Valuations on Division Rings

Abstract

AbstractIn this chapter we introduce the central object of study in this book: valuations on a division algebra D finite-dimensional over its center F. In §1.1 we define valuations and describe the associated structures familiar from commutative valuation theory: the valuation ring \(\mathcal {O}_{D}\), its unique maximal left and maximal right ideal \(\mathfrak {m}_{D}\), the residue division algebra \(\overline{D}\), and the value group Γ D . We also describe an important and distinctively noncommutative feature, namely a canonical homomorphism θ D from Γ D to the automorphism group \(\operatorname {\mathit{Aut}}(Z(\overline{D})\big/\,\overline{F}\,)\); θ D is induced by conjugation by elements of D ×. In §1.2, after proving the “Fundamental Inequality for valued division algebras, we look at valuations on D from the perspective of F. We show that a valuation on F has at most one extension to D, and prove a criterion for when such an extension exists. When this occurs, we show that the field \(Z(\overline{D})\) is finite-dimensional and normal over \(\overline{F}\) and that θ D is surjective. We also describe the technical adjustments needed to apply the classical method of “composition” of valuations to division algebras. The filtration on D induced by a valuation leads to an associated graded ring \(\operatorname {\mathsf {gr}}(D)\), which we describe in §1.3. Throughout the book we emphasize use of \(\operatorname {\mathsf {gr}}(D)\) to help understand the valuation on D. This chapter includes many examples of division algebras with valuations.KeywordsDivision AlgebraDivision RingValuation RingLaurent SeriesCanonical HomomorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.