SK_{1} OF GRADED DIVISION ALGEBRAS

R. HAZRAT AND A. R. WADSWORTH

Abstract. The reduced Whitehead group SK_{1} 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 SK_{1} of a tame valued division algebra over a henselian field coincides with SK_{1} of its associated graded division algebra. Furthermore, it is shown that SK_{1} of a graded division algebra is isomorphic to SK_{1} of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes SK_{1} for generic abelian crossed products.

Contents

1. Introduction  1
2. Graded division algebras  3
3. Reduced norm and reduced Whitehead group of a graded division algebra  7
4. SK_{1} of a valued division algebra and its associated graded division algebra  12
5. Stability of the reduced Whitehead group  17
Appendix A. The Wedderburn factorization theorem  26
Appendix B. The Congruence theorem for tame division algebras  28
References  33

1. Introduction

Let D be a division algebra with a valuation. To this one associates a graded division algebra gr(D) = ⨁_{γ∈Γ_{D}} gr(D)_{γ}, where Γ_{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 [HwW_{2}], 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,

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

notably by Boulagouaz [B], Hwang, Tignol and Wadsworth [HwW_{1}, HwW_{2}, 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 ([HwW_{2}], Cor. 1.2); so the reduced norm map exists for it, and one defines the reduced Whitehead group SK_{1} for gr(D) as the kernel of the reduced norm map and SH^{0} as its cokernel (see §3). In this paper we study these groups for a graded division algebra.

Apart from the work of Panin and Suslin [PS] on SH^{0} for Azumaya algebras over semilocal regular rings and [H_{4}] which studies SK_{1} 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 SK_{1}. Platonov [P_{1}] showed that SK_{1} 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 SK_{1}, see [P_{2}], [G], [Mer] or [W_{2}], §6.)

In this paper we first study the reduced Whitehead group SK_{1} 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 [W_{2}], or [W_{1}]), 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, SK_{1}(D) coincides with SK_{1}(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 group of certain valued division algebras, by passing to the graded setting; this shows the utility of the graded approach (see Cor. 4.10).

In the other direction, if E = ⨁_{γ∈Γ_{E}} E_{γ} is a graded division algebra whose grade group Γ_{E} is torsion-free abelian, then E has a quotient division algebra q(E) which has the same index as E. The same question on comparing the reduced Whitehead groups of these objects can also be raised here. It is known that when the grade group is ℤ, then E has the simple form of a skew Laurent polynomial ring D[x, x^{−1}, φ], where D is a division algebra and φ is an automorphism of D. In this setting the quotient division algebra of D[x, x^{−1}, φ] is D(x, φ). In [PY], Platonov and Yanchevskiĭ compared SK_{1}(D(x, φ)) with SK_{1}(D). In particular, they showed that if φ is an inner automorphism then SK_{1}(D(x, φ)) ≅ SK_{1}(D). In fact, if φ is inner, then D[x, x^{−1}, φ] is an unramified graded division algebra and we prove that SK_{1}(D[x, x^{−1}, φ]) ≅ SK_{1}(D) (Th. 3.6). By combining these, one concludes that the reduced Whitehead group of the graded division algebra D[x, x^{−1}, φ], where φ is inner, coincides with SK_{1} of its quotient division algebra. In Section 5, we show that this is a very special case of stability of SK_{1} for graded division algebras; namely, for any graded division algebra with torsion-free grade group, the reduced Whitehead group coincides with the reduced Whitehead group of its quotient division algebra. This allows us to give a formula for SK_{1} for generic abelian crossed product algebras.

The paper is organized as follows: In Section 2, we gather relevant background on the theory of graded division algebras indexed by a totally ordered abelian group and establish several homomorphisms needed in the paper. Section 3 studies the reduced Whitehead group SK_{1} of a graded division algebra. We establish analogues to Ershov’s linked exact sequences [E] in the graded setting, easily deducing formulas for SK_{1} of unramified, totally ramified, and semiramified graded division algebras. In Section 4, we prove that SK_{1} of a tame division algebra over a henselian field coincides with SK_{1} of its associated graded division algebra. Section 5 is devoted to proving that SK_{1} of a graded division algebra is isomorphic to SK_{1} of is quotient division algebra. We conclude the paper with two appendices. Appendix A establishes the Wedderburn factorization theorem on the setting of graded division rings, namely that the minimal polynomial of a homogenous element of a graded division ring E splits completely over E (Th. A.1). Appendix B provides a complete proof of the Congruence Theorem for all tame division algebras over henselian valued fields. This theorem was originally proved by Platonov for the case of complete discrete valuations of rank 1, and it was a key tool in his calculations of SK_{1} for certain valued division algebras.

2. Graded division algebras

In this section we establish notation and recall some fundamental facts about graded division algebras indexed by a totally ordered abelian group, and about their connections with valued division algebras. In addition, we establish some important homomorphisms relating the group structure of a valued division algebra to the group structure of its associated graded division algebra.

Let R = ⨁_{γ∈Γ} R_{γ} be a graded ring, i.e., Γ is an abelian group, and R is a unital ring such that each R_{γ} is a subgroup of (R, +) and R_{γ} · R_{δ} ⊆ R_{γ+δ} for all γ, δ ∈ Γ. Set

Γ_{R} = {γ ∈ Γ | R_{γ} ≠ 0}, the grade set of R;
R^{h} = ⋃_{γ∈Γ_{R}} R_{γ}, the set of homogeneous elements of R.

For a homogeneous element of R of degree γ, i.e., an r ∈ R_{γ} \ 0, we write deg(r) = γ. Recall that R_{0} is a subring of R and that for each γ ∈ Γ_{R}, the group R_{γ} is a left and right R_{0}-module. A subring S of R is a graded subring if S = ⨁_{γ∈Γ_{R}}(S ∩ R_{γ}). For example, the center of R, denoted Z(R), is a graded subring of R. If T = ⨁_{γ∈Γ} T_{γ} is another graded ring, a graded ring homomorphism is a ring homomorphism f: R → T with f(R_{γ}) ⊆ T_{γ} for all γ ∈ Γ. If f is also bijective, it is called a graded ring isomorphism; we then write R ≅_{gr} T.

For a graded ring R, a graded left R-module M is a left R-module with a grading M = ⨁_{γ∈Γ′} M_{γ}, where the M_{γ} are all abelian groups and Γ′ is a abelian group containing Γ, such that R_{γ} · M_{δ} ⊆ M_{γ+δ} for all γ ∈ Γ_{R}, δ ∈ Γ′. Then, Γ_{M} and M^{h} are defined analogously to Γ_{R} and R^{h}. We say that M is a graded free R-module if it has a base as a free R-module consisting of homogeneous elements.

A graded ring E = ⨁_{γ∈Γ} E_{γ} is called a graded division ring if Γ is a torsion-free abelian group and every non-zero homogeneous element of E has a multiplicative inverse. Note that the grade set Γ_{E} is actually a group. Also, E_{0} is a division ring, and E_{γ} is a 1-dimensional left and right E_{0} vector space for every γ ∈ Γ_{E}. The requirement that Γ be torsion-free is made because we are interested in graded division rings arising from valuations on division rings, and all the grade groups appearing there are torsion-free. Recall that every torsion-free abelian group Γ admits total orderings compatible with the group structure. (For example, Γ embeds in Γ ⊗_{ℤ} ℚ which can be given a lexicographic total ordering using any base of it as a ℚ-vector space.) By using any total ordering on Γ_{E}, it is easy to see that E has no zero divisors and that E^{∗}, the multiplicative group of units of E, coincides with E^{h} \ {0} (cf. [HwW_{2}], p. 78). Furthermore, the degree map

deg: E^{∗} → Γ_{E}  (2.1)

is a group homomorphism with kernel E_{0}^{∗}.

By an easy adaptation of the ungraded arguments, one can see that every graded module M over a graded division ring E is graded free, and every two homogenous bases have the same cardinality. We thus call M a graded vector space over E and write dim_{E}(M) for the rank of M as a graded free E-module. Let S ⊆ E be a graded subring which is also a graded division ring. Then, we can view E as a graded left S-vector space, and we write [E : S] for dim_{S}(E). It is easy to check the “Fundamental Equality,”

[E : S] = [E_{0} : S_{0}] |Γ_{E} : Γ_{S}|,  (2.2)

where [E_{0} : S_{0}] is the dimension of E_{0} as a left vector space over the division ring S_{0} and |Γ_{E} : Γ_{S}| denotes the index in the group Γ_{E} of its subgroup Γ_{S}.

A graded field T is a commutative graded division ring. Such a T is an integral domain, so it has a quotient field, which we denote q(T). It is known, see [HwW_{1}], Cor. 1.3, that T is integrally closed in q(T). An extensive theory of graded algebraic extensions of graded fields has been developed in [HwW_{1}]. For a graded field T, we can define a grading on the polynomial ring T [x] as follows: Let Δ be a totally ordered abelian group with Γ_{T} ⊆ Δ, and fix θ ∈ Δ. We have

T [x] = ⨁_{γ∈Δ} T [x]_{γ}, where T [x]_{γ} = {∑ a_{i}x^{i} | a_{i} ∈ T^{h}, deg(a_{i}) + iθ = γ}.  (2.3)

This makes T [x] a graded ring, which we denote T [x]^{θ}. Note that Γ_{T [x]^{θ}} = Γ_{T} + ⟨θ⟩. A homogeneous polynomial in T [x]^{θ} is said to be θ-homogenizable. If E is a graded division algebra with center T, and a ∈ E^{h} is homogeneous of degree θ, then the evaluation homomorphism ϵ_{a}: T [x]^{θ} → T [a] given by f ↦ f(a) is a graded ring homomorphism. Assuming [T [a] : T] < ∞, we have ker(ϵ_{a}) is a principal ideal of T [x] whose unique monic generator h_{a} is called the minimal polynomial of a over T. It is known, see [HwW_{1}], Prop. 2.2, that if deg(a) = θ, then h_{a} is θ-homogenizable.

If E is a graded division ring, then its center Z(E) is clearly a graded field. The graded division rings considered in this paper will always be assumed finite-dimensional over their centers. The finite-dimensionality assures that E has a quotient division ring q(E) obtained by central localization, i.e., q(E) = E ⊗_{T} q(T) where T = Z(E). Clearly, Z(q(E)) = q(T) and ind(E) = ind(q(E)), where the index of E is defined by ind(E)^{2} = [E : T]. If S is a graded field which is a graded subring of Z(E) and [E : S] < ∞, then E is said to be a graded division algebra over S.

A graded division algebra E with center T is said to be unramified if Γ_{E} = Γ_{T}. From (2.2), it follows then that [E : S] = [E_{0} : T_{0}]. At the other extreme, E is said to be totally ramified if E_{0} = T_{0}. In a case in the middle, E is said to be semiramified if E_{0} is a field and [E_{0} : T_{0}] = |Γ_{E} : Γ_{T} | = ind(E). These definitions are motivated by analogous definitions for valued division algebras ([W_{2}]). Indeed, if a valued division algebra is unramified, semiramified, or totally ramfied, then so is its associated graded division algebra (see §4).

A main theme of this paper is to study the correspondence between SK_{1} of a valued division algebra and that of its associated graded division algebra. We now recall how to associate a graded division algebra to a valued division algebra.

Let D be a division algebra finite dimensional over its center F, with a valuation v : D^{∗} → Γ. So Γ is a totally ordered abelian group, and v satisifies the conditions that for all a, b ∈ D^{∗},

(1) v(ab) = v(a) + v(b);

(2) v(a + b) ≥ min{v(a), v(b)} (b ≠ −a).

Let

V_{D} = {a ∈ D^{∗} : v(a) ≥ 0} ∪ {0}, the valuation ring of v;
M_{D} = {a ∈ D^{∗} : v(a) > 0} ∪ {0}, the unique maximal left (and right) ideal of V_{D};
\overline{D} = V_{D}/M_{D}, the residue division ring of v on D; and
Γ_{D} = im(v), the value group of the valuation.

For background on valued division algebras, see [JW] or the survey paper [W_{2}]. One associates to D a graded division algebra as follows: For each γ ∈ Γ_{D}, let

D^{≥γ} = {d ∈ D^{∗} : v(d) ≥ γ} ∪ {0}, an additive subgroup of D;
D^{>γ} = {d ∈ D^{∗} : v(d) > γ} ∪ {0}, a subgroup of D^{≥γ}; and
gr(D)_{γ} = D^{≥γ}/D^{>γ}.

Then define

gr(D) = ⨁_{γ∈Γ_{D}} gr(D)_{γ}.

Because D^{>γ}D^{≥δ} + D^{≥γ}D^{>δ} ⊆ D^{>(γ+δ)} for all γ, δ ∈ Γ_{D}, the multiplication on gr(D) induced by multiplication on D is well-defined, giving that gr(D) is a graded ring, called the associated graded ring of D. The multiplicative property (1) of the valuation v implies that gr(D) is a graded division ring. Clearly, we have gr(D)_{0} = \overline{D} and Γ_{gr(D)} = Γ_{D}. For d ∈ D^{∗}, we write d̃ for the image d + D^{>v(d)} of d in gr(D)_{v(d)}. Thus, the map given by d ↦ d̃ is a group epimorphism D^{∗} → gr(D)^{∗} with kernel 1 + M_{D}.

The restriction v|_{F} of the valuation on D to its center F, is a valuation on F, which induces a corresponding graded field gr(F). Then it is clear that gr(D) is a graded gr(F)-algebra, and by (2.2) and the Fundamental Inequality for valued division algebras,

[gr(D) : gr(F)] = [\overline{D} : \overline{F}] |Γ_{D} : Γ_{F} | ≤ [D : F] < ∞.

Let F be a field with a henselian valuation v. Recall that a field extension L of F of degree n < ∞ is said to be tamely ramified or tame over F if, with respect to the unique extension of v to L, the residue field \overline{L} is a separable field extension of \overline{F} and char(\overline{F}) ∤ n/[\overline{L} : \overline{F}]. Such an L is necessarily defectless over F, i.e., [L : F] = [\overline{L} : \overline{F}] |Γ_{L} : Γ_{F} | = [gr(L) : gr(F)]. Along the same lines, let D be a division algebra with center F (so, by convention, [D : F] < ∞); then v on F extends uniquely to a valuation on D. With respect to this valuation, D is said to be tamely ramified or tame if Z(\overline{D}) is separable over \overline{F} and char(\overline{F}) ∤ ind(D)/( ind(\overline{D})[Z(\overline{D}) : \overline{F}]). It is known (cf. Prop. 4.3 in [HwW_{2}]) that D is tame if and only if [gr(D) : gr(F)] = [D : F] and Z(gr(D)) = gr(F), if and only if D is split by the maximal tamely ramified extension of F, if and only if char(\overline{F}) = 0 or char(\overline{F}) = p ≠ 0 and the p-primary component of D is split by the maximal unramified extension of F. We say D is strongly tame if char(\overline{F}) ∤ ind(D). Note that strong tameness implies tameness. This is clear from the last characterization of tameness, or from (2.4) below. For a detailed study of the associated graded algebra of a valued division algebra refer to §4 in [HwW_{2}]. Recall also from [Mor], Th. 3, that for a valued division algebra D finite dimensional over its center F (here not necessarily henselian), we have the “Ostrowski theorem”

[D : F] = q^{k} [\overline{D} : \overline{F}] |Γ_{D} : Γ_{F} |  (2.4)

where q = char(\overline{D}) and k ∈ ℤ with k ≥ 0 (and q^{k} = 1 if char(\overline{D}) = 0). If q^{k} = 1 in equation (2.4), then D is said to be defectless over F.

Let E be a graded division algebra with, as we always assume, Γ_{E} a torsion-free abelian group. After fixing some total ordering on Γ_{E}, define a function

λ: E \ {0} → E^{∗} by λ(∑ c_{γ}) = c_{δ},

where δ is minimal among the γ ∈ Γ_{E} with c_{γ} ≠ 0. Note that λ(a) = a for a ∈ E^{∗}, and

λ(ab) = λ(a)λ(b) for all a, b ∈ E \ {0}.  (2.5)

Let Q = q(E). We can extend λ to a map defined on all of Q^{∗} as follows: for q ∈ Q^{∗}, write q = ac^{−1} with a ∈ E \ {0}, c ∈ Z(E) \ {0}, and set λ(q) = λ(a)λ(c)^{−1}. It follows from (2.5) that λ: Q^{∗} → E^{∗} is well-defined and is a group homomorphism. Since the composition E^{∗} ↪ Q^{∗} → E^{∗} is the identity, λ is a splitting map for the injection E^{∗} ↪ Q^{∗}. (In Lemma 5.5 below, we will observe that this map induces a monomorphism from SK_{1}(E) to SK_{1}(Q).)

Now, by composing λ with the degree map of (2.1) we get a map v,

\xymatrix{Q^{∗} \ar[r]^{λ} \ar[dr]^{v} & E^{∗} \ar[d]^{deg} \\  & Γ_{E}}  (2.6)

This v is in fact a valuation on Q: for a, b ∈ Q^{∗}, v(ab) = v(a) + v(b) as v is the composition of two group homomorphisms, and it is straightforward to check that v(a + b) ≥ min(v(a), v(b)) (check this first for a, b ∈ E \ {0}). It is easy to see that for the associated graded ring for this valuation on q(E), we have gr(q(E)) ≅_{gr} E; this is a strong indication of the close connection between graded and valued structures.

