Research Contribution
DOI:10.56994/ARMJ.011.001.001
Received: 10 March 2024; Accepted: 15 October
2024
A simple construction of the field of Witt vectors.
Abstract
Mathematics Subject Classification.
Key words and phrases.
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:
The following property is also an easy consequence of the definition: 1.
Introduction
2. Convolution
These
properties imply that the semi-group \(1+t\mathbb {F}[t]\) is a commutative
semi-ring with respect to the multiplication as a semi-ring addition and
convolution as a semi-ring multiplication. The multiplicativity property
6
is just the expression of the distributive law of the semi-ring.
This property implies that the convolution can be
extended to the group of formal power series \(1+t\mathbb {F}[[t]]\)
providing it with a ring structure. This ring is called the ring of the
universal or big Witt vectors and is denoted by \(W(\mathbb {F})\),
see [3] ,
section 9. 3.
Witt vectors
4. Relation to the standard definition of the Witt vectors
References