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SK1

OF GRADED DIVISION ALGEBRAS

R. HAZRAT AND A. R. WADSWORTH

Abstract. The reduced Whitehead group SK1 of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non- graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that SK1 of a tame valued division algebra over a henselian eld coincides with SK1 of its associated graded division algebra. Furthermore, it is shown that SK1 of a graded division algebra is isomorphic to SK1 of its quotient division algebra. The rst theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a uni ed manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK1 for generic abelian crossed products.

1. Introduction

Let

D

be

a

division

algebra

with

a

valuation.

To

this

one

associates

a

graded

division

algebra

gr(D) =

Lγ

D

gr(D)γ, where

D

is the value group of D and the summands gr(D)γ

arise from the

filtration on D induced by the valuation (see §2 for details). As is illustrated in [HwW2], even though computations in the graded setting are often easier than working directly with D, it seems that not much is lost in passage from D to its corresponding graded division algebra gr(D). This has provided motivation to systematically study this correspondence, notably by Boulagouaz [B], Hwang, Tignol and Wadsworth [HwW1, HwW2, TW], and to compare certain functors defined on these objects, notably the Brauer group.

In particular, the associated graded ring gr(D) is an Azumaya algebra ([HwW2], Cor. 1.2); so the reduced norm map exists for it, and one defines the reduced Whitehead group SK1 for gr(D) as the kernel of the reduced norm modulo the commutator subgroup of Dand SH0 as the cokernel of the reduced norm map (see (3.1) below). In this paper we study these groups for a graded division algebra.

Apart from the work of Panin and Suslin [PS] on SH0 for Azumaya algebras over semilocal regular rings and [H4] which studies SK1 for Azumaya algebras over henselian rings, it seems that not much is known about these groups in the setting of Azumaya algebras. Specializing to division algebras, however, there is an extensive literature on the group SK1. Platonov [P1] showed that SK1 could be non-trivial for certain division algebras over henselian valued fields. He thereby provided a series of counter-examples to questions raised in the setting of algebraic groups, notably the Kneser-Tits conjecture. (For surveys on this work and the group SK1, see [P3], [G], [Mer] or [W2], §6.)

In this paper we first study the reduced Whitehead group SK1 of a graded division algebra whose grade group is totally ordered abelian (see §3). It can be observed that the computations here are significantly easier and more transparent than in the non-graded setting. For a division algebra D finite-dimensional over a henselian valued field F , the valuation on F extends uniquely to D (see Th. 2.1 in [W2], or [W1]), and the filtration on D induced by the valuation yields an associated graded division algebra gr(D). Previous work on the subject has shown that this transition to graded setting is most “faithful” when the valuation is tame. Indeed, in Section 4, we show that for a tame valued division algebra D over a henselian field, SK1(D) coincides with SK1(gr(D)) (Th. 4.8). Having established this bridge between the graded setting and non-graded case, we will easily deduce known formulas in the literature for the reduced Whitehead

The rst author acknowledges the support of EPSRC rst grant scheme EP/D03695X/1. The second author would like to thank the rst author and Queen’s University, Belfast for their hospitality while the research for this paper was carried out.

Both authors thank the referee for his or her careful reading of the paper and constructive suggestions. 1

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