3. Reduced norm and reduced Whitehead group of a graded division algebra

Let A be an Azumaya algebra of constant rank n^{2} over a commutative ring R. Then there is a commutative ring S faithfully flat over R which splits A, i.e., A ⊗_{R} S ≅ M_{n}(S). For a ∈ A, considering a ⊗ 1 as an element of M_{n}(S), one then defines the reduced characteristic polynomial, the reduced trace, and the reduced norm of a by

char_{A}(x, a) = det(x − (a ⊗ 1)) = x^{n} − Trd_{A}(a)x^{n−1} + … + (−1)^{n}Nrd_{A}(a).

Using descent theory, one shows that char_{A}(x, a) is independent of S and of the choice of isomorphism A⊗_{R} S ≅ M_{n}(S), and that char_{A}(x, a) lies in R[x]; furthermore, the element a is invertible in A if and only if Nrd_{A}(a) is invertible in R (see Knus [K], III.1.2, and Saltman [S_{2}], Th. 4.3). Let A^{(1)} denote the set of elements of A with the reduced norm 1. One then defines the reduced Whitehead group of A to be SK_{1}(A) = A^{(1)}/A′, where A′ denotes the commutator subgroup of the group A^{∗} of invertible elements of A. The reduced norm residue group of A is defined to be SH^{0}(A) = R^{∗}/Nrd_{A}(A^{∗}). These groups are related by the exact sequence:

1 ⟶ SK_{1}(A) ⟶ A^{∗}/A′ ⟶^{Nrd} R^{∗} ⟶ SH^{0}(A) ⟶ 1

Now let E be a graded division algebra with center T. Since E is an Azumaya algebra over T ([B], Prop. 5.1 or[HwW_{2}], Cor. 1.2), its reduced Whitehead group SK_{1}(E) is defined.

Remark 3.1. The reduced norm for an Azumaya algebra is defined using a splitting ring, and in general splitting rings can be difficult to find. But for a graded division algebra E we observe that, analogously to the case of ungraded division rings, any maximal graded subfield L of E splits E. For, the centralizer C = C_{E}(L) is a graded subring of E containing L, and for any homogeneous c ∈ C, L[c] is a graded subfield of E containing L. Hence, C = L, showing that L is a maximal commutative subring of E. Thus, by Lemma 5.1.13(1), p. 141 of [K], as E is Azumaya, E ⊗_{T} L ≅ End_{L}(E) ≅ M_{n}(L). Thus, we can compute reduced norms for elements of E by passage to E ⊗_{T} L.

We have other tools as well for computing Nrd_{E} and Trd_{E}:

Proposition 3.2. Let E be a graded division ring with center T. Let q(T) be the quotient field of T, and let q(E) = E ⊗_{T} q(T), which is the quotient division ring of E. We view E ⊆ q(E). Let n = ind(E) = ind(q(E)). Then for any a ∈ E,

(i) char_{E}(x, a) = char_{q(E)}(x, a), so

Nrd_{E}(a) = Nrd_{q(E)}(a) and Trd_{E}(a) = Trd_{q(E)}(a).  (3.1)

(ii) If K is any graded subfield of E containing T and a ∈ K, then

Nrd_{E}(a) = N_{K/T} (a)^{n/[K:T]} and Trd_{E}(a) = \frac{n}{[K:T]} Tr_{K/T} (a).

(iii) For γ ∈ Γ_{E}, if a ∈ E_{γ} then Nrd_{E}(a) ∈ E_{nγ} and Trd(a) ∈ E_{γ}. In particular, E^{(1)} ⊆ E_{0}. (iv) Set δ = ind(E)/( ind(E_{0})[Z(E_{0}) : T_{0}]). If a ∈ E_{0}, then,

Nrd_{E}(a) = N_{Z(E_{0})/T_{0}}Nrd_{E_{0}}(a)^{δ} ∈ T_{0} and Trd_{E}(a) = δ Tr_{Z(E_{0})/T_{0}}Trd_{E_{0}}(a) ∈ T_{0}. (3.2)

Proof. (i) The construction of reduced characteristic polynonials described above is clearly compatible with scalar extension of the ground ring. Hence, char_{E}(x, a) = char_{q(E)}(x, a) (as we are identifying a ∈ E with a ⊗ 1 in E ⊗_{T} q(T) ). The formulas in (3.1) follow immediately.

(ii) Let h_{a} = x^{m} + t_{m−1}x^{m−1} + … + t_{0} ∈ q(T)[x] be the minimal polynomial of a over q(T). As noted in [HwW_{1}], Prop. 2.2, since the integral domain T is integrally closed and E is integral over T, we have h_{a} ∈ T [x]. Let ℓ_{a} = x^{k} + s_{k−1}x^{k−1} + … + s_{0} ∈ T [x] be the characteristic polynomial of the T -linear function on the free T -module K given by c ↦ ac. By definition, N_{K/T} (a) = (−1)^{k}s_{0} and Tr_{K/T} (a) = −s_{k−1}. Since q(K) = K ⊗_{T} q(T), we have [q(K) : q(T)] = [K : T] = k and ℓ_{a} is also the characteristic polynomial for the q(T)-linear transformation of q(K) given by q ↦ aq. So, ℓ_{a} = h_{a}^{k/m}. Since char_{q(E)}(x, a) = h_{a}^{n/m} (see [R], Ex. 1, p. 124), we have char_{q(E)}(x, a) = ℓ_{a}^{n/k}. Therefore, using (i),

Nrd_{E}(a) = Nrd_{q(E)}(a) = [(−1)^{k}s_{0}]^{n/k} = N_{K/T} (a)^{n/k}.

The formula for Trd_{E}(a) in (ii) follows analogously.

(iii) From the equalities char_{E}(x, a) = char_{q(E)}(x, a) = h_{a}^{n/m} noted in proving (i) and (ii), we have Nrd_{E}(a) = [(−1)^{m}t_{0}]^{n/m} and Trd_{E}(a) = −\frac{n}{m}t_{m−1}. As noted in [HwW_{1}], Prop. 2.2, if a ∈ E_{γ}, then its minimal polynomial h_{a} is γ-homogenizable in T [x] as in (2.3) above. Hence, t_{0} ∈ E_{mγ} and t_{m−1} ∈ E_{γ}. Therefore, Nrd_{E}(a) ∈ E_{nγ} and Trd(a) ∈ E_{γ}. If a ∈ E^{(1)} then a is homogeneous, since it is a unit of E, and since 1 = Nrd_{E}(a) ∈ E_{n deg(a)}, necessarily deg(a) = 0.

(iv) Suppose a ∈ E_{0}. Then, h_{a} is 0-homogenizable in T [x], i.e., h_{a} ∈ T_{0}[x]. Hence, h_{a} is the minimal polynomial of a over the field T_{0}. Therefore, if L is any maximal subfield of E_{0} containing a, we have N_{L/T_{0}}(a) = [(−1)^{m}t_{0}]^{[L:T_{0}]/m}. Now,

n/m = δ ind(E_{0})[Z(E_{0}) : T_{0}]/m = δ [L : T_{0}]/m.

Hence,

Nrd_{E}(a) = [(−1)^{m}t_{0}]^{n/m} = [(−1)^{m}t_{0}]^{δ[L:T_{0}]/m} = N_{L/T_{0}}(a)^{δ}
= N_{Z(E_{0})/T_{0}}N_{L/T_{0}}(a)^{δ} = N_{Z(E_{0})/T_{0}}Nrd_{E_{0}}(a)^{δ}.

The formula for Trd_{E}(a) is proved analogously. □

In the rest of this section we study the reduced Whitehead group SK_{1} of a graded division algebra. As we mentioned in the introduction, the motif is to show that working in the graded setting is much easier than in the non-graded setting.

The most successful approach to computing SK_{1} for division algebras over henselian fields is due to Ershov in [E], where three linked exact sequences were constructed involving a division algebra D, its residue division algebra \overline{D}, and its group of units U_{D} (see also [W_{2}], p. 425). From these exact sequences, Ershov recovered Platonov’s examples [P_{1}] of division algebras with nontrivial SK_{1} and many more examples as well. In this section we will easily prove the graded version of Ershov’s exact sequences (see diagram (3.4)), yielding formulas for SK_{1} of unramified, semiramified, and totally ramified graded division algebras. This will be applied in §4, where it will be shown that SK_{1} of a tame division algebra over a henselian field coincides with SK_{1} of its associated graded division algebra. We can then readily deduce from the graded results many established formulas in the literature for the reduced Whitehead groups of valued division algebras (see Cor. 4.10). This demonstrates the merit of the graded approach.

If N is a group, we denote by N^{n} the subgroup of N generated by all n-th powers of elements of N. A homogeneous multiplicative commutator of E where E is a graded division ring, is an element of the form aba^{−1}b^{−1} where a, b ∈ E^{∗} = E^{h} \ {0}. We will use the notation [a, b] = aba^{−1}b^{−1} for a, b ∈ E^{h}. Since a and b are homogeneous, note that [a, b] ∈ E_{0}. If H and K are subsets of E^{∗}, then [H, K] denotes the subgroup of E^{∗} generated by {[h, k] : h ∈ H, k ∈ K}. The group [E^{∗}, E^{∗}] will be denoted by E′.

Proposition 3.3. Let E = ⨁_{α∈Γ} E_{α} be a graded division algebra with graded center T, with ind(E) = n. Then,

(i) If N is a normal subgroup of E^{∗}, then N^{n} ⊆ Nrd_{E}(N)[E^{∗}, N].

(ii) SK_{1}(E) is n-torsion.

Proof. Let a ∈ N and let h_{a} ∈ q(T)[x] be the minimal polynomial of a over q(T), and let m = deg(h_{a}). As noted in the proof of Prop. 3.2, h_{a} ∈ T [x] and Nrd_{E}(a) = [(−1)^{m}h_{a}(0)]^{n/m}. By the graded Wedderburn Factorization Theorem A.1, we have h_{a} = (x − d_{1}ad_{1}^{−1}) … (x − d_{m}ad_{m}^{−1}) where each d_{i} ∈ E^{∗} ⊆ E^{h}. Note that [E^{∗}, N] is a normal subgroup of E^{∗}, since N is normal in E^{∗}. It follows that

Nrd_{E}(a) = (d_{1}ad_{1}^{−1} … d_{m}ad_{m}^{−1})^{n/m} = ([d_{1}, a]a[d_{2}, a]a … a[d_{m}, a]a)^{n/m}
= a^{n}d_{a} where d_{a} ∈ [E^{∗}, N].

Therefore, a^{n} = Nrd_{E}(a)d_{a}^{−1} ∈ Nrd_{E}(N)[E^{∗}, N], yielding (i). (ii) is immediate from (i) by taking N = E^{(1)}. □

The fact that SK_{1}(E) is n-torsion is also deducible from the injectivity of the map SK_{1}(E) → SK_{1}(q(E)) shown in Lemma 5.5 below.

We recall the definition of the group Ĥ^{−1}(G, A), which will appear in our description of SK_{1}(E). For any finite group G and any G-module A, define the norm map N_{G} : A → A as follows: for any a ∈ A, let N_{G}(a) = ∑_{g∈G} ga. Consider the G-module I_{G}(A) generated as an abelian group by {a − ga : a ∈ A and g ∈ G}. Clearly, I_{G}(A) ⊆ ker(N_{G}). Then,

Ĥ^{−1}(G, A) = ker(N_{G})/I_{G}(A).  (3.3)

Theorem 3.4. Let E be any graded division ring finite dimensional over its center T. Let δ = ind(E)/( ind(E_{0}) [Z(E_{0}) : T_{0}]), and let μ_{δ}(T_{0}) be the group of those δ-th roots of unity lying in T_{0}. Let G = 𝒢al(Z(E_{0})/T_{0}) and let Ñ = N_{Z(E_{0})/T_{0}} ∘Nrd_{E_{0}} : E_{0}^{∗} → T_{0}^{∗}. Then, the rows and column of the following diagram are exact:

\xymatrix{ & 1 \ar[d] \\ SK_{1}(E_{0}) \ar[r] & ker Ñ/[E_{0}^{∗}, E^{∗}] \ar[r]^{_{Nrd_{E_{0}}}} \ar[d] & Ĥ^{−1}(G, Nrd_{E_{0}}(E_{0}^{∗})) \ar[r] & 1 \\ Γ_{E}/Γ_{T} ∧ Γ_{E}/Γ_{T} \ar[r] & E^{(1)}/[E_{0}^{∗}, E^{∗}] \ar[r] \ar[d]^{Ñ} & SK_{1}(E) \ar[r] & 1 \\  & μ_{δ}(T_{0}) ∩ Ñ(E_{0}^{∗}) \ar[d] \\  & 1}  (3.4)

Proof. By Prop. 2.3 in [HwW_{2}], Z(E_{0})/T_{0} is a Galois extension and the map θ: E^{∗} → Aut(E_{0}), given by e ↦ (a ↦ eae^{−1}) for a ∈ E_{0}, induces an epimorphism E^{∗} → G = 𝒢al(Z(E_{0})/T_{0}). In the notation for (3.3) with A = Nrd_{E_{0}}(E_{0}^{∗}), we have N_{G} coincides with N_{Z(E_{0})/T_{0}} on A. Hence,

ker(N_{G}) = Nrd_{E_{0}}(ker(Ñ)).  (3.5)

Take any e ∈ E^{∗} and let σ = θ(e) ∈ Aut(E_{0}). For any a ∈ E_{0}^{∗}, let h_{a} ∈ Z(T_{0})[x] be the minimal polynomial of a over Z(T_{0}). Then σ(h_{a}) ∈ Z(T_{0})[x] is the minimal polynomial of σ(a) over Z(T_{0}). Hence, Nrd_{E_{0}}(σ(a)) = σ(Nrd_{E_{0}}(a)). Since σ|_{Z(T_{0})} ∈ G, this yields

Nrd_{E_{0}}([a, e]) = Nrd_{E_{0}}(aσ(a^{−1})) = Nrd_{E_{0}}(a)σ(Nrd_{E_{0}}(a))^{−1} ∈ I_{G}(A),  (3.6)

hence Ñ([a, e]) = 1. Thus, we have [E_{0}^{∗}, E^{∗}] ⊆ ker(Ñ) ⊆ E^{(1)} with the latter inclusion from Prop. 3.2(iv). The formula in Prop. 3.2(iv) also shows that Ñ(E^{(1)}) ⊆ μ_{δ}(T_{0}). Thus, the vertical maps in diagram (3.4) are well-defined, and the column in (3.4) is exact. Because Nrd_{E_{0}} maps ker(Ñ) onto ker(N_{G}) by (3.5) and it maps [E_{0}^{∗}, E^{∗}] onto I_{G}(A) by (3.6) (as θ(E^{∗}) maps onto G), the map labelled Nrd_{E_{0}} in diagram (3.4) is surjective with kernel E_{0}^{(1)} [E_{0}^{∗}, E^{∗}]/[E_{0}^{∗}, E^{∗}]. Therefore, the top row of (3.4) is exact. For the lower row, since [E^{∗}, E^{∗}] ⊆ E_{0}^{∗} and E^{∗}/(E_{0}^{∗} Z(E^{∗})) ≅ Γ_{E}/Γ_{T}, the following lemma yields an epimorphism Γ_{E}/Γ_{T} ∧Γ_{E}/Γ_{T} → [E^{∗}, E^{∗}]/[E_{0}^{∗}, E^{∗}]. Given this, the lower row in (3.4) is evidently exact. □

Lemma 3.5. Let G be a group, and let H be a subgroup of G with H ⊇ [G, G]. Let B = G/(H Z(G)). Then, there is an epimorphism B ∧ B → [G, G]/[H, G].

Proof. Since [G, G] ⊆ H, we have [[G, G], [G, G]] ⊆ [H, G], so [G, H] is a normal subgroup of [G, G] with abelian factor group. Consider the map β: G × G → [G, G]/[H, G] given by (a, b) ↦ aba^{−1}b^{−1}[H, G]. For any a, b, c ∈ G we have the commutator identity [a, bc] = [a, b] [b, [a, c]] [a, c]. The middle term [b, [a, c]] lies in [H, G]. Thus, β is multiplicative in the second variable; likewise, it is multiplicative in the first variable. As [H Z(G), G] ⊆ [H, G], this β induces a well-defined group homomorphism β′: B ⊗_{ℤ}B → [G, G]/[H, G], which is surjective since im(β) generates [G, G]/[H, G]. Since β′(η ⊗ η) = 1 for all η ∈ B, there is an induced epimorphism B ∧ B → [G, G]/[H, G]. □

Corollary 3.6. Let E be a graded division ring with graded center T.

(i) If E is unramified, then SK_{1}(E) ≅ SK_{1}(E_{0}).

