SK_{1}-LIKE FUNCTORS FOR DIVISION ALGEBRAS

R. Hazrat

Abstract. We introduce a Dieudonne/reduced Dieudonne functor for central simple algebras A, which embraces different groups which are associated to A in different contexts, including the reduced Whitehead group SK_{1}(A). We show that a reduced Dieudonne functor has most important functorial properties of the reduced Whitehead group SK_{1}(A). Specializing a Dieudonne functor to the group G(A) = A^{∗}/F^{∗}A′, where F is the center of A and A′ its commutator subgroup, we establish a fundamental connection between this group, its residue version and relative valued group when A is a henselian division algebra. The structure of G(A) turns out to carry significant information about the arithmetic of A. Along these lines, we employ G(A) to compute the group SK_{1}(A). As an application, we obtain theorems of reduced K-theory which requires heavy machinery, as simple examples of our method.

AMS Classification 12E15.

1. Introduction

Let D be a division algebra with center F. The non-triviality of the important group SK_{1}(D) is shown by V. P. Platonov who developed a so-called Reduced K-Theory to compute SK_{1}(D) for certain division algebras. The group SK_{1}(D) enjoys some interesting properties which makes it distinguish from the K-Theory functor K_{1}(D). Among interesting characteristic of the torsion group SK_{1}(D) is its behavior under extension of the ground field. Also in the case of valued division algebra, its stability under reduction, namely SK_{1}(D) = SK_{1}(\overline{D}), where D is unramified division algebra. Also the primary decomposition of a division algebra induces a corresponding decomposition of SK_{1}(D).

On the other hand, there have been other groups which are associated with a division algebra D, in order to study the group and arithmetic structure of D, e.g, D^{∗}/Nrd(D^{∗})D′ in [1], or D^{∗}/D^{∗}^{2} in [8]. Also see [5]. It can be seen that these groups are in close connection with the group SK_{1}(D).

On this note we introduce a “Dieudonne” functor for division algebras which embraces groups that have been already associated with division algebra in different contexts. We then show that a “reduced” Dieudonne functor (See Definition 2.2) shares most important functorial properties of the reduced Whitehead group. Notice that a reduced Dieudonne functor covers a wide range of groups of different nature. In section 2 we will show that a reduced Dieudonne functor is torsion of bounded exponent. we then obtain some SK_{1}-like properties for this functor. we show that a reduced Dieudonne functor may grow “pathologically” for algebraic extension of ground field whose degree is prime to the index of D. It is then shown that this functor satisfies a decomposition property analog to one for SK_{1}(D) (Theorem 2.10).

Key words and phrases. Division ring, Reduced K-theory.

We then specialize a Dieudonne functor to the group G(D) = D^{∗}/F^{∗}D′ where D is a division algebra over its center F and D′ is its commutator subgroup. It turns out that there is a close connection between the group structure of G(D) and algebraic structure of D. For example in section 3, after establishing a fundemental connection between G(D), its residue version and relative valued group when D enjoys a henselian valuation, we show that if D is a totally ramified division algebra, then there is a one to one correspondence between isomorphism classes of F -subalgebra of D and the subgroups of G(D). We then use G(D) to compute SK_{1}(D) for certain division algebras. We show that if G(D) canonically coincides with the relative valued group, then there is a explicit formula for the group SK_{1}(D) (Theorem 3.7). It turns out that many theorems and examples of reduced K-theory which require heavy machinery can all be viewed as simple examples of our case. Section 3 is devoted to the unitary version of the group G(D).

We fix some notations. Let D be a division algebra over its center F with index i(D) = n. Then Nrd_{D/F} : D^{∗} ⟶ F^{∗} is the reduced norm function and SK_{1}(D) = D^{(1)}/D′ is the reduced Whitehead group where D^{(1)} is the kernel of Nrd_{D/F}. Put SH^{0}(D) for the cokernel of Nrd_{D/F}. we take μ_{n}(F) for the group of n − th roots of unity in F, and Z(D′) for the center of the group D′. Observe that μ_{n}(F) = F^{∗} ∩ D^{(1)} and Z(D′) = F^{∗} ∩ D′. If G is a group, by G^{n} we denote the subgroup of G generated by all elements of n−th power of G. Let exp(G) stands for the exponent of the group G. Let also det : GL_{n}(D)/SL_{n}(D) ⟶ D^{∗}/D′ denote the Dieudonne determinant, where GL_{n}(D) is the general linear group and SL_{n}(D) is its commutator subgroup.

2. Dieudonne Functor

Let C be the class of all central simple algebras and 𝔊 : C ⟶ Ab be a functor from C to the category of abelian groups.

Definition 2.1. The functor 𝔊 is called a Dieudonne functor if for any division algebra D and a nonnegative integer n, there is a homomorphism d : 𝔊(M_{n}(D)) ⟶ 𝔊(D) such that for any x ∈ 𝔊(A), di(x) = x^{n}, where i : 𝔊(D) ⟶ 𝔊(M_{n}(D)) induced by the natural embedding D ⟶ M_{n}(D).

