A simple construction of the field of Witt vectors.

In the wikipedia article on Witt vectors (july 2024) it is indicated that "they have a highly non-intuitive structure". The aim of this note is to refute this claim.

A very similar construction is essentially contained in the notes by Michiel Hazewinkel [3] , mainly in the section 9 and the section 14. This review contains a lot of information for further reading on the subject.

Let \(f(t)=1+a_1t+\cdots \) and \(g(t)=1+b_1t+\cdots \) belong to the multiplicative semi-group \(1+t\mathbb {F}[t]\) of polynomials with coefficients in a field \(\mathbb {F}\) and the constant term equal to 1. Define a convolution \(f\star g\) as a polynomial with the constant term 1 and having as roots the products of one root of \(f\) and one of \(g\). In other words, suppose that \(f(t)=\prod _i(1-\lambda _it)\) and \(g(t)=\prod _j(1-\mu _jt)\) with \(\lambda _i,\mu _j\) belonging to the algebraic closure of the field \(\mathbb F\). Then

\[f\star g\,(t) = \prod _{ij}(1-t\lambda _i\mu _j)=\prod _ig(\lambda _it)=\prod _jf(\mu _jt).\]

The convolution can also be expressed in term of the resultant, namely

\[f\star g\,(t)=\mathtt {res}_z(f(z),z^{\deg g}g(t/z)).\]

To give an equivalent definition, consider the ring \(\mathbb {F}[x,y]/(f(x))+(g(y))\) and denote by \(\hat {x}^{-1}\) and \(\hat {y}^{-1}\) the multiplication in this ring by \(x^{-1}\) and \(y^{-1}\), respectively. In the standard basis they are given by matrices with entries in \(\mathbb {F}\). Then

\[f\star g\,(t)=\det (1-t\hat {x}^{-1}\hat {y}^{-1}).\]

In this definition it is explicit that the coefficients of \(f\star g\) are polynomial functions of those of \(f\) and \(g\).

The fourth definition works for \(\mathbb {F}=\mathbb {C}\) and shows the relation to the tame symbol. Consider a curve \(\gamma \) around zero on the complex plane sufficiently small in order not to surround any root of \(f(z)\). The convolution can be defined by the formula (see P.Deligne [2] , formula 2.7.2)

\[f\star g\,(t)=\{f(z),g(t/z)\}_\gamma =\exp \left (\frac {1}{2\pi i}\int _{\gamma }\ln f(z) d\ln g(t/z)\right )\]

valid for \(t\) so small that all roots of \(g(t/z)\) are inside the curve \(\gamma \).

The convolution enjoys the following properties obvious from the definitions:

• \(\deg (f\star g)=\deg f \deg g\),
• \(f\star 1=1,\)
• \(f\star (1-t)=f,\)
• \((1-at)\star (1-bt)=(1-abt),\)
• \(f\star g = g\star f,\)
• \(f\star (g_1g_2)=(f\star g_1)(f\star g_2)\).

The following property is also an easy consequence of the definition:

• A. The set \(1+t^{n}\mathbb {F}[t]\) is an ideal.

For that we need just another property of the universal Witt ring obviously following from the definition of the convolution:

• The set \(1+t^{n}\mathbb {F}[[t^n]]\) is also an ideal.

Let \(\mathbb {F}\) be a field of characteristic \(p\) and let \(W(\mathbb {F})\) be the corresponding universal Witt ring.

Define the Witt ring \(W_\mathbb {F}\) as a quotient

\[W_\mathbb {F}= W(\mathbb {F})/\prod _{n>1|(n,p)=1}(1+t^n\mathbb {F}[[t^n]]).\]

Here we used the property 3 and denoted the sum of ideals multiplicatively since it corresponds to the product of the series.

Observe that any element of the group \(1+t\mathbb {F}[[t]]\) can be presented either as a sum \(1+\alpha _1t+\alpha _2t^2+\cdots \) or as a product \((1-a_1t)(1-a_2t^2)(1-a_3t^3)\cdots \).

Using the latter presentation the elements of the ring \(W_\mathbb {F}\) can be uniquely represented as formal products

\[f(t)=\prod _{i=0}^\infty (1-a_it^{p^i}).\]

In this presentation certain properties of the Witt vectors become obvious. In particular, it follows from the property 4 that the correspondence \(a\mapsto (1-at)\) gives an embedding of multiplicative groups \(\mathbb {F}^\times \to W_{\mathbb {F}}^\times \). The images of the elements of \(\mathbb {F}^\times \) are called their Teichmüller representatives. It is also obvious that the ring multiplication by \(p\) in the ring \(W_{\mathbb {F}}\) amounts to the shift of the coefficients \(a_i\) composed with the Frobenius automorphism:

\[ \prod _{i=0}^\infty (1-a_it^{p^i})\mapsto \prod _{i=1}^\infty (1-a^p_{i-1}t^{p^{i}}).\]

This property allows to identify the field of fractions of the ring \(W_\mathbb {F}\) with the expressions of the form

\[ \prod _{i=N}^\infty (1-a_it^{p^i})\]

with possibly negative \(N\).

Recall that for a field \(\mathbb {F}_p\) of \(p\) elements the ring \(W_\mathbb {F}\) coincides with the ring \(\mathbb {Z}_p\) of \(p\)-adic integers.

\[(\sum _{k=0}^\infty a_kt^k)\circ (\sum _{k=0}^\infty b_kt^k)=\sum _{k=0}^\infty a_kb_kt^k\]

Clearly this ring is just a direct sum of infinitely many copies of the ring \(\mathbb {C}\).

The map \(f\mapsto -f'/f\) gives an isomorphism between the rings \((1+\mathbb {C}[[t]])^{\cdot \star }\) and \(\mathbb {C}[[t]]^{+\circ }\). Indeed

\[\begin {aligned} (-f'/f)\circ (-g'/g) &=\left (\sum _{k=1}^{\infty }(\sum _i\lambda _i^k)t^{k-1}\right )\circ \left (\sum _{k=0}^{\infty }(\sum _j\mu _j^k){t^{k-1}}\right )=\\&=\left (\sum _{k}(\sum _{ij}\lambda _i^k\mu _j^k)t^{k-1}\right )=-(f\star g)'/(f\star g), \end {aligned}\]

and obviously

\[-f'/f-g'/g=-(fg)'/(fg).\]

In the explicit coordinates we have

\[ \prod (1-a_it^{i})\mapsto \sum _i\sum _ka_i^{k}t^{ik-1}=\sum _{l}\sum _{i|l}ia^{l/i}_{i}t^{l-1}.\]

The expressions

\[S_l(a_1,a_2,\ldots )=\sum _{i|l}{ia^{i/l}_{i}}\]

are called the universal Witt polynomials (P. Cartier [1] and [3] section 9). We see that each of the Witt polynomials gives a homomorphism from the universal Witt ring to the ring of complex numbers and a collection of all such polynomials gives an isomorphism of the universal Witt ring to the infinite sum of complex numbers. The standard construction uses this isomorphism to define the ring structure in terms of the coefficients \(a_i\). Then one proves that the product and the sum are in fact given by algebraic expressions with integer coefficients and thus are defined over any field.