\equiv \omega (i)+\omega (j){\bmod {p}}$ in $\mathbb {Z} _{p},$ then $i+j=k$ in $\mathbb {F} _{p}.$

Representing elements in Z_p as elements in the ring of Witt vectors W(F_p)

Because of this one-to-one correspondence given by $\omega$, one can expand every $p$-adic integer as a power series in $p$ with coefficients taken from the Teichmüller representatives. An explicit algorithm can be given, as follows. Write the Teichmüller representative as $\omega (t_{0})=t_{0}+t_{1}p^{1}+t_{2}p^{2}+\cdots .$ Then, if one has some arbitrary $p$-adic integer of the form $x=x_{0}+x_{1}p^{1}+x_{2}p^{2}+\cdots ,$ one takes the difference Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-\omega(x_0)=x'_1 p^1 + x'_2 p^2 + \cdots,} leaving a value divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} . Hence, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-\omega(x_0) = 0 \bmod p} . The process is then repeated, subtracting Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(x'_1)p} and proceed likewise. This yields a sequence of congruences

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} x &\equiv \omega(x_0) && \bmod p \\ x &\equiv \omega(x_0) + \omega(x'_1)p && \bmod p^2 \\ &\cdots \end{align}}

So that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \equiv \sum_{j = 0}^i \omega(\bar{x}_j)p^j \bmod p^{i+1}}

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i' > i} implies:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j = 0}^{i'} \omega(\bar{x}_j)p^j \equiv \sum_{j=0}^i \omega(\bar{x}_j)p^j \bmod p^{i+1}}

for

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{x}_{i} := m\left(\frac{x-\sum_{j=0}^{i-1} \omega(\bar{x}_j)p^j}{p^i}\right).}

Hence we have a power series for each residue of x modulo powers of p, but with coefficients in the Teichmüller representatives rather than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{0, \ldots, p-1\}} . It is clear that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j = 0}^\infty \omega(\bar{x}_j)p^j=x,}

since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^{i+1} \mid x - \sum_{j = 0}^i \omega(\bar{x}_j)p^j}

for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\to\infty,} so the difference tends to 0 with respect to the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic metric. The resulting coefficients will typically differ from the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} modulo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^i} except the first one.

Additional properties of elements in the ring of Witt vectors motivating general definition

The Teichmüller coefficients have the key additional property that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(\bar{x}_i)^p=\omega(\bar{x}_i),} which is missing for the numbers in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,p-1]} . This can be used to describe addition, as follows. Consider the equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle c = a+b} in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathbb{Z}_p} and let the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_i, b_i, c_i \in \mathbb{Z}_p} now be as in the Teichmüller expansion. Since the Teichmüller character is not additive, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0=a_0+b_0} is not true in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Z_p} . But it holds in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_p,} as the first congruence implies. In particular,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0^p\equiv (a_0+b_0)^p \bmod p^2,}

and thus

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0-a_0-b_0\equiv (a_0+b_0)^p-a_0-b_0\equiv \binom{p}{1} a_0^{p-1}b_0+\cdots+ \binom{p}{p-1} a_0 b_0^{p-1} \bmod p^2.}

Since the binomial coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{p}{i}} is divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , this gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1\equiv a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-\cdots- a_0 b_0^{p-1} \bmod p.}

This completely determines Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1} by the lift. Moreover, the congruence modulo Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} indicates that the calculation can actually be done in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{F}_p,} satisfying the basic aim of defining a simple additive structure.

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_2} this step is already very cumbersome. Write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1=c_1^p \equiv \left(a_1+b_1- a_0^{p-1}b_0-\frac{p-1}{2}a_0^{p-2}b_0^2-\cdots - a_0 b_0^{p-1}\right)^p \bmod p^2.}

Just as for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0,} a single Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} th power is not enough: one must take

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_0=c_0^{p^2}\equiv(a_0+b_0)^{p^2} \bmod p^3.}

However, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \binom{p^2}{i}} is not in general divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^2,} but it is divisible when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=pd,} in which case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^ib^{p^2-i}=a^db^{p-d}} combined with similar monomials in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_1^p} will make a multiple of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^2} .

At this step, it becomes clear that one is actually working with addition of the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} c_0 &\equiv a_0+b_0 && \bmod p \\ c_0^p+c_1 p &\equiv a_0^p+a_1 p+b_0^p+b_1 p && \bmod p^2 \\ c_0^{p^2}+c_1^p p+c_2 p^2 &\equiv a_0^{p^2}+a_1^p p+a_2 p^2+b_0^{p^2}+b_1^p p+b_2 p^2 && \bmod p^3 \end{align}}

This motivates the definition of Witt vectors.