(ii) If E is totally ramified, then SK_{1}(E) ≅ μ_{n}(T_{0})/μ_{e}(T_{0}) where n = ind(E) and e is the

exponent of Γ_{E}/Γ_{T}. (iii) If E is semiramified, then for G = 𝒢al(E_{0}/T_{0}) ≅ Γ_{E}/Γ_{T} there is an exact sequence

G ∧ G → Ĥ^{−1}(G, E_{0}^{∗}) → SK_{1}(E) → 1.  (3.7)

(iv) If E has maximal subfields L and K which are respectively unramified and totally ramified over T, then E is semiramified and SK_{1}(E) ≅ Ĥ^{−1}(𝒢al(E_{0}/T_{0}), E_{0}^{∗}).

Proof. (i) Since E is unramified over T, we have E_{0} is a central T_{0}-division algebra, ind(E_{0}) = ind(E), and E^{∗} = E_{0}^{∗}T^{∗}. It follows that G = 𝒢al(Z(E_{0})/T_{0}) is trivial, and thus Ĥ^{−1}(G, Nrd_{E_{0}}(E_{0})) is trivial; also, δ = 1, and from (3.2), Nrd_{E_{0}}(a) = Nrd_{E}(a) for all a ∈ E_{0}. Furthermore, [E_{0}^{∗}, E^{∗}] = [E_{0}^{∗}, E_{0}^{∗}T^{∗}] = [E_{0}^{∗}, E_{0}^{∗}] as T^{∗} is central. Plugging this information into the exact top row of diagram (3.4) and noting that the exact sequence extends to the left by 1 → [E_{0}^{∗}, E^{∗}]/[E_{0}^{∗}, E_{0}^{∗}] → SK_{1}(E_{0}), part (i) follows.

(ii) When E is totally ramified, E_{0} = T_{0}, δ = n, Ñ is the identity map on T_{0}, and [E^{∗}, E_{0}^{∗}] = [E^{∗}, T_{0}^{∗}] = 1. Plugging all this into the exact column of diagram (3.4), it follows that E^{(1)} ≅ μ_{n}(T_{0}). Also by [HwW_{2}] Prop. 2.1, E′ ≅ μ_{e}(T_{0}) where e is the exponent of the torsion abelian group Γ_{E}/Γ_{T}. Part (ii) now follows.

(iii) As recalled at the beginning of the proof of Th. 3.4, for any graded division algebra E with center T, we have Z(E_{0}) is Galois over T_{0}, and there is an epimorphism θ: E^{∗} → 𝒢al(Z(E_{0})/T_{0}). Clearly, E_{0}^{∗} and T^{∗} lie in ker(θ), so θ induces an epimorphism θ′: Γ_{E}/Γ_{T} → 𝒢al(Z(E_{0})/T_{0}). When E is semiramified, by definition [E_{0} : T_{0}] = |Γ_{E} : Γ_{T} | = ind(E) and E_{0} is a field. Let G = 𝒢al(E_{0}/T_{0}). Because |G| = [E_{0} : T_{0}] = |Γ_{E} : Γ_{T} |, the map θ′ must be an isomorphism. In diagram (3.4), since SK_{1}(E_{0}) = 1 and clearly δ = 1, the exact top row and column yield E^{(1)}/[E_{0}^{∗}, E^{∗}] ≅ Ĥ^{−1}(G, E_{0}^{∗}). Therefore, the exact row (3.7) follows from the exact second row of diagram (3.4) and the isomorphism Γ_{E}/Γ_{T} ≅ G given by θ′.

(iv) Since L and K are maximal subfields of E, we have ind(E) = [L : T] = [L_{0} : T_{0}] ≤ [E_{0} : T_{0}] and ind(E) = [K : T] = |Γ_{K} : Γ_{T} | ≤ |Γ_{E} : Γ_{T} |. It follows from (2.2) that these inequalities are equalities, so E_{0} = L_{0} and Γ_{E} = Γ_{K}. Hence, E is semiramified, and (iii) applies. Take any η, ν ∈ Γ_{E}/Γ_{T}, and any inverse images a, b of η, ν in E^{∗}. The left map in (3.7) sends η ∧ ν to aba^{−1}b^{−1} mod I_{G}(E_{0}^{∗}). Since Γ_{E} = Γ_{K}, these a and b can be chosen in K^{∗}, so they commute. Thus, the left map of (3.7) is trivial here, yielding the isomorphism of (iv). □

For a graded division algebra E with center T, define

CK_{1}(E) = E^{∗}/(T^{∗}E′).  (3.8)

This is the graded analogue to CK_{1}(D) for a division algebra D, which is defined as CK_{1}(D) = D^{∗}/(F^{∗}D′), where F = Z(D). That is, CK_{1}(D) is the cokernel of the canonical map K_{1}(F) → K_{1}(D). See [H_{1}] for background on CK_{1}(D). Notably, it is known that CK_{1}(D) is torsion of bounded exponent n = ind(D), and CK_{1} has functorial properties similar to SK_{1}. The CK_{1} functor was used in [HW] in showing that for “nearly all” division algebras D, the multiplicative group D^{∗} has a maximal proper subgroup. It is conjectured (see [HW] and its references) that if CK_{1}(D) is trivial, then D is a quaternion division algebra (necessarily over a real Pythagorean field).

Now, for the graded division algebra E with center T, the degree map (2.1) induces a surjective map E^{∗} → Γ_{E}/Γ_{T} which has kernel T^{∗}E_{0}^{∗}. One can then observe that there is an exact sequence

1 ⟶ E_{0}^{∗}/T_{0}^{∗}E′ ⟶ CK_{1}(E) ⟶ Γ_{E}/Γ_{T} ⟶ 1.

Thus if E is unramified, CK_{1}(E) ≅ E_{0}^{∗}/(T_{0}^{∗}E′) and E^{∗} ≅ T^{∗}E_{0}^{∗}. It then follows that E′ ≅ E_{0}′, yielding CK_{1}(E) ≅ CK_{1}(E_{0}). At the other extreme, when E is totally ramified then E_{0}^{∗}/(T_{0}^{∗}E′) = 1, so the exact sequence above yields CK_{1}(E) ≅ Γ_{E}/Γ_{T}.

4. SK_{1} of a valued division algebra and its associated graded division algebra

The aim of this section is to study the relation between the reduced Whitehead group (and other related functors) of a valued division algebra with that of its corresponding graded division algebra. We will prove that SK_{1} of a tame valued division algebra over a henselian field coincides with SK_{1} of its associated graded division algebra. We start by recalling the concept of λ-polynomials introduced in [MW]. We keep the notations introduced in §2.

Let F be a field with valuation v, let gr(F) be the associated graded field, and F^{alg} the algebraic closure of F. For a ∈ F^{∗}, let ã ∈ gr(F)_{v(a)} be the image of a in gr(F), let ˜0 = 0_{gr(F)}, and for f = ∑ a_{i}x^{i} ∈ F [x], let f̃ = ∑ ã_{i}x^{i} ∈ gr(F)[x].

Definition 4.1. Take any λ in the divisible hull of Γ_{F} and let f = a_{n}x^{n} + … + a_{i}x^{i} + … + a_{0} ∈ F [x] with a_{n}a_{0} ≠ 0. Take any extension of v to F^{alg}. We say that f is a λ-polynomial if it satisfies the following equivalent conditions:

(a) Every root of f in F^{alg} has value λ;

(b) v(a_{i}) ≥ (n − i)λ + v(a_{n}) for all i and v(a_{0}) = nλ + v(a_{n});

(c) Take any c ∈ F^{alg} with v(c) = λ and let h = \frac{1}{a_{n}c^{n}} f(cx) ∈ F^{alg}[x]; then h is monic in V_{F^{alg}}[x] and h(0) ≠ 0 (so h is a 0-polynomial).

If f is a λ-polynomial, let

f^{(λ)} = ∑_{i=0}^{n} a′_{i}x^{i} ∈ gr(F)[x],  (4.1)

where a′_{i} is the image of a_{i} in gr(F)_{(n−i)λ+v(a_{n})} (so a′_{0} = ã_{0} and a′_{n} = ã_{n}, but for 1 ≤ i ≤ n − 1, a′_{i} = 0 if v(a_{i}) > (n − i)λ + v(a_{n}) ). Note that f^{(λ)} is a homogenizable polynomial in gr(F)[x], i.e., f^{(λ)} is homogeneous (of degree v(a_{0})) with respect to the the grading on gr(F)[x] as in (2.3) with θ = λ. Also, f^{(λ)} has the same degree as fas a polynomial in x.

The λ-polynomials are useful generalizations of polynomials h ∈ V_{F} [x] with h(0) ≠ 0—these are 0-polynomials. The following proposition collects some basic properties of λ-polynomials over henselian fields, which are analogous to well-known properties for 0-polynomials, and have similar proofs. See, e.g., [EP], Th. 4.1.3, pp. 87–88 for proofs for 0-polynomials, and [MW] for proofs for λ-polynomials.

Proposition 4.2. Suppose the valuation v on F is henselian. Then,

(i) If f is a λ-polynomial and f = gh in F [x], then g and h are λ-polynomials and f^{(λ)} = g^{(λ)}h^{(λ)} in gr(F)[x]. So, if f^{(λ)} is irreducible in gr(F)[x], then f is irreducible in F [x].

(ii) If f = ∑_{i=0}^{n} a_{i}x^{i} is irreducible in F [x] with a_{n}a_{0} ≠ 0, then f is a λ-polynomial for λ = (v(a_{0}) − v(a_{n}))/n. Furthermore, f^{(λ)} = ã_{n}h^{s} for some irreducible monic λhomogenizable polynomial h ∈ gr(F)[x].

(iii) If f is a λ-polynomial in F [x] and if f^{(λ)} = g′h′ in gr(F)[x] with gcd(g′, h′) = 1, then

there exist λ-polynomials g, h ∈ F [x] such that f = gh and g^{(λ)} = g′ and h^{(λ)} = h′. (iv) If f is a λ-polynomial in F [x] and if f^{(λ)} has a simple root b in gr(F), then f has a

simple root a in F with ã = b.

(v) Suppose k is a λ-polynomial in gr(F)[x] with k(0) ≠ 0, and suppose f ∈ F [x] with f̃ = k. Then f is a λ-polynomial and f^{(λ)} = k.

Lemma 4.3. Let F ⊆ K be fields with [K : F] < ∞. Let v be a henselian valuation on F such that K is defectless over F. Then, for every a ∈ K^{∗}, with ã its image in gr(K)^{∗},

{ \atop N_{K/F} (a)} = N_{gr(K)/gr(F)}(ã).

Proof. Let n = [K : F]. Note that [gr(K) : gr(F)] = n as K is defectless over F. Let f = x^{ℓ} + c_{ℓ−1}x^{ℓ−1} + … + c_{0} ∈ F [x] be the minimal polynomial of a over F. Then f is irreducible in F [x] and since v is henselian, f is a λ-polynomial, where λ = v(a) = v(c_{0})/n (see Prop. 4.2(ii)). Let f^{(λ)} be the corresponding λ-homogenizable polynomial in gr(F)[x] as in (4.1). Then f^{(λ)}(ã) = 0 in gr(K) (by Prop. 4.2(i) with g = x − a), and by Prop. 4.2(ii) f^{(λ)} has only one monic irreducible factor in gr(F)[x], say f^{(λ)} = h^{s}, with deg(h) = ℓ/s. Since f^{(λ)}(ã) = 0, h must be the minimal polynomial of ã over gr(F) and over q(gr(F)). (Recall that since gr(F) is integrally closed, a monic polynomial in gr(F)[x] is irreducible in gr(F)[x] iff it is irreducible in q(gr(F))[x].) We have N_{K/F} (a) = (−1)^{n}c_{0}^{n/ℓ}. Hence, as q(gr(K)) ≅ gr(K) ⊗_{gr(F)} q(gr(F)),

N_{gr(K)/gr(F)}(ã) = N_{q(gr(K))/q(gr(F))}(ã) = (−1)^{n}h(0)^{ns/ℓ} = (−1)^{n}(h(0)^{s})^{n/ℓ}
= (−1)^{n}(c̃_{0}^{n/ℓ}) = { \atop (−1)^{n}c_{0}^{n/ℓ}} = { \atop N_{K/F} (a)}.  □

Remark. The preceding lemma is still valid if v on F is not assumed to be henselian, but merely assumed to have a unique and defectless extension to K. This can be proved by scalar extension to the henselization F^{h} of F. (Since v extends uniquely and defectlessly to K, K ⊗_{F} F^{h} is a field, and gr(K ⊗_{F} F^{h}) ≅_{gr} gr(K).)

Corollary 4.4. Let F be a field with henselian valuation v, and let D be a tame F -central division algebra. Then for every a ∈ D^{∗}, Nrd_{gr(D)}(ã) = { \atop Nrd_{D}(a)}.

Proof. Recall from §2 that the assumption D is tame over F means that [D : F] = [gr(D) : gr(F)] and gr(F) = Z(gr(D)). Take any maximal subfield L of D containing a. Then L/F is defectless as D/F is defectless, so [gr(L) : gr(F)] = [L : F] = ind(D) = ind(gr(D)). Hence, using Lemma 4.3 and Prop. 3.2(ii), we have,

{ \atop Nrd_{D}(a)} = { \atop N_{L/F} (a)} = N_{gr(L)/gr(F)}(ã) = Nrd_{gr(D)}(ã).  □

Remarks 4.5. (i) Again, we do not need that v be henselian for Cor. 4.4. It suffices that the valuation v on F extends to D and D is tame over F.

(ii) Analogous results hold for the trace and reduced trace, with analogous proof. In the setting of Lemma 4.3, we have: if v(Tr_{K/F} (a)) = v(a), then Tr_{K/F} (ã) = { \atop Tr_{K/F} (a)}, but if v(Tr_{K/F} (a)) > v(a), then Tr_{K/F} (ã) = 0.

(iii) By combining Cor. 4.4 with equation (3.2), for a tame valued division algebra D over henselian field F, we can relate the reduced norm of D with the reduced norm of \overline{D} as follows:

\overline{Nrd_{D}(a)} = N_{Z(\overline{D})/\overline{F}} Nrd\overline{D}(\overline{a})^{δ},  (4.2)

for any a ∈ V_{D}\M_{D} (thus, Nrd_{D}(a) ∈ V_{F} \M_{F}) and δ = ind(D)/( ind(\overline{D}) [Z(\overline{D}) : \overline{F}]) (cf. [E], Cor. 2).

The next proposition will be used several times below. It was proved by Ershov in [E], Prop. 2, who refers to Yanchevskiĭ [Y] for part of the argument. We give a proof here for the convenience of the reader, and also to illustrate the utility of λ-polynomials.

Proposition 4.6. Let F ⊆ K be fields with henselian valuations v such that [K : F] < ∞ and K is tamely ramified over F. Then N_{K/F} (1 + M_{K}) = 1 + M_{F}.

Proof. If s ∈ 1 + M_{K} then s̃ = 1 in gr(K). So, as K is defectless over F by Lemma. 4.3, { \atop N_{K/F} (s)} = N_{gr(K)/gr(F)}(s̃) = 1 in gr(F), i.e., N_{K/F} (s) ∈ 1+M_{F}. Thus N_{K/F} (1+M_{K}) ⊆ 1+M_{F}. To prove that this inclusion is an equality, we can assume [K : F] > 1. We have [gr(K) : gr(F)] = [K : F] > 1, since tamely ramified extensions are defectless. Also, the tame ramification implies that q(gr(K)) is separable over q(gr(F)). For, q(gr(F)) · gr(K)_{0} is separable over q(gr(F)) since gr(K)_{0} = \overline{K} and \overline{K} is separable over gr(F)_{0} = \overline{F}. But also, q(gr(K)) is separable over q(gr(F)) · gr(K)_{0} because [q(gr(K)) : q(gr(F) · gr(K)_{0}] = |Γ_{K} : Γ_{F} |, which is not a multiple of char(\overline{F}). Now, take any homogenous element b ∈ gr(K), b ∉ gr(F), and let g be the minimal polynomial of b over q(gr(F)). Then g ∈ gr(F)[x], b is a simple root of g, and g is λ-homogenizable where λ = deg(b), by [HwW_{1}], Prop. 2.2. Take any monic λ-polynomial f ∈ F [x] with f^{(λ)} = g. Since f^{(λ)} has the simple root b in gr(K) and the valuation on K is henselian, by Prop. 4.2(iv) there is a ∈ K such that a is a simple root of f and ã = b. Let L = F (a) ⊆ K. Write f = x^{n} + c_{n−1}x^{n−1} + … + c_{0}. Take any t ∈ 1 + M_{F}, and let h = x^{n} + c_{n−1}x^{n−1} + … + c_{1}x + tc_{0} ∈ F [x]. Then h is a λ-polynomial (because f is) and h^{(λ)} = f^{(λ)} = g in gr(F)[x]. Since h^{(λ)} has the simple root b in gr(L), h has a simple root d in L with d̃= b = ã by Prop. 4.2(iv). So, da^{−1} ∈ 1 + M_{L}. The polynomials f and h are irreducible F [x] by Prop. 4.2(i), as g is irreducible in gr(F)[x]. Since f (resp. h), is the minimal polynomial of a (resp. d) over F, we have N_{L/F} (a) = (−1)^{n}c_{0} and N_{L/F} (d) = (−1)^{n}c_{0}t. Thus, N_{L/F} (da^{−1}) = t, showing that N_{L/F} (1 + M_{L}) = 1 + M_{F}. If L = K, we are done. If not, we have [K : L] < [K : F], and K is tamely ramified over L. So, by induction on [K : F], we have N_{K/L}(1 + M_{K}) = 1 + M_{L}. Hence,

N_{K/F} (1 + M_{K}) = N_{L/F} (N_{K/L}(1 + M_{K})) = N_{L/F} (1 + M_{L}) = 1 + M_{F} .  □

Corollary 4.7. Let F be a field with henselian valuation v, and let D be an F -central division algebra which is tame with respect to v. Then, Nrd_{D}(1 + M_{D}) = 1 + M_{F}.

