Reduced norm map of division algebras over complete discrete valuation fields of certain type. (English) Zbl 0990.11072

Yamazaki, Takao

Compos. Math. 112, No. 2, 127-145 (1998).

The main object of the paper under review is a finite dimensional central division algebra \(D\) over a complete discrete valuation field \(K\). It is assumed that the residue field \(F\) of \(K\) is of characteristic \(p>0\), \([F:F^p]=p\), and the residue algebra \(C\) of \(D\) is commutative and purely inseparable over \(F\). The author’s goal is to develop a ramification theory for such division algebras generalizing the classical ramification theory of finite field extensions of \(K\).
The first step is to define Herbrand’s function measuring how wild the ramification of \(D\) is. The definition is based on the notion of Swan conductor sw: \(Br(K)\to{\mathbb Z}_{\geq 0}\) introduced by K. Kato [Contemp. Math. 83, 101-131 (1989; Zbl 0716.12006)]. For an algebra \(D\) of exponent \(p^n\) and for each integer \(0\leq j\leq n\), one can define \(s_j=\text{sw}(p^j[D])\), where \([D]\) stands for the Brauer class of \(D\). Then Herbrand’s function \(\psi:{\mathbb Z}_{\geq 0}\to{\mathbb Z}_{\geq 0}\) is defined as follows: \(\psi (0)=0, \psi (i)=\psi (s_j)+p^{n-j}(i-s_j)\) for \(s_j\leq i\leq s_{j-1}\).
The first main result of the paper (Theorem 4.1) says that this Herbrand’s function defines a filtration on the unit group \(U_D\) of \(D\) that is compatible (with respect to the action of the reduced norm \(D^*\to K^*\)) with the natural filtration on the unit group \(U_K\) of \(K\).
Next, the author defines the ramification numbers of \(D\) as \(\psi (s_j)\). The second main result (Theorem 5.1) expresses these numbers in terms of the normalized valuation \(v_D\) of \(D\). In particular, it turns out that the least ramification number equals inf{\(v_D(aba^{-1}b^{-1}-1) |a,b\in D^*\}\). This is a generalization of a theorem of K. Kato [J. Fac. Sci., Univ. Tokyo, Sec. I A 26, 303-376 (1979; Zbl 0428.12013)].

Reviewer: B.Kunyavskii (Ramat Gan)

MSC:

11S15	Ramification and extension theory
16K20	Finite-dimensional division rings
12F15	Inseparable field extensions

Keywords:

division algebra; reduced norm; Swan conductor; Herbrand’s function

Citations:

Zbl 0716.12006; Zbl 0428.12013