Construction of Witt rings

Fix a prime number p. A Witt vector over a commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} (relative to the prime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ) is a sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_0,X_1,X_2,\ldots)} of elements of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} . Define the Witt polynomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_i} by

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_0=X_0}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1=X_0^p+pX_1}
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2=X_0^{p^2}+pX_1^p+p^2X_2}

and in general

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n=\sum_{i=0}^np^iX_i^{p^{n-i}}.}

The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n} are called the ghost components of the Witt vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_0,X_1,X_2,\ldots)} , and are usually denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{(n)}} ; taken together, the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n} define the ghost map to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \prod_{i=0}^\infty R} . If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R} is p-torsionfree, then the ghost map is injective and the ghost components can be thought of as an alternative coordinate system for the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} -module of sequences (though note that the ghost map is not surjective unless Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R} is p-divisible).

The ring of (p-typical) Witt vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(R)} is defined by componentwise addition and multiplication of the ghost components. That is, that there is a unique way to make the set of Witt vectors over any commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} into a ring such that:

1. the sum and product are given by polynomials with integral coefficients that do not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , and
2. projection to each ghost component is a ring homomorphism from the Witt vectors over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} , to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

In other words,

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X+Y)_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (XY)_i} are given by polynomials with integral coefficients that do not depend on R, and

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{(i)}+Y^{(i)}=(X+Y)^{(i)}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X^{(i)}Y^{(i)}=(XY)^{(i)}.}

The first few polynomials giving the sum and product of Witt vectors can be written down explicitly. For example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_0,X_1,\ldots)+(Y_0,Y_1,\ldots)=(X_0+Y_0, X_1+Y_1-((X_0+Y_0)^p - X_0^p -Y_0^p)/p,\ldots)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_0,X_1,\ldots)\times(Y_0,Y_1,\ldots)=(X_0 Y_0, X_0^p Y_1+X_1 Y_0^p+p X_1 Y_1,\ldots)}

These are to be understood as shortcuts for the actual formulas. If for example the ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} has characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , the division by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} in the first formula above, the one by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^2} that would appear in the next component and so forth, do not make sense. However, if the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -power of the sum is developed, the terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_0^p+Y_0^p} are cancelled with the previous ones and the remaining ones are simplified by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , no division by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} remains and the formula makes sense. The same consideration applies to the ensuing components.

Examples of addition and multiplication

As would be expected, the unit in the ring of Witt vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(A)} is the element

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{1} = (1,0,0,\ldots)}

Adding this element to itself gives a non-trivial sequence, for example in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\mathbb{F}_5)} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{1} + \underline{1} = (2,4,\ldots)}

since

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 2 &= 1 + 1\\ 4 &= -\frac{32 - 1 - 1}{5} \mod 5 \\ &\cdots \end{align}}

which is not the expected behavior, since it doesn't equal Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{2}} . But, when we reduce with the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m:W(\mathbb{F}_5) \to \mathbb{F}_5} , we get Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m(\omega(1) + \omega(1)) = m(\omega(2))} . Note if we have an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in A} and an element Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a \in W(A)} then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{x}a = (xa_0,x^pa_1,\ldots,x^{p^n}a_n,\ldots)}

showing multiplication also behaves in a highly non-trivial manner.

Examples

• The Witt ring of any commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in which $p$ is invertible is just isomorphic to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\N} (the product of a countable number of copies of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ). In fact the Witt polynomials always give a homomorphism from the ring of Witt vectors to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^\N} , and if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is invertible this homomorphism is an isomorphism.

• The Witt ring $W(\mathbb {F} _{p})\cong \mathbb {Z} _{p}$ of the finite field of order $p$ is the ring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic integers written in terms of the Teichmüller representatives, as demonstrated above.

• The Witt ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\mathbb{F}_q) \cong \mathcal{O}_K} of a finite field of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^n} is the ring of integers of the unique unramified extension of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} of the ring of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adic numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K/\mathbb{Q}_p} . Note Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \cong \mathbb{Q}_p(\mu_{q-1})} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_{q-1}} the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (q-1)} -th root of unity, hence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(\mathbb{F}_q) \cong \mathbb{Z}_p[\mu_{q-1}]} .

Universal Witt vectors

The Witt polynomials for different primes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} are special cases of universal Witt polynomials, which can be used to form a universal Witt ring (not depending on a choice of prime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ). Define the universal Witt polynomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\geq1} by

1. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_1=X_1}
2. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_2=X_1^2+2X_2}
3. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_3=X_1^3+3X_3}
4. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_4=X_1^{4}+2X_2^2+4X_4}

and in general

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n=\sum_{d|n}dX_d^{n/d}.}

Again, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (W_1,W_2,W_3,\ldots)} is called the vector of ghost components of the Witt vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_1,X_2,X_3,\ldots)} , and is usually denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X^{(1)},X^{(2)},X^{(3)},\ldots)} .