Proof. Take any a ∈ 1+M_{D} and any maximal subfield K of D with a ∈ K. Then, K is defectless over F, since D is defectless over F. So, a ∈ 1 + M_{K}, and Nrd_{D}(a) = N_{K/F} (a) ∈ 1 + M_{F} by the first part of the proof of Prop. 4.6, which required only defectlessness, not tameness. Thus, Nrd_{D}(1 + M_{D}) ⊆ 1 + M_{F}. For the reverse inclusion, recall from [HwW_{2}], Prop. 4.3 that as D is tame over F, it has a maximal subfield L with L tamely ramified over F. Then by Prop. 4.6,

1 + M_{F} = N_{L/F} (1 + M_{L}) = Nrd_{D}(1 + M_{L}) ⊆ Nrd_{D}(1 + M_{D}) ⊆ 1 + M_{F} ,

so equality holds throughout. □

We can now prove the main result of this section:

Theorem 4.8. Let F be a field with henselian valuation v and let D be a tame F -central division algebra. Then SK_{1}(D) ≅ SK_{1}(gr(D)).

Proof. Consider the canonical surjective group homomorphism ρ: D^{∗} → gr(D)^{∗} given by a ↦ ã. Clearly, ker(ρ) = 1 + M_{D}. If a ∈ D^{(1)} ⊆ V_{D} then ã ∈ gr(D)_{0} and by Cor. 4.4,

Nrd_{gr(D)}(ã) = { \atop Nrd_{D}(a)} = 1.

This shows that ρ(D^{(1)}) ⊆ gr(D)^{(1)}. Now consider the diagram

\xymatrix{1 \ar[r] & (1 + M_{D}) ∩ D′ \ar[r] \ar[d] & D′ \ar[d] \ar[r]^{ρ} & gr(D)′ \ar[d] \ar[r] & 1 \\ 1 \ar[r] & (1 + M_{D}) ∩ D^{(1)} \ar[r] & D^{(1)} \ar[r] & gr(D)^{(1)} \ar@{.>}[r] & 1}  (4.3)

The top row of the above diagram is clearly exact. The Congruence Theorem (see Th. B.1 in Appendix B), implies that the left vertical map in the diagram is an isomorphism. Once we prove that ρ(D^{(1)}) = gr(D)^{(1)}, we will have the exactness of the second row of diagram (4.3), and the theorem follows by the exact sequence for cokernels.

To prove the needed surjectivity, take any b ∈ gr(D)^{∗} with Nrd_{gr(D)}(b) = 1. Thus b ∈ gr(D)_{0} by Th. 3.3. Choose a ∈ V_{D} such that ã = b. Then we have,

\overline{Nrd_{D}(a)} = { \atop Nrd_{D}(a)} = Nrd_{gr(D)}(b) = 1.

Thus Nrd_{D}(a) ∈ 1 + M_{F}. By Cor. 4.7, since Nrd_{D}(1 + M_{D}) = 1 + M_{F}, there is c ∈ 1 + M_{D} such that Nrd_{D}(c) = Nrd(a)^{−1}. Then, ac ∈ D^{(1)} and ρ(ac) = ρ(a) = b. □

Recall from §2 that starting from any graded division algebra E with center T and any choice of total ordering ≤ on the torsion-free abelian group Γ_{E}, there is an induced valuation v on q(E), see (2.6). Let h(T) be the henselization of T with respect to v, and let h(E) = q(E) ⊗_{q(T)} h(T). Then, h(E) is a division ring by Morandi’s henselization theorem ([Mor], Th. 2 or see [W_{2}], Th. 2.3), and with respect to the unique extension of the henselian valuation on h(T) to h(E), h(E) is an immediate extension q(E), i.e., gr(h(E)) ≅_{gr} gr(q(E)). Furthermore, as

[h(E) : h(T)] = [q(E) : q(T)] = [E : T] = [gr(q(E)) : gr(q(T))] = [gr(h(E) : gr(h(T))]

and

Z(gr(h(E))) ≅_{gr} Z(gr(q(E))) ≅_{gr} T ≅_{gr} gr(h(T)) = gr(Z(h(E))),

h(E) is tame (see the characterizations of tameness in §2).

Corollary 4.9. Let E be a graded division algebra. Then SK_{1}(h(E)) ≅ SK_{1}(E).

Proof. Since h(E) is a tame valued division algebra, by Th. 4.8, SK_{1}(h(E)) ≅ SK_{1}(gr(h(E))). But gr(h(E)) ≅_{gr} gr(q(E)) ≅_{gr} E, so the corollary follows. □

Having now established that the reduced Whitehead group of a division algebra coincides with that of its associated graded division algebra, we can easily deduce stability of SK_{1} for unramified valued division algebra, due originally to Platonov (Cor. 3.13 in [P_{1}]), and also a formula for SK_{1} for a totally ramified division algebra ([LT], p. 363, see also [E], p. 70), and also a formula for SK_{1} in the nicely semiramfied case ([E], p. 69), as natural consequences of Th. 4.8:

Corollary 4.10. Let F be a field with Henselian valuation, and let D be a tame division algebra with center F.

(i) If D is unramified then SK_{1}(D) ≅ SK_{1}(\overline{D})

(ii) If D is totally ramified then SK_{1}(D) ≅ μ_{n}(\overline{F})/μ_{e}(\overline{F}) where n = ind(D) and e is the

exponent of Γ_{D}/Γ_{F}. (iii) If D is semiramified, let G = 𝒢al(\overline{D}/\overline{F}) ≅ Γ_{D}/Γ_{F}. Then, there is an exact sequence

G ∧ G → Ĥ^{−1}(G, \overline{D}^{∗}) → SK_{1}(D) → 1.

(iv) If D is nicely semiramfied, then SK_{1}(D) ≅ Ĥ^{−1}(𝒢al(\overline{D}/\overline{F}), \overline{D}^{∗}).

Proof. Because D is tame, Z(gr(D)) = gr(F) and ind(gr(D)) = ind(D). Therefore, for D in each case (i)–(iv) here, gr(D) is in the corresponding case of Cor. 3.6. (In case (iii), that D is semiramified means [\overline{D} : \overline{F}] = |Γ_{D} : Γ_{F} | = ind(D) and \overline{D} is a field. Hence gr(D) is semiramified. In case (iv), since D is nicely semiramified, by definition (see [JW], p. 149) it contains maximal subfields K and L, with K unramified over F and L totally ramified over F. (In fact, by [M_{1}], Th. 2.4, D is nicely semiramified if and only if it has such maximal subfields.) Then, gr(K) and gr(L) are maximal graded subfields of gr(D) by dimension count and the graded double centralizer theorem,[HwW_{2}], Prop. 1.5(b), with gr(K) unramified over gr(F) and gr(L) totally ramified over gr(F). So, gr(D) is then in case (iv) of Cor. 3.6.) Thus, in each case Cor. 4.10 for D follows from Cor. 3.6 for gr(D) together with the isomorphism SK_{1}(D) ≅ SK_{1}(gr(D)) given by Th. 4.8. □

Recall that the reduced norm residue group of D is defined as SH^{0}(D) = F^{∗}/Nrd_{D}(D^{∗}). It is known that SH^{0}(D) coincides with the first Galois cohomology group H^{1}(F, D^{(1)}) (see [KMRT], §29). We now show that for a tame division algebra D over a henselian field, SH^{0}(D) coincides with SH^{0} of its associated graded division algebra.

Theorem 4.11. Let F be a field with a henselian valuation v and let D be a tame F -central division algebra. Then SH^{0}(D) ≅ SH^{0}(gr(D)).

Proof. Consider the diagram with exact rows,

\xymatrix{1 \ar[r] & 1 + M_{D} \ar[r] \ar[d] & D^{∗} \ar[d]^{_{Nrd_{D}}} \ar[r]^{ρ} & gr(D)^{∗} \ar[d]^{_{Nrd_{gr(D)}}} \ar[r] & 1 \\ 1 \ar[r] & 1 + M_{F} \ar[r] & F^{∗} \ar[r] & gr(F)^{∗} \ar[r] & 1}  (4.4)

where Cor. 4.4 guarantees that the diagram is commutative. By Cor. 4.7, the left vertical map is an epimorphism. The theorem follows by the exact sequence for cokernels. □

Remark. As with SK_{1}, if D is tame and unramified, then

SH^{0}(D) ≅ SH^{0}(gr(D)) ≅ SH^{0}(gr(D)_{0}) ≅ SH^{0}(\overline{D}).

We conclude this section by establishing a similar result for the CK_{1} functor of (3.8) above. Note that here, unlike the situation with SK_{1} (Th. 4.8) or with SH^{0} (Th. 4.11), we need to assume strong tameness here.

Theorem 4.12. Let F be a field with henselian valuation v and let D be a strongly tame F -central division algebra. Then CK_{1}(D) ≅ CK_{1}(gr(D)).

Proof. Consider the canonical epimorphism ρ: D^{∗} → gr(D)^{∗} given by a ↦ ã, with kernel 1 + M_{D}. Since ρ maps D′ onto gr(D)′ and F^{∗} onto gr(F)^{∗}, it induces an isomorphism D^{∗}/(F^{∗}D′(1 + M_{D})) ≅ gr(D)^{∗}/(gr(F)^{∗}gr(D)′). We have gr(F) = Z(gr(D)) and by Lemma 2.1 in [H_{3}], as D is strongly tame, 1 + M_{D} = (1 + M_{F})[D^{∗}, 1 + M_{D}] ⊆ F^{∗}D′. Thus, CK_{1}(D) ≅ CK_{1}(gr(D)). □

5. Stability of the reduced Whitehead group

The goal of this section is to prove that if E is a graded division ring (with Γ_{E} a torsionfree abelian group), then SK_{1}(E) ≅ SK_{1}(q(E)), where q(E) is the quotient division ring of E. When Γ_{E} ≅ ℤ, this was essentially proved by Platonov and Yanchevskiĭ in [PY], Th. 1 (see the Introduction). Their argument was based on properties of twisted polynomial rings, and our argument is based on their approach. So, we will first look at twisted polynomial rings. For these, an excellent reference is Ch. 1 in [J].

Let D be a division ring finite dimensional over its center Z(D). Let σ be an automorphism of D whose restriction to Z(D) has finite order, say ℓ. Let T = D[x, σ] be the twisted polynomial ring, with multiplication given by xd = σ(d)x, for all d ∈ D. By Skolem-Noether, there is w ∈ D^{∗} with σ^{ℓ} = int(w^{−1}) (= conjugation by w^{−1}); moreover, w can be chosen so that σ(w) = w (by a Hilbert 90 argument, see [J], Th. 1.1.22(iii) or [PY], Lemma 1). Then Z(T) = K[y] (a commutative polynomial ring), where K = Z(D)^{σ}, the fixed field of Z(D) under the action of σ, and y = wx^{ℓ}. Let Q = q(T) = D(x, σ), the division ring of quotients of T. Note that Z(Q) = q(Z(T)) = K(y), and ind(Q) = ℓ ind(D). Observe that within Q we have the twisted Laurent polynomial ring T [x^{−1}] = D[x, x^{−1}, σ] which is a graded division ring, graded by degree in x, and T ⊆ T [x^{−1}] ⊆ q(T), so that q(T [x^{−1}]) = Q. Recall that, since we have left and right division algorithms for T, T is a principal left (and right) ideal domain.

Let 𝒮 denote the set of isomorphism classes [S] of simple left T -modules S, and set

Div(T) = ⨁_{[S]∈𝒮} ℤ[S],

the free abelian group with base 𝒮. For any T -module M satisfying both ACC and DCC, the Jordan-Hölder Theorem yields a well-defined element jh(M) ∈ Div(T), given by

jh(M) = ∑_{[S]∈𝒮} n_{[S]}(M)[S],

where n_{[S]}(M) is the number of appearances of simple factor modules isomorphic to S in any composition series of M. Note that for any f ∈ T \ {0}, the division algorithm shows that dim_{D}(T/T f) = deg(f) < ∞. Hence, T/T f has ACC and DCC as a T -module. Therefore, we can define a divisor function

δ: T \ {0} → Div(T), given by δ(f) = jh(T/T f).

Remark 5.1. Note the following properties of δ:

(i) For any f, g ∈ T \ {0}, δ(fg) = δ(f) + δ(g). This follows from the isomorphism T g/T fg ≅ T/T f (as T has no zero divisors).

(ii) We can extend δ to a map δ: Q^{∗} → Div(T), where Q = q(T), by δ(fh^{−1}) = δ(f)−δ(h) for any f ∈ T \ {0}, h ∈ Z(T) \ {0}. It follows from (i) that δ is well-defined and is a group homomorphism on Q^{∗}. Clearly, δ is surjective, as every simple T -module is cyclic.

(iii) For all q, s ∈ Q^{∗}, δ(sqs^{−1}) = δ(q). This is clear, as δ is a homomorphism into an

abelian group. (iv) For all q ∈ Q^{∗}, δ(Nrd_{Q}(q)) = n δ(q), where n = ind(Q). This follows from (iii), since

Wedderburn’s factorization theorem applied to the minimal polynomial of q over Z(Q) shows that Nrd_{Q}(q) = ∏_{i=1}^{n} s_{i}qs_{i}^{−1} for some s_{i} ∈ Q^{∗}.

(v) If Nrd_{Q}(q) = 1, then δ(q) = 0. This is immediate from (iv), as Div(T) is torsion-free.

Lemma 5.2. Take any f, g ∈ T \{0} with T/T f ≅ T/T g, so deg(f) = deg(g). If deg(f) ≥ 1, there exist s, t ∈ T \ {0} with deg(s) = deg(t) < deg(f) such that fs = tg.

Proof. (cf. [J], Prop. 1.2.8) We have deg(f) = dim_{D}(T/T f) = dim_{D}(T/T g) = deg(g). Let α: T/T f → T/T g be a T -module isomorphism, and let α(1 + T f) = s + T g. By the division algorithm, s can be chosen with deg(s) < deg(g). We have

fs + T g = f(s + T g) = fα(1 + T f) = α(f + T f) = α(0) = 0 in T/T g.

Hence, fs = tg for some t ∈ T. Since deg(f) = deg(g), we have

deg(t) = deg(s) < deg(g) = deg(f).  □

Proposition 5.3. Consider the group homomorphism δ: Q^{∗} → Div(T) defined in Remark 5.1(ii) above. Then ker(δ) = D^{∗}Q′.

Proof. (cf. [PY], proof of Lemma 5) Clearly, D^{∗} ⊆ ker(δ) and Q′ ⊆ ker(δ), so D^{∗}Q′ ⊆ ker(δ). For the reverse inclusion take h ∈ ker(δ) and write h = f/g with f, g ∈ T \ {0}. Since δ(f/g) = 0, we have δ(f) = δ(g), so deg(f) = deg(g). If deg(f) = 0, then h ∈ D^{∗}, and we’re done. So, assume deg(f) > 1. Write f = pf_{1} with p irreducible. Then, T/T p is one of the simple composition factors of T/T f. If g = q_{1}q_{2} … q_{k} with each q_{i} irreducible, then the composition factors of T/T g are (up to isomorphism) T/T q_{1}, …, T/T q_{k}. Because δ(f) = δ(g), i.e. jh(T/T f) = jh(T/T g), we must have T/T p ≅ T/T q_{j} for some j. Write g = g_{1}qg_{2} where q = q_{j}. By Lemma 5.2, there exist s, t ∈ T \ {0} with deg(s) = deg(t) < deg(p) = deg(q) and ps = tq. Then, working modulo Q′, we have

h = fg^{−1} = (pf_{1})(g_{1}qg_{2})^{−1} ≡ f_{1}(pq^{−1})(g_{1}g_{2})^{−1} ≡ f_{1}(ts^{−1})(g_{1}g_{2})^{−1} ≡ (f_{1}t)(g_{1}g_{2}s)^{−1}.

Let h′ = (f_{1}t)(g_{1}g_{2}s)^{−1}. Since h′ ≡ h (mod Q′), we have δ(h′) = δ(h) = 0, while deg(f_{1}t) < deg(f). By iterating this process we can repeatedly lower the degree of numerator and denominator to obtain h′′ ∈ D^{∗} with h′′ ≡ h′ ≡ h (mod Q′). Hence, h ∈ D^{∗}Q′, as desired. □

Remark. Since K_{1}(Q) = Q^{∗}/Q′, Prop. 5.3 can be stated as saying that there is an exact sequence

K_{1}(D) ⟶ K_{1}(Q) ⟶^{δ} Div(T) ⟶ 0.  (5.1)

This can be viewed as part of an exact localization sequence in K-Theory. We prefer the explicit description of Div(T) and δ given here, as it helps to understand the maps associated with Div(T).