We can use these polynomials to define the ring of universal Witt vectors or big Witt ring of any commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} in much the same way as above (so the universal Witt polynomials are all homomorphisms to the ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} ).

Generating Functions

Witt also provided another approach using generating functions.

Definition

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X} be a Witt vector and define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_X(t)=\prod_{n\ge 1}(1-X_n t^n)=\sum_{n\ge 0}A_n t^n}

For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\ge 1} let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{I}_n} denote the collection of subsets of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1,2,\ldots,n\}} whose elements add up to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} . Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n=\sum_{I\in\mathcal{I}_n}(-1)^{|I|}\prod_{i\in I}{X_i}.}

We can get the ghost components by taking the logarithmic derivative:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} -t\frac{d}{dt}\log f_X(t)&= -t\frac{d}{dt} \sum_{n\ge 1} \log(1-X_n t^n) \\ &=t \frac{d}{dt}\sum_{n\ge 1}\sum_{d\ge 1}\frac{X_n^d t^{nd}}{d}\\ &=\sum_{n\ge 1}\sum_{d\ge 1} n X_n^d t^{nd} \\ &=\sum_{m\ge 1}\sum_{d|m}dX_{d}^{m/d}t^m \\ &=\sum_{m\ge 1}X^{(m)}t^m \end{align}}

Sum

Now we can see Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_{Z}(t)=f_X(t) f_Y(t)} if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z=X+Y} . So that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C_n=\sum_{0\le i\le n}A_n B_{n-i},}

if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n,B_n,C_n} are the respective coefficients in the power series Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_X(t),f_Y(t),f_Z(t)} . Then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_n=\sum_{0\le i\le n}A_n B_{n-i}-\sum_{I\in\mathcal{I}_n,I\ne\{n\}}(-1)^{|I|}\prod_{i\in I}{Z_i}.}

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_n} is a polynomial in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1, \ldots, X_n} and likewise for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_n} , we can show by induction that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_n} is a polynomial in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1, \ldots, X_n, Y_1, \ldots, Y_n.}

Product

If we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=XY} then

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -t\frac{d}{dt}\log f_W(t)=-\sum_{m\ge 1}X^{(m)}Y^{(m)}t^m.}

But

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{m\ge 1}X^{(m)}Y^{(m)}t^m=\sum_{m\ge 1}\sum_{d|m}d X_d^{m/d}\sum_{e|m}e Y_e^{m/e}t^m} .

Now 3-tuples Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {m,d,e}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\in\Z^+,d|m,e|m} are in bijection with 3-tuples Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {d,e,n}} with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d,e,n\in\Z^+} , via Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=m/[d,e]} (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [d,e]} is the least common multiple), our series becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{d,e\ge 1}d e\sum_{n\ge 1} \left (X_d^{\frac{[d,e]}{d}} Y_e^{\frac{[d,e]}{e}} t^{[d,e]} \right )^n=-t\frac{d}{dt}\log\prod_{d,e\ge 1} \left (1-X_d^{\frac{[d,e]}{d}}Y_e^{\frac{[d,e]}{e}} t^{[d,e]} \right )^{\frac{de}{[d,e]}}}

So that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_W(t)=\prod_{d,e\ge 1} \left (1-X_d^{\frac{[d,e]}{d}}Y_e^{\frac{[d,e]}{e}} t^{[d,e]} \right )^{\frac{de}{[d,e]}}=\sum_{n\ge 0}D_n t^n,}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_n} are polynomials of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1, \ldots, X_n, Y_1, \ldots, Y_n.} So by similar induction, suppose

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_W(t)=\prod_{n\ge 1}(1-W_n t^n),}

then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_n} can be solved as polynomials of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1,\ldots, X_n, Y_1,\ldots, Y_n.}

Ring schemes

The map taking a commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to the ring of Witt vectors over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} (for a fixed prime Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ) is a functor from commutative rings to commutative rings, and is also representable, so it can be thought of as a ring scheme, called the Witt scheme, over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Spec}(\Z).} The Witt scheme can be canonically identified with the spectrum of the ring of symmetric functions.

Similarly, the rings of truncated Witt vectors, and the rings of universal Witt vectors correspond to ring schemes, called the truncated Witt schemes and the universal Witt scheme.

Moreover, the functor taking the commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} to the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} is represented by the affine space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}_{\Z}^n} , and the ring structure on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R^n} makes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}_{\Z}^n} into a ring scheme denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \underline{\mathcal{O}}^n} . From the construction of truncated Witt vectors, it follows that their associated ring scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{W}_n} is the scheme Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{A}_{\Z}^n} with the unique ring structure such that the morphism Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{W}_n\to \underline{\mathcal{O}}^n} given by the Witt polynomials is a morphism of ring schemes.

Commutative unipotent algebraic groups

Over an algebraically closed field of characteristic 0, any unipotent abelian connected algebraic group is isomorphic to a product of copies of the additive group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_a} . The analogue of this for fields of characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is false: the truncated Witt schemes are counterexamples. (We make them into algebraic groups by forgetting the multiplication and just using the additive structure.) However, these are essentially the only counterexamples: over an algebraically closed field of characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} , any unipotent abelian connected algebraic group is isogenous to a product of truncated Witt group schemes.

Universal property

André Joyal explicated the universal property of the (p-typical) Witt vectors. The basic intuition is that the formation of Witt vectors is the universal way to deform a characteristic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} ring to characteristic 0 together with a lift of its Frobenius endomorphism. To make this precise, define a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (R, \delta)} to consist of a commutative ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R} together with a map of sets Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \delta: R \to R} that is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -derivation, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \delta} satisfies the relations

• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(0) = \delta(1) = 0} ;
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x y) = x^p \delta(y) + y^p \delta(x) + p \delta(x) \delta(y)} ;
• Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(x + y) = \delta(x) + \delta(y) + \frac{x^p+y^p-(x+y)^p}{p}} .

The definition is such that given a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -ring Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle (R, \delta)} , if one defines the map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \phi: R \to R} by the formula Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \phi(x) = x^p + p \delta(x)} , then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \phi} is a ring homomorphism lifting Frobenius on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R/p} . Conversely, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -torsionfree, then this formula uniquely defines the structure of a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -ring on ${\textstyle R}$ from that of a Frobenius lift. One may thus regard the notion of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -ring as a suitable replacement for a Frobenius lift in the non Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -torsionfree case.

The collection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -rings and ring homomorphisms thereof respecting the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -structure assembles to a category Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathrm{CRing}_{\delta}} . One then has a forgetful functorFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U: \mathrm{CRing}_{\delta} \to \mathrm{CRing}} whose right adjoint identifies with the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle W} of Witt vectors. In fact, the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle U} creates limits and colimits and admits an explicitly describable left adjoint as a type of free functor; from this, it is not hard to show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \mathrm{CRing}_{\delta}} inherits local presentability from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{CRing}} so that one can construct the functor Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle W} by appealing to the adjoint functor theorem.

One further has that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle W} restricts to a fully faithful functor on the full subcategory of perfect rings of characteristic p. Its essential image then consists of those Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} -rings that are perfect (in the sense that the associated map Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \phi} is an isomorphism) and whose underlying ring is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -adically complete.

See also

• p-derivation
• Formal group
• Artin–Hasse exponential
• Necklace ring

References

Introductory

• Notes on Witt vectors: a motivated approach - Basic notes giving the main ideas and intuition. Best to start here!
• The Theory of Witt Vectors - Elementary introduction to the theory.
• Complexe de de Rham-Witt et cohomologie cristalline - Note he uses a different but equivalent convention as in this article. Also, the main points in the introduction are still valid.

Applications

• Mumford, David (1966-08-21), Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59, Princeton, NJ: Princeton University Press, ISBN 978-0-691-07993-6
• Serre, Jean-Pierre (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90424-5, MR 0554237, section II.6
• Serre, Jean-Pierre (1988), Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1035-1, ISBN 978-0-387-96648-9, MR 0918564
• Greenberg, Marvin J. (1969). Lectures on Forms in Many Variables. New York and Amsterdam: Benjamin. ASIN B0006BX17M. MR 0241358.

References

• Dolgachev, Igor V. (2001) [1994], "Witt vector", Encyclopedia of Mathematics, EMS Press
• Hazewinkel, Michiel (2009), "Witt vectors. I.", Handbook of algebra. Vol. 6, Amsterdam: Elsevier/North-Holland, pp. 319–472, arXiv:0804.3888, doi:10.1016/S1570-7954(08)00207-6, ISBN 978-0-444-53257-2, MR 2553661
• Witt, Ernst (1936), "Zyklische Körper und Algebren der Characteristik p vom Grad pⁿ. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pⁿ", Journal für die Reine und Angewandte Mathematik (in German), 1937 (176): 126–140, doi:10.1515/crll.1937.176.126{{citation}}: CS1 maint: unrecognized language (link)