Let R = Z(T) = K[y]. So, q(R) = Z(Q). We define Div(R) just as we defined Div(T) above. Note that this Div(R) coincides canonically with the usual divisor group of fractional ideals of the PID R, since for a ∈ R \ {0}, the simple composition factors of R/Ra are the simple modules R/P as P ranges over the prime ideal factors of the ideal Ra.

Proposition 5.4. For R = Z(T) = K[Y], there is a map Nrd : Div(T) → Div(R) such that the following diagram commutes:

\xymatrix{D^{∗} \ar[r] \ar[d]_{_{Nrd_{D}}} & Q^{∗} \ar[dd]^{_{Nrd_{Q}}} \ar[r]^{δ_{T}} & Div(T) \ar[dd]^{Nrd} \\ Z(D)^{∗} \ar[d]_{N_{Z(D)/K}} \\ K^{∗} \ar[r] & q(R)^{∗} \ar[r]^{δ_{R}} & Div(R)}  (5.2)

Moreover, Nrd is injective.

Proof. Let E = T [x^{−1}] = D[x, x^{−1}, σ], which with its grading by degree in x is a graded division ring with E_{0} = D and q(E) = Q. Since ind(Q) = ind(D) [Z(D) : K], by (3.2), for d ∈ D^{∗} = E_{0}^{∗}, Nrd_{Q}(d) = N_{Z(D)/K}(Nrd_{D}(d)). This gives the commutativity of the left rectangle in the diagram.

For the right vertical map in diagram (5.2), note that there is a canonical map, call it N: Div(T) → Div(R) given by taking a T -module M (with ACC and DCC) and viewing it as an R-module; that is N(jh_{T} (M)) = jh_{R}(M). But, this is not the map Nrd : Div(T) → Div(R) we need here! (Consider N a norm map, while our Nrd is a reduced norm map.) Note that as T is integral over R and R is integrally closed, Nrd_{Q} maps T into R. Define a function

ψ: T \ {0} → Div(R) by ψ(f) = δ_{R}(Nrd_{Q}(f)) = jh_{R}(R/R Nrd_{Q}(f)).

Since Nrd_{Q} is multiplicative and δ_{R} is a group homomorphism, we have

ψ(fg) = ψ(f) + ψ(g) for all f, g ∈ T \ {0}.  (5.3)

We then extend ψ to Q^{∗} by defining ψ(fr^{−1}) = ψ(f) − ψ(r) for all f ∈ T \ {0}, r ∈ R \ {0}. Equation (5.3) shows that ψ is well-defined on Q^{∗} and is a group homomorphism. Since Nrd_{Q}(D^{∗}) ⊆ K^{∗} ⊆ R^{∗} by (3.2), D^{∗} ⊆ ker(ψ). Also, Q′ ⊆ ker(ψ) as Div(R) is abelian. Thus, by Prop. 5.3, ker(δ_{T}) ⊆ ker(ψ), so there is an induced homomorphism Nrd : Div(T) → Div(R) such that Nrd ∘ δ_{T} = ψ on Q^{∗}. This is the map we need. Since for every f ∈ T \ {0}, Nrd(δ_{T} (f)) = ψ(f) = δ_{R}(Nrd_{Q}(f)), the right rectangle in (5.2) is commutative.

We have a scalar extension map from R-modules to T -modules given by M → T ⊗_{R} M. This induces a map ρ: Div(R) → Div(T) given by ρ(jh_{R}(M)) = jh_{T} (T ⊗_{R} M). For any r ∈ R, we have T ⊗_{R} (R/Rr) ≅ T/T r. Thus for any g ∈ T \ {0},

ρ(Nrd(δ_{T} (g))) = ρ(δ_{R}(Nrd_{Q}(g))) = ρ(jh_{R}(R/RNrd_{Q}(g)))
= jh_{T} (T/T Nrd_{Q}(g)) = δ_{T} (Nrd_{Q}(g)) = n δ_{T} (g),

using Remark 5.1(iv). This shows that ρ∘Nrd : Div(T) → Div(T) is multiplication by n, which is an injection, as Div(T) is a torsion-free abelian group. Hence Nrd must be injective. □

Remark. Here is a description of how the maps Nrd : Div(T) → Div(R) and N: Div(T) → Div(R) and ρ: Div(R) → Div(T) are related, and a formula for Nrd on generators of Div(T).

Proofs are omitted. We have

ρ ∘ Nrd = n id_{Div(T)};  (5.4)

and

N = n · Nrd.  (5.5)

Let S be any simple left T -module, and [S] the corresponding basic generator of Div(T). Let M = ann_{T} (S), and let P = ann_{R}(S), which is a maximal ideal of R. Let k = matrix of size of T/M = dim_{Δ}(S), where Δ = End_{T} (S), so T/M ≅ M_{k}(Δ). Then,

Nrd([S]) = n_{S}[R/P], where n_{S} = \frac{1}{nk} dim_{R/P} (T/M) = ind(T/M).  (5.6)

We now consider an arbitrary graded division ring E. As usual, we assume throughout that Γ_{E} is a torsion-free abelian group and [E : Z(E)] < ∞.

Lemma 5.5. Let E be a graded division ring, and let Q = q(E). Then, the canonical map SK_{1}(E) → SK_{1}(Q) is injective.

Proof. Recall from Prop. 3.2(i) that Nrd_{E}(a) = Nrd_{Q}(a) for all a ∈ E, so the inclusion E^{∗} ↪ Q^{∗} yields a map SK_{1}(E) = E^{(1)}/E′ → Q^{(1)}/Q′ = SK_{1}(Q). Also recall the homomorphism λ: Q^{∗} → E^{∗} of (2.6), which maps Q′ to E′. Since the composition E^{∗} ↪ Q^{∗} →^{λ} E^{∗} is the identity map, for any a ∈ E^{(1)} ∩Q′, we have a = λ(a) ∈ E′. Thus, the map SK_{1}(E) → SK_{1}(Q) is injective. □

Proposition 5.6. Let E be a graded division ring, and let Q = q(E). Then,

Q^{(1)} = (Q^{(1)} ∩ E_{0})Q′.

Once this proposition is proved, it will quickly yield the main theorem of this section:

Theorem 5.7. Let E be a graded division ring. Then, SK_{1}(E) ≅ SK_{1}(q(E)).

Proof. Set Q = q(E). Since the reduced norm respects scalar extensions, Q^{(1)}∩E_{0} ⊆ E^{(1)}. The image of the map ξ: SK_{1}(E) → SK_{1}(Q) is E^{(1)}Q′/Q′, which thus contains (Q^{(1)} ∩E_{0})Q′/Q′ = Q^{(1)}/Q′ = SK_{1}(Q) (using Prop. 5.6). Thus ξ is surjective, as well as being injective by Lemma 5.5, proving the theorem. □

Proof of Prop. 5.6. We first treat the case where Γ_{E} is finitely generated.

Case I. Suppose Γ_{E} = ℤ^{n} for some n ∈ ℕ.

Let F = Z(E), a graded field, and let ε_{i} = (0, …, 0, 1, 0, …, 0) (1 in the i-th position), so Γ_{E} = ℤε_{1} ⊕ … ⊕ ℤε_{n}. For 1 ≤ i ≤ n, let Δ_{i} = ℤε_{1} ⊕ … ⊕ ℤε_{i} ⊆ Γ_{E}; and let S_{i} = E_{Δ_{i}} = ⨁_{γ∈Δ_{i}} E_{γ}, which is a graded sub-division ring of E. Let Q_{i} = Q(S_{i}), the quotient division ring of S_{i}; so Q_{n} = Q as S_{n} = E. Set R_{0} = Q_{0} = E_{0}. Note that [S_{i} : (S_{i} ∩ F)] < ∞, so Q_{i} is obtainable from S_{i} by inverting the nonzero elements of S_{i} ∩ F. This makes it clear that Q_{i} ⊆ Q_{i+1}, for each i.

For each j, 1 ≤ j ≤ n, choose and fix a nonzero element x_{j} ∈ E_{ε_{j}}. Let φ_{j} = int(x_{j}) ∈ Aut(E) (i.e., φ_{j} is conjugation by x_{j}). Since φ_{j} is a degree-preserving automorphism of E, φ_{j} maps each S_{i} to itself. Hence, φ_{j} extends uniquely to an automorphism to Q_{i}, also denoted φ_{j}. Since each Γ_{E}/Γ_{F} is a torsion abelian group, there is ℓ_{j} ∈ ℕ such that ℓ_{j}ε_{j} ∈ Γ_{F}. Then, if we choose any nonzero z_{j} ∈ F_{ℓ_{j}ε_{j}}, we have x_{j}^{ℓ_{j}} ∈ E_{ℓ_{j}ε_{j}} = E_{0}z_{j}. So, x_{j}^{ℓ_{j}} = c_{j}z_{j} for some c_{j} ∈ E_{0}^{∗}, and z_{j} ∈ F = Z(E). Then φ_{j}^{ℓ_{j}} = int(x_{j}^{l_{j}}) = int(c_{j}z_{j}) = int(c_{j}). Thus, φ_{j}^{ℓ_{j}}|_{S_{i}} is an inner automorphism of S_{i} for each i, as c_{j} ∈ E_{0}^{∗} ⊆ S_{i}.

Now, fix i with 1 ≤ i ≤ n. We will prove:

Q_{i}^{∗} ∩ Q^{(1)} ⊆ (Q_{i−1}^{∗} ∩ Q^{(1)})[Q_{i}^{∗}, Q^{∗}].  (5.7)

We have S_{i} = S_{i−1}[x_{i}, x_{i}^{−1}] ≅ S_{i−1}[x_{i}, x_{i}^{−1}, φ_{i}] (twisted Laurent polynomial ring). Likewise, within Q_{i} we have Q_{i−1}[x_{i}] ≅ Q_{i−1}[x_{i}, φ_{i}] (twisted polynomial ring), with φ_{i}^{ℓ_{i}} an inner automorphism of Q_{i−1}. In order to invoke Prop. 5.4, let

T = Q_{i−1}[x_{i}] ≅ Q_{i−1}[x_{i}, φ_{i}] and let R = Z(T).

Since S_{i−1}[x_{i}] ⊆ T ⊆ Q_{i} = q(S_{i−1}[x_{i}]), we have q(T) = Q_{i}. Let 𝒢 ⊆ Aut(Q_{i}) be the subgroup of automorphisms of Q_{i} generated by φ_{i+1}, …, φ_{n}, and let G = 𝒢/(𝒢∩Inn(Q_{i})), where Inn(Q_{i}) is the group of inner automorphisms of Q_{i}. Since Skolem-Noether shows that Inn(Q_{i}) is the kernel of the restriction map Aut(Q_{i}) → Aut(Z(Q_{i})), this G maps injectively into Aut(Z(Q_{i})). For σ ∈ G, we write σ|_{Z(Q_{i})} for the automorphism of Z(Q_{i}) determined by σ. Note that G is a finite abelian group, since the images of the φ_{i} have finite order in G and commute pairwise. (For, we have x_{j}x_{k} = c_{jk}x_{k}x_{j} for some c_{jk} ∈ E_{0}^{∗}. Hence φ_{j}φ_{k} = int(c_{jk})φ_{k}φ_{j} and int(c_{jk}) ∈ Inn(Q_{i}), as c_{jk} ∈ E_{0}^{∗} ⊆ Q_{i}^{∗}). Every element of 𝒢 is an automorphism of S_{i−1}[x_{i}] preseerving degree in x_{i}, so an automorphism of T, since this is true of each φ_{j}. Therefore we have a group action of 𝒢 on T by ring automorphisms, and an induced action of 𝒢 on Div(T). Note that as any ψ ∈ 𝒢 permutes the maximal left ideals of T, the action of ψ on Div(T) arises from an action on the base of Div(T) consisting of isomorphism classes of simple T - modules. That is, Div(T) is a permutation 𝒢-module. 𝒢 also acts on R = Z(T) by ring automorphisms, and on Div(R), and all the maps in the commutative diagram below (see Prop. 5.4) are 𝒢-module homomorphisms.

\xymatrix{Q_{i}^{∗} \ar[r]^{δ_{T}} \ar[d]_{_{Nrd_{Q_{i}}}} & Div(T) \ar[d]^{Nrd} \\ Z(Q_{i})^{∗} \ar[r]^{δ_{R}} & Div(R)}  (5.8)

Since inner automorphisms of Q_{i} act trivially on Div(T) (see Remark 5.1(iii)), and on Z(Q_{i}) and Div(R), these 𝒢-modules are actually G-modules. Let

𝔑 = Nrd(Div(T)) ⊆ Div(R).

Because Nrd : Div(T) → Div(R) is injective (see Prop. 5.4), 𝔑 is a G-module isomorphic to Div(T), so 𝔑 is a permutation G-module. In 𝔑 we have two distinguished 𝒢-submodules,

𝔑_{0} = ker(N_{G}), where N_{G}: 𝔑 → 𝔑 is the norm, given by N_{G}(b) = ∑_{σ∈G} σ(b); and
I_{G}(𝔑) = ⟨{β − σ(β) | β ∈ 𝔑, σ ∈ G}⟩ ⊆ 𝔑_{0}.

By definition, Ĥ^{−1}(G, 𝔑) = 𝔑_{0}/I_{G}(𝔑). But, because 𝔑 is a permutation G-module, Ĥ^{−1}(G, 𝔑) = 0. (This is well known, and is an easy calculation, as 𝔑 is a direct sum of G-modules of the form ℤ[G/H] for subgroups H of G.) That is, 𝔑_{0} = I_{G}(𝔑).

Take any generator β − σ(β) of I_{G}(𝔑), where σ ∈ G and β ∈ 𝔑, say β = Nrd(η), where η ∈ Div(T). Take any b ∈ Q_{i}^{∗} with δ_{T} (b) = η, and choose u ∈ E^{∗} which is some product of the φ_{j} (i+1 ≤ j ≤ n), such that int(u)|_{Z(Q_{i})} = σ|_{Z(Q_{i})}. Then, δ_{R}(Nrd_{Q_{i}}(b)) = Nrd(δ_{T} (b)) = β (see (5.8)). Also, because int(u)|_{Q_{i}} is an automorphism of Q_{i}, we have Nrd_{Q_{i}}(ub^{−1}u^{−1}) = u Nrd_{Q_{i}}(b^{−1})u^{−1}. Thus, bub^{−1}u^{−1} ∈ [Q_{i}^{∗}, Q^{∗}] ∩ Q_{i} and

Nrd_{Q_{i}}(bub^{−1}u^{−1}) = Nrd_{Q_{i}}(b) Nrd_{Q_{i}}(ub^{−1}u^{−1})
= Nrd_{Q_{i}}(b) u Nrd_{Q_{i}}(b^{−1})u^{−1} = Nrd_{Q_{i}}(b)/σ(Nrd_{Q_{i}}(b)).

Hence, in Div(R),

δ_{R}(Nrd_{Q_{i}}(bub^{−1}u^{−1})) = δ_{R}(Nrd_{Q_{i}}(b)/σNrd_{Q_{i}}(b)) = β − σ(β).

Since such β−σ(β) generate I_{G}(𝔑), it follows that for any γ ∈ I_{G}(𝔑), there is c ∈ [Q_{i}^{∗}, Q^{∗}]∩Q_{i}, with γ = δ_{R}(Nrd_{Q_{i}}(c)) = Nrd(δ_{T} (c)) (see(5.8)).

To prove (5.7), we need a formula for Nrd_{Q} for an element of Q_{i}. For this, note that E = S_{i}[x_{i+1}, x_{i+1}^{−1}, …, x_{n}, x_{n}^{−1}] which can be considered a graded ring over S_{i}. Now, let C = Q_{i}[x_{i+1}, x_{i+1}^{−1}, …, x_{n}, x_{n}^{−1}] ⊆ Q. This C is a graded division ring with C_{0} = Q_{i} and Γ_{C} = ℤε_{i+1} ⊕ … ⊕ ℤε_{n}. Since E ⊆ C ⊆ Q = q(E), we have q(C) = Q. For the graded field Z(C) we have Z(C)_{0} consists of those elements of Z(C_{0}) = Z(Q_{i}) centralized by x_{i+1}, …, x_{n}, i.e., Z(C)_{0} is the fixed field Z(Q_{i})^{𝒢} = Z(Q_{i})^{G}. Since, as noted earlier G injects into Aut(Z(Q_{i}), we have G ≅ 𝒢al(Z(Q_{i})/Z(C)_{0}). Thus, for any q ∈ Q_{i} = C_{0}, by Prop. 3.2(i) and (iv),

Nrd_{Q}(q) = Nrd_{q(C)}(q) = Nrd_{C}(q) = N_{Z(C_{0})/Z(C_{0})^{G}}(Nrd_{C_{0}}(q))^{m}
= N_{Z(Q_{i})/Z(Q_{i})^{G}}(Nrd_{Q_{i}}(q))^{m},

where m = ind(Q)/ ind(Q_{i})[Z(Q_{i}) : Z(Q_{i})^{G}].

To verify (5.7), take any a ∈ Q_{i}^{∗} ∩ Q^{(1)}. Thus,

1 = Nrd_{Q}(a) = N_{Z(Q_{i})/Z(Q_{i})^{G}}(Nrd_{Q_{i}}(a))^{m}.

Hence, for α = δ_{T} (a) ∈ Div(T), using the identification of G with 𝒢al(Z(Q_{i})/Z(C)_{0}) and commutative diagram (5.8),

0 = δ_{R}(Nrd_{Q}(a)) = δ_{R}(N_{Z(Q_{i})/Z(Q_{i})^{G}}(Nrd_{Q_{i}}(a))^{m}) = ∑_{σ∈G} σ(δ_{R}(Nrd_{Q_{i}}(a)^{m}))
= N_{G}(δ_{R}(Nrd_{Q_{i}}(a))^{m}) = m N_{G}(Nrd(δ_{T} (a))) = m N_{G}(Nrd(α)).

Since Div(R) is torsion-free, we have N_{G}(Nrd(α)) = 0, i.e., Nrd(α) ∈ ker(N_{G}) = 𝔑_{0} = I_{G}(𝔑). Therefore, as we saw above, there is c ∈ [Q_{i}^{∗}, Q^{∗}] ∩ Q_{i}^{∗} with Nrd(α) = Nrd(δ_{T} (c)). Let a′ = a/c ∈ Q_{i}^{∗}. Then,

Nrd(δ_{T} (a′)) = Nrd(δ_{T} (a)) − Nrd(δ_{T} (c)) = Nrd(α) − Nrd(α) = 0.

Because Nrd : Div(T) → Div(R) is injective (see Prop. 5.4), it follows that δ_{T} (a′) = 0 in Div(T). Therefore, as T = Q_{i−1}[x_{,}φ_{i}] and q(T) = Q_{i}, by Prop. 5.3 there is a′′ ∈ Q_{i−1} with a′′ ≡ a′ (mod Q′_{i}). So, a′′ ≡ a (mod [Q_{i}^{∗}, Q^{∗}]), and hence Nrd_{Q}(a′′) = Nrd_{Q}(a) = 1, i.e., a′′ ∈ Q_{i−1}^{∗} ∩ Q^{(1)}. Thus, a ∈ (Q_{i−1}^{∗} ∩ Q^{(1)})[Q_{i}^{∗}, Q^{∗}], proving (5.7).

The inclusion (5.7) shows that for any i, 1 ≤ i ≤ n and any a ∈ Q^{(1)} ∩ Q_{i} there is b ∈ Q^{(1)} ∩ Q_{i−1} with b ≡ a (mod Q′). Hence, by downward induction on i, for any q ∈ Q^{(1)} = Q^{(1)} ∩ Q_{n} there is d ∈ Q_{0} ∩ Q^{(1)} = E_{0} ∩ Q^{(1)} with d ≡ q mod Q′). So, Q^{(1)} ⊆ (Q^{(1)} ∩ E_{0})Q′. The reverse inclusion is clear, completing the proof of Case I.

Case II. Suppose Γ_{E} is not a finitely generated abelian group.

The basic point is that E is a direct limit of sub-graded division algebras with finitely generated grade group, so we can reduce to Case I. But we need to be careful about the choice of the sub-division algebras to assure that they have the same index as E, so that the reduced norms are compatible.

Let F = Z(E). Since |Γ_{E}/Γ_{F} | < ∞, there is a finite subset, say {γ_{1}, …, γ_{k}} of Γ_{E} whose images in Γ_{E}/Γ_{F} generate this group. Let Δ_{0} be any finitely generated subgroup of Γ_{E}, and let Δ be the subgroup of Γ_{E} generated by Δ_{0} and γ_{1}, …, γ_{k}. Then, Δ is also a finitely generated subgroup of Γ_{E}, but with the added property that Δ + Γ_{F} = Γ_{E}. Let

E_{Δ} = ⨁_{δ∈Δ} E_{δ},

which is a graded sub-division ring of E, with E_{Δ,0} = E_{0} and Γ_{E_{Δ}} = Δ. Since Δ + Γ_{F} = Γ_{E}, we have E_{Δ}F = E. (For, take any γ ∈ Γ_{E} and write γ = δ + η with δ ∈ Δ and η ∈ Γ_{F}, and any nonzero d ∈ E_{Δ,δ} and c ∈ F_{η}. Then, E_{γ} = dcE_{0} ⊆ E_{Δ}F.) Because E_{Δ}F = E, we have Z(E_{Δ}) = F ∩ E_{Δ} = F_{Δ∩Γ_{F}}. Note that

[E_{Δ} : Z(E_{Δ})] = [E_{Δ,0} : F_{Δ∩Γ_{F} ,0}] |Γ_{Δ} : (Γ_{Δ} ∩ Γ_{F})| = [E_{0} : F_{0}] |(Γ_{Δ} + Γ_{F}) : Γ_{F} |
= [E_{0} : F_{0}] |Γ_{E} : Γ_{F} | = [E : F].

The graded homomorphism E_{Δ} ⊗_{Z(E_{Δ})} F → E is onto as E_{Δ}F = E, and is then also injective by dimension count (or by the graded simplicity of E_{Δ} ⊗_{Z(E_{Δ})} F). Thus, E_{Δ} ⊗_{Z(E_{Δ})} F ≅ E. It follows that q(E_{Δ}) ⊗_{q(Z(E_{Δ}))} q(F) ≅ q(E). Specifically,

q(E_{Δ}) ⊗_{q(Z(E_{Δ}))} q(F) ≅ (E_{Δ} ⊗_{Z(E_{Δ})} q(Z(E_{Δ}))) ⊗_{q(Z(E_{Δ}))} q(F) ≅ E_{Δ} ⊗_{Z(E_{Δ})} q(F)
≅ (E_{Δ} ⊗_{Z(E_{Δ})} F) ⊗_{F} q(F) ≅ E ⊗_{F} q(F) ≅ q(E).

Therefore, for any a ∈ q(E_{Δ}), Nrd_{q(E_{Δ})}(a) = Nrd_{q(E)}(a).

Now, if we take any a ∈ Q^{(1)} where Q = q(E), there is a subgroup Δ ⊆ Γ_{E} with Δ finitely generated and Δ + Γ_{F} = Γ_{E} and a ∈ E_{Δ}. Since Nrd_{q(E_{Δ})}(a) = Nrd_{Q}(a) = 1, we have, by Case I applied to E_{Δ}, a ∈ (q(E_{Δ})^{(1)} ∩ E_{0})q(E_{Δ})′ ⊆ (Q^{(1)} ∩ E_{0})Q′, completing the proof for Case II. □

Remark. (i) Prop. 5.6 for those E with Γ_{E} ≅ ℤ was proved in [PY], and our proof of this is essentially the same as theirs, expressed in a somewhat different language. Platonov and Yanchevskiĭ also in effect assert Prop. 5.6 for E with Γ_{E} finitely generated, expressed as a result for iterated quotient division rings of twisted polynomial rings. (See [PY], Lemma 8.) By way of proof of [PY], Lemma 8, the authors say nothing more than that it follows by induction from the rank 1 case. It is not clear whether the proof given here coincides with their unstated proof, since the transition from rank 1 to finite rank is not transparent.

(ii) So far the functor CK_{1} has manifested properties similar to SK_{1}. However, the similarity does not hold here, since the functor CK_{1} is not (homotopy) stable. In fact, for a division algebra D over its center F of index n, one has the following split exact sequence,

1 → CK_{1}(D) → CK_{1}(D(x)) → ⨁_{p} ℤ/(n/n_{p})ℤ → 1

where p runs over irreducible monic polynomials of F [x] and n_{p} is the index of central simple algebra D⊗_{F} (F [x]/(p)) (see Th. 2.10 in [H_{1}]). This is provable by mapping the exact sequence (5.1) with T = F [x] to the sequence for T = D[x] and taking cokernels.

Example 5.8. Let E be a semiramified graded division ring with Γ_{E} ≅ ℤ^{n}, and let T = Z(E). Since Γ_{E}/Γ_{T} is a torsion group, there are a base {γ_{1}, …, γ_{n}} of the free abelian group Γ_{E} and some r_{1}, …, r_{n} ∈ ℕ such that {r_{1}γ_{1}, …, r_{n}γ_{n}} is a base of Γ_{T}. Choose any nonzero z_{i} ∈ E_{γ_{i}} and x_{i} ∈ T_{r_{i}γ_{i}}, 1 ≤ i ≤ n. Let F = T_{0} and M = E_{0}, and let G = 𝒢al(M/F). Because E is semiramified, M is Galois over F with [M : F] = |Γ_{E} : Γ_{T} | = ind(E) = r_{1} … r_{n}, and G ≅ Γ_{E}/Γ_{T}. Since z_{i}^{r_{i}} ∈ E_{r_{i}γ_{i}} = E_{0}x_{i}, there is b_{i} ∈ M with z_{i}^{r_{i}} = b_{i}x_{i}. Let u_{ij} = z_{i}z_{j}z_{i}^{−1}z_{j}^{−1} ∈ M. Let σ_{i} ∈ G be the automorphism of M determined by conjugation by z_{i}. From the isomorphism G ≅ Γ_{E}/Γ_{T}, each σ_{i} has order r_{i} in G and G ≅ ⟨σ_{1}⟩ × … × ⟨σ_{n}⟩. Clearly, T = F [x_{1}, x_{1}^{−1}, … x_{n}, x_{n}^{−1}], an iterated Laurent polynomial ring, and E = M[z_{1}, z_{1}^{−1}, …, z_{n}, z_{n}^{−1}], an iterated twisted Laurent polynomial ring whose multiplication is completely determined by the b_{i} ∈ M, the u_{ij} ∈ M, and the action of the σ_{i} on M.

Let D = q(E), which is a division ring with center q(T) = F (x_{1}, …, x_{n}), a rational function field over F. Then, D is the generic abelian crossed product determined by M/F, the base {σ_{1}, …, σ_{n}} of G, the b_{i} and the u_{ij}, as defined in [AS]. As was pointed out in [BM], all generic abelian crossed products arise this way as rings of quotients of semiramified graded division algebras. Generic abelian crossed products were used in [AS] to give the first examples of noncyclic p-algebras, and in [S_{1}] to prove the existence of noncrossed product p-algebras. It is known by [T], Prop. 2.1 that D is determined up to F -isomorphism by M and the u_{ij}. By Cor. 3.6(iii) and Th. 5.7, there is an exact sequence

G ∧ G → Ĥ^{−1}(G, M^{∗}) → SK_{1}(D) → 1,  (5.9)

where the left map is determined by sending σ_{i} ∧ σ_{j} to u_{ij} mod I_{G}(M^{∗}). An important condition introduced by Amitsur and Saltman in [AS] was nondegeneracy of {u_{ij}}. This condition was essential for the noncyclicity results in [AS], and is also key to the results on noncyclicity and indecomposability of generic abelian crossed products in recent work of McKinnie in [Mc_{1}], [Mc_{2}] and Mounirh [M_{2}]. The original definition of nondegeneracy in [AS] was somewhat mysterious. A cogent characterization was given recently in [Mc_{3}], Lemma 5.1: A family {u_{ij}} in M^{∗} (meeting the conditions to appear in a generic abelian crossed product) is nondegenerate iff for every rank 2 subgroup H of G, the map H ∧H → Ĥ^{−1}(H, M^{∗}) appearing in the complex (5.9) for the generic abelian crossed product C_{D}(M^{H}) is nonzero. In the first nontrivial case, where G ≅ ℤ_{p} × ℤ_{p} with p a prime number, we have {u_{ij}} is nondegenerate iff the map G ∧ G → Ĥ^{−1}(G, M^{∗}) is nonzero, iff the epimorphism Ĥ^{−1}(G, M^{∗}) → SK_{1}(D) is not injective. Thus, the nondegeneracy is encoded in SK_{1}(D), and it occurs just when SK_{1}(D) is not “as large as possible.”

Appendix A. The Wedderburn factorization theorem

In a division ring, additive and multiplicative commutators play important roles and there are extensive results in the literature known as commutativity theorems. The main theme in these results is that, additive and multiplicative commutators are “dense” in a division ring. For example, if an element commutes with all additive commutators, then it is already a central element. It seems that this trend continues for the additive commutators for a graded division ring. However the multiplicative commutators are too “isolated” to determine the structure of a graded division ring.

Let E be a graded division ring with graded center T. A homogeneous additive commutator of E is an element of the form ab−ba where a, b ∈ E^{h}. We will use the notation [a, b]_{ad} = ab−ba for a, b ∈ E^{h} and let [H, K]_{ad} be the additive group generated by {dk − kd : d ∈ H^{h}, k ∈ K^{h}} where H and K are graded subrings of E. Parallel to the theory of division rings, one can show that if all the homogenous additive commutators of graded division ring E are central, then E is a graded field. To observe this, one can carry over the non-graded proof, mutatismutandis, to the graded setting, see, e.g., [L], Prop. 13.4. Alternatively, let y ∈ E^{h} be an element which commutes with homogeneous additive commutators of E. Then y commutes with all (non-homogeneous) commutators of E. Consider [x_{1}, x_{2}]_{ad} where x_{1}, x_{2} ∈ q(E). Since q(E) = E⊗_{T} q(T), it follows that y[x_{1}, x_{2}]_{ad} = [x_{1}, x_{2}]_{ad}y. So y commutes with all commutators of q(E), a division ring, thus y ∈ q(T). But E^{h} ∩ q(T) ⊆ T^{h}, proving that y ∈ T^{h}. Thus, E is commutative. Again parallel to the theory of division rings, one can prove that if K ⊆ E are graded division rings, with [E, K]_{ad} ⊆ K and char(K) ≠ 2, then K ⊆ Z(E). However, for this one it seems there is no shortcut, and one needs to carry out a proof similar to the one for ungraded division rings, as in ([L], Prop. 3.7).

The paragraph above shows some similar behavior between the Lie algebra structure of division rings and that of graded division rings. However, this analogy often fails for the multiplicative structure of graded division algebras. For example, the Cartan-Brauer-Hua theorem (the multiplicative analogue of the statement above that if K ⊆ E are graded division rings, with [E, K]_{ad} ⊆ K and char(K) ≠ 2, then K ⊆ Z(E)) is not valid in the graded setting. Also, the multiplicative group E^{∗} of a totally ramified graded division algebra E is nilpotent (since E′ ⊆ E_{0}^{∗} = T_{0}^{∗} ⊆ Z(E^{∗})), while the multiplicative group of a noncommutative division ring is not even solvable, cf. [St]. Furthermore, a totally ramified graded division algebra E^{∗} is radical over its center T (since E^{∗ exp(Γ_{E}/Γ_{T})} ⊆ T^{∗}), but this is not the case for any non-commutative division ring ([L], Th. 15.15). Nonetheless, one significant theorem involving conjugates that can be extended to the graded setting is the Wedderburn factorization theorem. (This is used in proving Th. 3.3.)

Theorem A.1 (Wedderburn Factorization Theorem). Let E be a graded division ring with center T (with Γ_{E} torsion-free abelian). Let a be a homogenous element of E which is algebraic over T with minimal polynomial h_{a} ∈ T [x]. Then, h_{a} splits completely in E. Furthermore, there exist n conjugates a_{1}, …, a_{n} of a such that h_{a} = (x − a_{n})(x − a_{n−1}) … (x − a_{1}) in E[x].

Proof. The proof is similar to Wedderburn’s original proof for a division ring ([We], see also [L] for a nice account of the proof). We sketch the proof for the convenience of the reader. For f = ∑ c_{i}x^{i} ∈ E[x] and a ∈ E, our convention is that f(a) means ∑ c_{i}a^{i}. Since Γ_{E} is torsion-free, we have E^{∗} = E^{h} \ {0}.

I: Let f ∈ E[x] with factorization f = gk in E[x]. If a ∈ E such that k(a) ∈ T · E^{∗}, then f(a) = g(a′)k(a), for some conjugate a′ of a. (Here E could be any ring with T ⊆ Z(E).)

Proof. Let g = ∑ b_{i}x^{i}. Then, f = ∑ b_{i}kx^{i}, so f(a) = ∑ b_{i}k(a)a^{i}. But, k(a) = te, where t ∈ T and e ∈ E^{∗}. Thus, f(a) = ∑ b_{i}tea^{i} = ∑ b_{i}ea^{i}e^{−1}te = ∑ b_{i}(eae^{−1})^{i}te = g(eae^{−1})k(a). □

II: Let f ∈ E[x] be a non-zero polynomial (here E could be, in fact, any ring). Then r ∈ E is a root of f if and only if x − r is a right divisor of f in E[x]. (Here E could be any ring.)

Proof. We have x^{i} − r^{i} = (x^{i−1} + x^{i−2}r + … + r^{i−1})(x − r) for any i ≥ 1. Hence,

f − f(r) = g · (x − r)  (A.1)

for some g ∈ E[x]. So, if f(r) = 0, then f = g · (x − r). Conversely, if x − r is a right divisor of f, then equation (A.1) shows that x − r is a right divisor of the constant f(r). Since x − r is monic, this implies that f(r) = 0. □

III: If a non-zero monic polynomial f ∈ E[x] vanishes identically on the conjugacy class A of a (i.e., h(b) = 0 for all b ∈ A),then deg(f) ≥ deg(h_{a}).

Proof. Consider f = x^{m} +d_{1}x^{m−1} +…+d_{m} ∈ E[x] such that f(A) = 0 and m < deg(h_{a}) with m as small as possible. Suppose a ∈ E_{γ}, so A ⊆ E_{γ}, as the units of E are all homogeneous. Since the E_{mγ}-component of f(b) is 0 for each b ∈ A, we may assume that each d_{i} ∈ E_{iγ}. Because f ∉ T [x], some d_{i} ∉ T. Choose j minimal with d_{j} ∉ T, and some e ∈ E^{∗} such that ed_{j} ≠ d_{j}e. For any c ∈ E, write c′ := ece^{−1}. Thus d′_{j} ≠ d_{j} but d′_{ℓ} = d_{ℓ} for ℓ < j. Let f′ = x^{m} + d′_{1}x^{m−1} + … + d′_{m} ∈ E[x]. Now, for all b ∈ A, we have f′(b′) = [f(b)]′ = 0′ = 0. Since, eAe^{−1} = A, this shows that f′(A) = 0. Let g = f − f′, which has degree j < m with leading coefficient d_{j} − d′_{j}. Then, g(A) = 0. But, d_{j} − d′_{j} ∈ E_{jγ} \ {0} ⊆ E^{∗}. Thus, (d_{j} − d′_{j})^{−1}g is monic of degree j < m in E[x], and it vanishes on A. This contradicts the choice of f; hence, m ≥ deg(h_{a}). □

We now prove the theorem. Since h_{a}(a) = 0, by (II), h_{a} ∈ E[x]·(x−a). Take a factorization

h_{a} = g · (x − a_{r}) … (x − a_{1})

where g ∈ E[x], a_{1}, …, a_{r} ∈ A and r is as large as possible. Let k = (x−a_{r}) … (x−a_{1}) ∈ E[x]. We claim that k(A) = 0, where A is the conjugacy class of a. For, suppose there exists b ∈ A such that k(b) ≠ 0. Since k(b) is homogenous, we have k(b) ∈ E^{∗}. But, h_{a} = gk, and h_{a}(b) = 0, as b ∈ A; hence, (I) implies that g(b′) = 0 for some conjugate b′ of b. We can then write g = g_{1} · (x − b′), by (II). So h_{a} has a right factor (x − b′)k = (x − b′)(x − a_{r}) … (x − a_{1}), contradicting our choice of r. Thus k(A) = 0, and using (III), we have r ≥ deg(h_{a}), which says that h_{a} = (x − a_{r}) … (x − a_{1}). □

Remark (Dickson Theorem). One can also see that, with the same assumptions as in Th. A.1, if a, b ∈ E have the same minimal polynomial h ∈ T [x], then a and b are conjugates. For, h = (x − b)k where k ∈ T [b][x]. But then by (III), there exists a conjugate of a, say a′, such that k(a′) ≠ 0. Since h(a′) = 0, by (I) some conjugate of a′ is a root of x − b. (This is also deducible using the graded version of the Skolem-Noether theorem, see [HwW_{2}], Prop. 1.6.)

Appendix B. The Congruence theorem for tame division algebras

For a valued division algebra D, the congruence theorem provides a bridge for relating the reduced Whitehead group of D to the reduced Whitehead group of its residue division algebra. This was used by Platonov [P_{1}] to produce non-trivial examples of SK_{1}(D), by carefully choosing D with a suitable residue division algebra. Keeping the notations of Section 2, Platonov’s congruence theorem states that for a division algebra D with a complete discrete valuation of rank 1, such that Z(\overline{D}) is separable over \overline{F}, (1 + M_{D}) ∩ D^{(1)} ⊆ D′. This crucial theorem was established with a lengthy and rather complicated proof in [P_{1}]. In [E], Ershov states that the “same” proof will go through for tame valued division algebras over henselian fields. However, this seems highly problematical, as Platonov’s original proof used properties of maximal orders over discrete valuation rings which have no satisfactory analogues for more general valuation rings. For the case of strongly tame division algebras, i.e., char(\overline{F}) ∤ [D : F], a short proof of the congruence theorem was given in [H_{2}] and another (in the case of discrete rank 1 valuations) in [Sus]. In this appendix, we provide a complete proof for the general situation of a tame valued division algebra.

Theorem B.1 (Congruence Theorem). Let F be a field with a henselian valuation v, and let D be a tame F -cental division algebra. Then (1 + M_{D}) ∩ D^{(1)} ⊆ D′.

Tameness is meant here, as in the main body of the article, in the weaker sense used in [JW] and [E]. Among the several characterizations of tameness mentioned in §2, the ones we use here are that D is tame if and only if D is split by the maximal tamely ramified extension of F, if and only if char(\overline{F}) = 0 or char(\overline{F}) = \overline{p} ≠ 0 and the \overline{p}- primary component of D is inertially split, i.e., split by the maximal unramified extension of F.

The proof of the theorem will use the following well-known lemma:

Lemma B.2. Let D be a division ring with center F and let L be a field extension of F with [L : F] = ℓ. If a ∈ D and a ⊗ 1 ∈ (D ⊗_{F} L)′, then a^{ℓ} ∈ D′.

Proof. The regular representation L → M_{ℓ}(F) yields a ring monomorphism D⊗_{F} L → M_{ℓ}(D). Therefore, we have a composition of group homomorphisms

(D ⊗_{F} L)^{∗} → GL_{ℓ}(D) → D^{∗}/D′, a ↦ ( \begin{matrix}a & 0 & ... & 0 \\ 0 & a & ... & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ... & a \end{matrix} )_{ℓ×ℓ} ↦ a^{ℓ}D′,

where the second map is the Dieudonné determinant. (See [D], §20 for properties of the Dieudonné determinant.) The lemma follows at once, since the image of the composition is abelian, so its kernel contains (D ⊗_{F} L)′. □

Note that in the preceding lemma, there is no valuation present, and D could be of infinite dimension over F.

Proof of Theorem B.1. The proof is carried out in four steps.

Step 1. We prove the theorem if D is inertially split of prime power degree over F. This is a direct adaptation of Platonov’s argument in [P_{1}] for discrete (rank 1) valuations. (When v is discrete, every tame division algebra is inertially split.)

Suppose ind(D) = p^{k}, p prime and D is inertially split. Then, D has a maximal subfield K which is unramified over F (cf. [JW], Lemma 5.1, or [W_{2}], Th. 3.4) Take any a ∈ (1 + M_{D}) ∩ D^{(1)}. We first push a into K. Since \overline{K} is separable over \overline{F}, there is y ∈ \overline{K} with \overline{K} = \overline{F} (y). Choose any z ∈ V_{K} with \overline{z} = y. So K = F (z), by dimension count, as \overline{F (z)} ⊇ \overline{F} (y). Note that \overline{az} = \overline{z} in \overline{D}. If f is the minimal polynomial of az over F, then f ∈ V_{F} [x] as az ∈ V_{D}, and \overline{z} = \overline{az} is a root of the image \overline{f} of f in \overline{F} [x]. We have deg(\overline{f}) = deg(f) = [F (az) : F] ≤ [K : F] = [\overline{F} (\overline{z}) : \overline{F}]. Hence, \overline{f} is the minimal polynomial of \overline{z} over \overline{F}, so \overline{z} is a simple root of \overline{f}. By Hensel’s lemma applied over K, K contains a root b of f with \overline{b} = \overline{z}. Since b and az have the same minimal polynomial f over F, by Skolem–Noether there is t ∈ D^{∗} with b = tazt^{−1}. So az = t^{−1}bt. Then,

a = t^{−1}btz^{−1} = (t^{−1}btb^{−1})(bz^{−1}).

We have bz^{−1} ∈ K, as b, z ∈ K, and bz^{−1} ≡ a (mod D′); so, Nrd_{D}(bz^{−1}) = Nrd_{D}(a) = 1, and bz^{−1} ∈ 1 + M_{D}, as \overline{b} = \overline{z}. Therefore, we may replace a by bz^{−1}, so we may assume a ∈ K.

Let N be the normal closure of K over F, and let G = 𝒢al(N/F). Since K is unramified over F and the maximal unramified extension F_{nr} of F is Galois over F (cf. [EP], Th. 5.2.7, Th. 5.2.9, pp. 124–126), N ⊆ F_{nr}; so N is also unramified over F. Let P be a p-Sylow subgroup of G and let L = N^{P}, the fixed field of P. Thus, [L : F] = |G : P |, which is prime to p, and N is Galois over L with 𝒢al(N/L) = P. Since gcd ([L : F], ind(D)) = 1, D_{1} = L ⊗_{F} D is a division ring and K_{1} = L ⊗_{F} K is a field with K_{1} ≅ L · K ⊆ N. So, K_{1} is unramified over F and hence over L. We have Nrd_{D_{1}}(1 ⊗ a) = Nrd_{D}(a) = 1 and 1 ⊗ a ∈ 1 + M_{D}, so if we knew the result for D_{1}, we would have 1 ⊗ a ∈ D′_{1}. But then by Lemma B.2, a^{[L:F]} ∈ D′. But we also have a^{ind(D)} ∈ D′, since SK_{1}(D) is ind(D)-torsion (by [D], p. 157, Lemma 2 or Lemma B.2 above with L a maximal subfield of D). Since gcd ([L : F], ind(D)) = 1, it would follow that a ∈ D′, as desired. Thus, it suffices to prove the result for D_{1}.

To simplify notation, replace D_{1} by D, K_{1} by K, 1 ⊗ a by a, and L by F. Because F ⊆ K ⊆ N with N Galois over F any subfield T of K minimal over F corresponds to a maximal subgroup of 𝒢al(N/F) containing 𝒢al(N/K). Since [N : F] is a power of p, by p-group theory such a maximal subgroup is normal in 𝒢al(N/F) and of index p. Thus, T is Galois over F and [T : F] = p. So 𝒢al(T/F) is a cyclic group , say 𝒢al(T/F) = ⟨σ⟩. Let E = C_{D}(T), so F ⊆ T ⊆ K ⊆ E ⊆ D. Note that K is a maximal subfield of E, since it is a maximal subfield of D.

Let c = N_{K/T} (a) = Nrd_{E}(a). Because K is unramified over T and a ∈ V_{K}, we have c ∈ V_{T} and \overline{c} = N_{\overline{K}/\overline{T}} (\overline{a}) = N_{\overline{K}/\overline{T}} (\overline{1}) = \overline{1}, so c ∈ 1 + M_{T}. We have N_{T/F} (c) = N_{T/F} (N_{K/T} (a)) = N_{K/F} (a) = Nrd_{D}(a) = 1. By Hilbert 90, c = b/σ(b) for some b ∈ T. This equation still holds if we replace b in it by any F^{∗}-multiple of b. Thus, as Γ_{T} = Γ_{F} since T is unramified over F, we may assume that v(b) = 0. But further, since T is unramified and cyclic Galois over F, its residue field \overline{T} is cyclic Galois of degree p over \overline{F}, with 𝒢al(\overline{T} /\overline{F}) = ⟨\overline{σ}⟩ where \overline{σ} is the automorphism of \overline{T} induced by σ on T. In \overline{T} we have \overline{b}/\overline{σ}(\overline{b}) = \overline{b/σ(b)} = \overline{c} = \overline{1}. Therefore, \overline{b} lies in the fixed field of \overline{σ} in \overline{T}, which is \overline{F} . Hence, there is η ∈ V_{F} with \overline{η} = \overline{b} in \overline{T}. By replacing b by bη^{−1}, we can assume \overline{b} = \overline{1}, i.e., b ∈ 1 + M_{T}.

Since K is unramified and hence tame over T, Prop. 4.6 shows N_{K/T} (1 + M_{K}) = 1 + M_{T}. So, there is s ∈ 1 + M_{K} with N_{K/F} (s) = b. Now, by Skolem–Noether, there is an inner automorphism φ of D such that φ(T) = T and φ|_{T} = σ. Since E = C_{D}(T), we have φ is a (non-inner) automorphism of E, and φ(K) is a maximal subfield of E (since K is a maximal subfield of E). We have Nrd_{E}(φ(s)) = N_{φ(K)/φ(T)}(φ(s)) = φ(N_{K/T} (s)) = σ(b). Thus,

Nrd_{E}(s/φ(s)) = b/σ(b) = c.

Now, as φ is inner, there is u ∈ D^{∗} with φ(s) = usu^{−1}. So, φ(s) ∈ 1 + M_{D}. Let a′ = a/(s/φ(s)) = a/(sus^{−1}u^{−1}) ∈ E. So a′ ≡ a (mod D′). But further, a′ ∈ E ∩ (1 + M_{D}) = 1 + M_{E} (as a, s, φ(s) ∈ (1 + M_{D}) ∩ E). Also,

Nrd_{E}(a′) = Nrd_{E}(a)/ Nrd_{E}(s/φ(s)) = N_{K/T} (a)/c = 1.

Since [E : T] < [D : F] and E is inertially split over T (since it is split by its maximal subfield K which is unramified over T), by induction on index the theorem holds for T over E. Hence, a′ ∈ E′. Since a ≡ a′ (mod D′), we thus have a ∈ D′, as desired. This completes the proof of Step 1.

Step 2. The theorem is true if D is strongly tame over F, i.e., char(\overline{F}) ∤ [D : F]. This has a short proof given in [H_{2}] and another (in the case of discrete valuation of rank 1) in [Sus], Lemma 1.6. For the convenience of the reader, we recall the argument from [H_{2}]:

Let n = ind(D), so char(\overline{F}) ∤ n. Take any s ∈ D^{∗}, and let f = x^{k}+c_{k−1}x^{k−1}+…+c_{0} ∈ F [x] be the minimal polynomial of s over F. By applying the Wedderburn factorization theorem to f (see [L], (16.9), pp. 251–252, or Appendix A above), we see that there exist d_{1}, …, d_{k} ∈ D^{∗} with (−1)^{k}c_{0} = (d_{1}sd_{1}^{−1}) … (d_{k}sd_{k}^{−1}). Hence, as D^{∗}/D′ is abelian,

Nrd_{D}(s) = [(−1)^{k}c_{0}]^{n/k} ≡ [s^{k}(d_{1}sd_{1}^{−1}s^{−1}) … (d_{k}sd_{k}^{−1}s^{−1})]^{n/k} ≡ s^{n} (mod D′). (B.1)

Now, take any a ∈ 1+M_{D} with Nrd_{D}(a) = 1. Since char(\overline{F}) ∤ n, Hensel’s Lemma applied over F (a) shows that there is s ∈ 1+M_{F} (a) ⊆ 1+M_{D} with s^{n} = a. Then, Nrd_{D}(s) = 1+m ∈ 1+M_{F} by Cor. 4.7. But,

(1 + m)^{n} = Nrd_{D}(a^{n}) = Nrd_{D}(s) = 1.

If m ≠ 0, then we have 1 = (1+m)^{n} = 1+n m+r with v(r) ≥ 2v(m), which would imply that v(n m) = v(r) > v(m). This cannot occur since char(\overline{F}) ∤ n; hence, m = 0. Thus, by (B.1)

a = s^{n} ≡ Nrd_{D}(s) = 1 + m = 1 (mod D′),

i.e., a ∈ D′. This completes Step 2.

Step 3. Suppose D = P ⊗_{F} Q, where gcd(ind(P), ind(Q)) = 1, and suppose the theorem is true for P_{L} and Q_{L} for any subfield L of D, L ⊇ F, where P_{L} (resp. Q_{L}) is the division algebra Brauer equivalent to L ⊗_{F} P (resp. L ⊗_{F} Q). Then we show using Prop. B.3 below that the theorem is true for D.

Let L be a maximal subfield of P, and let C = C_{D}(L). Then, C ≅ C_{L}(P) ⊗_{F} Q = L ⊗_{F} Q; since C is a division ring, Q_{L} ≅ C. Also,

L ⊗_{F} D ≅ (L ⊗_{F} P) ⊗_{L} (L ⊗_{F} Q) ≅ M_{ℓ}(L) ⊗_{L} C ≅ M_{ℓ}(C),

where ℓ = [L : F] = ind(P). Take any a ∈ (1 + M_{D}) ∩ D^{(1)}. For 1 ⊗ a ∈ L ⊗ D = M_{ℓ}(C), Prop. B.3 shows that there is c ∈ 1 + M_{C} with ddet(a) ≡ c (mod C′), where ddet denotes the Dieudonné determinant. Then,

1 = Nrd_{D}(a) = Nrd_{M_{ℓ}(C)}(1 ⊗ a) = Nrd_{C}(ddet(1 ⊗ a)) = Nrd_{C}(c).

Hence, c ∈ (1+M_{C})∩C^{(1)} which lies in C′ by hypothesis as C = Q_{L}. That is, ddet(1 ⊗ a) = 1 ∈ C^{∗}/C′. Hence, 1 ⊗ a ∈ ker(ddet) = (L ⊗_{F} D)′. Therefore, by Lemma B.2, a^{ℓ} ∈ D′. Likewise, we can take a maximal subfield K of Q, and by looking at 1 ⊗ a ∈ K ⊗_{F} D, we obtain a^{k} ∈ D′ where k = [K : F] = ind(Q). Since gcd(ℓ, k) = 1, it follows that a ∈ D′, completing Step 3.

Step 4. We now prove the theorem in full. Let F be a henselian field, and let D be a tame F -central division algebra. If char(\overline{F}) = 0, then D is strongly tame over F, so the theorem holds for D by Step 2. If char(\overline{F}) = \overline{p} ≠ 0 we have D ≅ P ⊗_{F} Q where P is the \overline{p}-primary component of D and Q is the tensor product of all the other primary components of D. So, gcd(ind(P), ind(Q)) = 1. For any subfield L of F, Q_{L} is tame over L with ind(Q_{L})| ind(Q), which is prime to \overline{p}. So, Q_{L} is strongly tame over L, and the theorem holds for Q_{L} by Step 2. On the other hand, P_{L} is tame over L and ind(P_{L}) is a power of \overline{p}; hence, P_{L} is inertially split. Hence, by Step 1 the theorem holds for P_{L}. Thus, by Step 3 the theorem holds for D. □

The following proposition will complete the proof of the Congruence Theorem.

Proposition B.3. Let F be a henselian valued field, and let D be an F -central division algebra which is defectless over F. Let L be a field, F ⊆ L ⊆ D, and let C = C_{D}(L), so L⊗_{F} D ≅ M_{ℓ}(C) where ℓ = [L : F]. Take any a ∈ 1+M_{D}. Then for 1⊗a ∈ L⊗_{F} D ≅ M_{ℓ}(C),

ddet(1 ⊗ a) ∈ 1 + M_{C} (mod C′),

where ddet denotes the Dieudonné determinant.

Proof. D is an L-D bimodule via multiplication in D. Hence (as L is commutative) D is a right L ⊗_{F} D-module, with module action given by a(∑ ℓ_{i} ⊗ d_{i}) = ∑ ℓ_{i}ad_{i}. D is a simple right L ⊗_{F} D-module, since it is already a simple right D-module. Hence, by Wedderburn’s Theorem, L ⊗_{F} D ≅ End_{Δ}(D), where Δ = End_{L⊗_{F} D}(D) (acting on D on the left). Since (for D acting on D on the right) End_{D}(D) ≅ D (elements of D acting on D by left multiplication) End_{L⊗_{F} D}(D) consists of left multiplication by elements of D which commute with the left action of L on D, i.e., Δ ≅ C_{D}(L) = C. So, L ⊗_{F} D ≅ End_{Δ}(D) ≅ End_{C}(D) ≅ M_{ℓ}(C) where ℓ = [D : C] = [L : F]. The last isomorphism is obtained by choosing a base {b_{1}, …, b_{ℓ}} of D as a left C-vector space (D = Cb_{1} ⊕ … ⊕ Cb_{ℓ}) and writing the matrix for an element of L ⊗_{F} D acting C-linearly on D (on the right) relative to this base, with matrix entries in C.

Because D is defectless over F, D is also defectless over C, i.e., [D : C] = [gr(D) : gr(C)]; thus, the valuation w on D extending v on F is a w|_{C}-norm by [RTW], Cor. 2.3. This means that we can choose our base {b_{1}, …, b_{ℓ}} to be a splitting base for w over w|_{C}, i.e., satisfying, for all c_{1}, …, c_{ℓ} ∈ C,

w( ∑_{i=1}^{ℓ} c_{i}b_{i}) = min_{1≤i≤ℓ} (w(c_{i}) + w(b_{i})).  (B.2)

Let γ_{i} = w(b_{i}) for 1 ≤ i ≤ ℓ.

Let

R = {A = (a_{ij}) ∈ M_{ℓ}(C) : w(a_{ij}) ≥ γ_{i} − γ_{j} for all i, j};
J = {A = (a_{ij}) ∈ M_{ℓ}(C) : w(a_{ij}) > γ_{i} − γ_{j} for all i, j};
1 + J = {I_{ℓ} + A : A ∈ J}, where I_{ℓ} ∈ M_{ℓ}(c) is the identity matrix.

Because w is a valuation, it is easy to check that R is a subring of M_{ℓ}(C) and J is an ideal of R. Therefore, 1 + J is closed under multiplication. Take any f ∈ End_{C}(D) (which acts on D on the right), and let A = (a_{ij}) be the matrix of f relative to the C-base {b_{1}, … b_{ℓ}} of D, i.e., b_{i}f = ∑_{j=1}^{ℓ} a_{ij}b_{j} for all i. So,

w(b_{i}f) = w( ∑_{j=1}^{ℓ} a_{ij}b_{j}) = min_{1≤j≤ℓ} (w(a_{ij}) + γ_{j}).

Thus, w(b_{i}f) ≥ w(b_{i}) = γ_{i} iff w(a_{ij}) ≥ γ_{i} − γ_{j} for 1 ≤ j ≤ ℓ. From this it is clear that A = (a_{ij}) ∈ R iff w(b_{i}f) ≥ w(b_{i}) for all i. Analogously, A ∈ J iff w(b_{i}f) > w(b_{i}) for all i.

Now, take any u ∈ 1 + M_{D}, say u = 1 + m with m ∈ M_{D}. Then, 1 ⊗ m ∈ L ⊗_{F} D corresponds to ρ_{m} ∈ End_{C}(D), where dρ_{m} = dm for all d ∈ D. Let S ∈ M_{ℓ}(C) be the matrix for ρ_{m}. Since w(m) > 0, we have

w(b_{i}ρ_{m}) = w(b_{i}m) = w(b_{i}) + w(m) > w(b_{i}) for all i.

Hence, S ∈ J, as we saw above.

Claim. For any matrix T ∈ 1 + J, we have ddet(T) ∈ 1 + M_{C} (mod C′).

The Proposition follows at once from this claim, since the matrix for 1 ⊗ (1 + m) is I_{ℓ} + S ∈ 1 + J.

Proof of Claim. Take T ∈ 1 + J. The idea is that the process of bringing T to upper triangular form by row operations is carried out entirely within 1 + J. Write T = I_{ℓ} + Z with Z = (z_{ij}) ∈ J. So, w(z_{ii}) > γ_{i} − γ_{i} = 0 for all i, i.e., z_{ii} ∈ M_{C}. Thus, for all i, j, we have

t_{ii} = 1 + z_{ii} ∈ 1 + M_{C} and t_{ij} = z_{ij}, so w(t_{ij}) > γ_{i} − γ_{j} when i ≠ j.

Fix k with 1 ≤ k ≤ ℓ−1. Since t_{kk} ∈ 1+M_{C}, w(t_{kk}) = 0, so t_{kk} ≠ 0. Let Y = (y_{ij}) ∈ M_{ℓ}(C) be the matrix for the row operations to bring 0’s to all entries in the k-th column of T below the main diagonal, i.e., the i-th row of Y T is: (the i-th row of T) − (t_{ik}t_{kk}^{−1}· the k-th row of T) for k < i ≤ ℓ (with the first k rows unchanged). So, y_{ii} = 1 for all i; y_{ik} = −t_{ik}t_{kk}^{−1} for our fixed k and all i with k < i ≤ ℓ; and y_{ij} = 0 otherwise. For i > k,we have

w(y_{ik}) = w(t_{ik}) − w(t_{kk}) > γ_{i} − γ_{k}.

Hence, Y ∈ 1 + J and Y is a unipotent lower triangular matrix. Since 1 + J is closed under multiplication, we have Y T ∈ 1 + J. To bring T to upper triangular form we apply the row operations successively for columns 1 to ℓ − 1. We end up with an upper triangular matrix T′ = Y_{ℓ−1}Y_{ℓ−2} … Y_{2}Y_{1}T ∈ 1 + J, where each Y_{k} ∈ 1 + J is the matrix for zeroing the k-th column as described above, but applied to the matrix Y_{k−1} … Y_{1}T ∈ 1 + J (not to T). Say T′ = (t′_{ij}). Each Y_{k} is unipotent and lower triangular, so ddet(Y_{k}) = 1 ∈ C^{∗}/C′, So, ddet(T′) = ddet(Y_{k−1}) … ddet(Y_{1}) ddet(T) = ddet(T) in C^{∗}/C′. Since T′ is upper triangular with each t′_{ii} ∈ 1 + M_{C}, we have

ddet(T) = ddet(T′) = t′_{11} … t′_{ℓℓ} ∈ 1 + M_{C} (equality modulo C′),

proving the Claim. □

References

[AS] S. A. Amitsur, D. J. Saltman, Generic Abelian crossed products and p-algebras, J. Algebra, 51 (1978), 76–87. 25

[B] M. Boulagouaz, Le gradué d’une algèbre à division valuée, Comm. Algebra, 23 (1995), 4275–4300. 2, 7

[BM] M. Boulagouaz, K. Mounirh, Generic abelian crossed products and graded division algebras, pp. 33–47 in Algebra and Number Theory, eds. M. Boulagouaz and J.-P. Tignol, Lecture Notes in Pure and Appl. Math., 208, Dekker, New York, 2000. 25

[D] P. Draxl, Skew Fields, London Math. Soc. Lecture Note Series, 81, Cambridge University Press, Cambridge, 1983. 28, 29

[EP] A. J. Engler, A. Prestel, Valued Fields, Springer-Verlag, Berlin, 2005. 13, 29

[E] Yu. Ershov, Henselian valuations of division rings and the group SK_{1}, Mat. Sb. (N.S.), 117 (1982), 60–68 (in Russian); English trans., Math USSR-Sb. 45 (1983), 63–71. 3, 8, 14, 16, 28

[G] P. Gille, Le probléme de Kneser-Tits, exposé Bourbaki, No. 983, to appear in Astérisque; preprint

      available at: http://www.dma.ens.fr/∼gille/ . 2

[H_{1}] R. Hazrat, SK_{1}-like functors for division algebras, J. Algebra, 239 (2001), 573–588. 12, 25

[H_{2}] R. Hazrat, Wedderburn’s factorization theorem, application to reduced K-theory, Proc. Amer. Math. Soc., 130 (2002), 311–314. 28, 30

[H_{3}] R. Hazrat, On central series of the multiplicative group of division rings, Algebra Colloq., 9 (2002), 99–106. 17

[H_{4}] R. Hazrat, SK_{1} of Azumaya algebras over Hensel pairs, to appear in Math. Z.; preprint available

      (No. 282) at: http://www.math.uni-bielefeld.de/LAG/ . 2

[HW] R. Hazrat, A. R Wadsworth, On maximal subgroups of the multiplicative group of a division algebra, to appear in J. Algebra; preprint available (No. 260) at: http://www.math.uni-bielefeld.de/LAG/ . 12

[HwW_{1}] Y.-S. Hwang, A. R. Wadsworth, Algebraic extensions of graded and valued fields, Comm. Algebra, 27 (1999), 821–840. 2, 4, 8, 14

[HwW_{2}] Y.-S. Hwang, A. R. Wadsworth, Correspondences between valued division algebras and graded division algebras, J. Algebra, 220 (1999), 73–114. 1, 2, 4, 6, 7, 10, 11, 15, 16, 28 [JW] B. Jacob, A. Wadsworth, Division algebras over Henselian fields., J. Algebra, 128 (1990), 126–179. 5,

16, 28, 29

[J] N. Jacobson, Finite-Dimensional Division Algebras over Fields, Springer-Verlag, Berlin, 1996. 17, 18

[K] M. -A. Knus, Quadratic and Hermitian Forms over Rings, Springer-Verlag, Berlin, 1991. 7

[KMRT] M. -A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol, The Book of Involutions, AMS Coll. Pub., 44, 1998. 17

[L] T.-Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics, 131, Springer-Verlag, New York, 1991. 26, 30

[LT] D. W. Lewis, J.-P. Tignol, Square class groups and Witt rings of central simple algebras, J. Algebra, 154 (1993), 360–376. 16

[Mc_{1}] K. McKinnie, Prime to p extensions of the generic abelian crossed product, J. Algebra, 317 (2007), 813–832. 25

[Mc_{2}] K. McKinnie, Indecomposable p-algebras and Galois subfields in generic abelian crossed products, J. Algebra, 320 (2008), 1887–1907. 25

[Mc_{3}] K. McKinnie, Degeneracy and decomposability in abelian crossed products, preprint, arXiv: 0809.1395. 25

[Mer] A. S. Merkurjev, K-theory of simple algebras, pp. 65–83 in K-theory and Algebraic Geometry: connections with quadratic forms and division algebras, eds. B. Jacob and A. Rosenberg, Proc. Sympos. Pure Math., 58, Part 1, Amer. Math. Soc., Providence, RI, (1995), 65–83. 2

[Mor] P. Morandi, The Henselization of a valued division algebra, J. Algebra, 122 (1989), 232–243. 6, 16

[M_{1}] K. Mounirh, Nicely semiramified division algebras over Henselian fields, Int. J. Math. Math. Sci., 2005 (2005), 571–577. 16

[M_{2}] K. Mounirh, Nondegenerate semiramified valued and graded division algebras, to appear in Comm. Algebra; an earlier version is part of preprint No. 256 at: http://www.math.uni-bielefeld.de/LAG/ . 25

[MW] K. Mounirh, A. R. Wadsworth, Subfields of nondegenerate tame semiramified division algebras, preprint in preparation. 12, 13

[PS] I. A. Panin, A. A. Suslin, On a conjecture of Grothendieck concerning Azumaya algebras, St. Petersburg Math. J., 9 (1998), 851–858. 2

[P_{1}] V. P. Platonov, The Tannaka-Artin problem and reduced K-theory, Izv. Akad. Nauk SSSR Ser. Mat., 40 (1976), 227–261 (in Russian); English trans., Math. USSR-Izv., 10 (1976), 211–243. 2, 8, 16, 28, 29

[P_{2}] V. P. Platonov, Algebraic groups and reduced K-theory, pp. 311–317 in Proceedings of the ICM (Helsinki 1978), ed. O. Lehto, Acad. Sci. Fennica, Helsinki. 2

[PY] V. P. Platonov, V. I. Yanchevskiĭ, SK_{1} for division rings of noncommutative rational functions. Dokl. Akad. Nauk SSSR, 249 (1979), 1064–1068 (in Russian); English trans., Soviet Math. Doklady, 20 (1979), 1393–1397. 2, 17, 18, 19, 24

[R] I. Reiner, Maximal Orders, Academic Press, New York, 1975. 8

[RTW] J.-F. Renard, J.-P. Tignol., A. R. Wadsworth, Graded Hermitian forms and Springer’s theorem, Indag. Math., N.S., 18 (2007), 97–134. 31

[S_{1}] D. J. Saltman, Noncrossed product p-algebras and Galois p-extensions, J. Algebra, 52 (1978), 302–314. 25

[S_{2}] D. J. Saltman, Lectures on Division Algebras, Reg. Conf. Series in Mathematics, no. 94, AMS, Providence, 1999. 7 [St] C. Stuth, A generalization of the Cartan-Brauer-Hua theorem, Proc. Amer. Math Soc., 15 (1964),

211–217. 26

[Sus] A. A. Suslin, SK_{1} of division algebras and Galois cohomology, pp. 75–99 in Algebraic K-theory, ed. A. A. Suslin, Adv. Soviet Math., 4, Amer. Math. Soc., Providence, RI, 1991. 28, 30

[T] J.-P. Tignol, Produits croisés abéliens, J. Algebra, 70 (1981), 420–436. 25

[TW] J.-P. Tignol, A. R. Wadsworth, Value functions and associated graded rings for semisimple algebras, to appear in Trans. Amer. Math. Soc; preprint available (No. 247) at:

      http://www.math.uni-bielefeld.de/LAG/ . 2

[W_{1}] A. R. Wadsworth, Extending valuations to finite dimensional division algebras, Proc. Amer. Math. Soc., 98 (1986), 20–22. 2

[W_{2}] A. R. Wadsworth, Valuation theory on finite dimensional division algebras, Fields Inst. Commun. 32, Amer. Math. Soc., Providence, RI, (2002), 385–449. 2, 5, 8, 16, 29

[We] J. H. M. Wedderburn, On division algebras, Trans. Amer. Math. Soc. 15, (1921), 162–166. 26

[Y] V. I. Yanchevskiĭ, Reduced unitary K-Theory and division rings over discretely valued Hensel fields, Izv. Akad. Nauk SSSR Ser. Mat., 42 (1978), 879–918 (in Russian); English trans., Math. USSR Izvestiya, 13 (1979), 175–213. 14

Department of Pure Mathematics, Queen’s University, Belfast BT7 1NN, United Kingdom

   E-mail address: r.hazrat@qub.ac.uk

Department of Mathematics, University of California at San Diego, La Jolla, California 92093-0112, U.S.A.

   E-mail address: arwadsworth@ucsd.edu