The functor K_{1} with the Dieudonne determinant det as d, clearly forms a Dieudonne functor.

Let A be a central simple algebra and (A^{∗})^{2} be the subgroup of A^{∗} generated by the squares of A^{∗}. It is easy to see that 𝔊(A) = A^{∗}/(A^{∗})^{2} with det as above is a Dieudonne functor. This group is studied in [8] in connection with Witt rings of division algebras.

Definition 2.2. A Dieudonne functor 𝔊 is said to be reduced if for any field F, 𝔊(F) is trivial and if x ∈ Ker(𝔊(M_{n}(D)) ⟶ 𝔊(D)), where D is a division algebra and n ∈ ℕ, then x^{n} = 1.

Example 2.3. Let A ∈ C with center F, then the functors 𝔊(A) = A^{∗}/F^{∗}A′, 𝔊(A) = (A^{∗})^{2}/(F^{∗})^{2}A′, and 𝔊(A) = A^{∗}/F^{∗}A′_{r} where A′_{r} = {x ∈ A^{∗}|x^{r} ∈ A′} and r ∈ ℕ can be all viewed as reduced Dieudonne functors.

Example 2.4. Let A ∈ C be a central simple algebra finite over its center. In order to consider the reduced Whitehead group SK_{1}(A) and SH^{0}(A) as a reduced Dieudonne functors, we shall limit the morphisms of our category (See chapters 22 and 23 in [3]). The following commutative diagram shows that SK_{1} and SH^{0} are Dieudonne functors.

\xymatrix{1 \ar[r] & SK_{1}(D) \ar[d] \ar[r] & K_{1}(D) \ar[r]^{Nrd_{D/F}} \ar[d] & F^{∗} \ar[d]^{η_{n}} \ar[r] & SH^{0}(D) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & SK_{1}(M_{n}(D)) \ar[d] \ar[r] & K_{1}(M_{n}(D)) \ar[r]^{Nrd_{D/F}} \ar[d]_{det} & F^{∗} \ar[d]^{1} \ar[r] & SH^{0}(M_{n}(D)) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & SK_{1}(D) \ar[r] & K_{1}(D) \ar[r]^{Nrd_{D/F}} & F^{∗} \ar[r] & SH^{0}(D) \ar[r] & 1}

where η_{n}(x) = x^{n} for any x ∈ F^{∗} and D is a division algebra with center F. Now it is easy to see that SK_{1}(A) = A^{(1)}/A′ and SH^{0}(A) = F^{∗}/Nrd_{A/F} (A^{∗}) satisfy conditions of being reduced Dieudonne functor.

In the same way, it can be seen that 𝔊(A) = A^{∗}/Nrd_{A/F} (A^{∗})A′ and 𝔊(A) = A^{∗}/F^{∗}A^{(1)} ≃ Nrd(A^{∗})/F^{∗}^{DegA} are also reduced Dieudonne functors.

Our primary aim in this section is to show that a reduced Dieudonne functor 𝔊 shares almost all important functorial properties of SK_{1}. Note that a reduced Dieudonne functor covers groups of different nature which are associated to a division algebra. We will show that this functor is torsion of bounded exponent.

We begin by showing that a reduced Dieudonne functor may grow pathologically for algebraic extension of ground field whose degree is prime to the index of division algebra D.

Clearly the natural embedding of D in D ⊗_{F} L where L is a finite field extension of F, induce a group homomorphism I : 𝔊(D) ⟶ 𝔊(D ⊗_{F} L). The following proposition provides us with a homomorphism in the opposite direction.

Proposition 2.5. Let 𝔊 be a Dieudonne functor and D be a division ring with center F. If L is a finite extension of F then there is a homomorphism P : 𝔊(D ⊗_{F} L) ⟶ 𝔊(D) such that PI = η_{[L:F]}, where η_{m}(x) = x^{m}.

Proof. Let [L : F] = m. Consider the regular representation L ⟶^{ι} M_{m}(F) and the corresponding sequence when we tensor over F with D:

D ⟶ D ⊗_{F} L ⟶^{1⊗ι} D ⊗_{F} M_{m}(F) ⟶ M_{m}(D)

(2.1)  a ⟼ a ⊗ 1 ⟼ a ⊗ 1 ⟼ aI_{m}
1 ⊗ ℓ ⟼ 1 ⊗ ι(ℓ) ⟼ ι(ℓ).

Since 𝔊 is a Dieudonne functor, we obtain a homomorphism P : 𝔊(D ⊗_{F} L) ⟶ 𝔊(M_{m}(D)) ⟶^{d} 𝔊(D). Again the sequence (2.1) shows that, thanks to definition of a Dieudonne functor, PI(x) = x^{m}. □

Note that in the above proposition D could be an infinite dimensional division algebra. If D is finite dimension over its center F, then it turns out that 𝔊(D), where 𝔊 is reduced Dieudonne functor, is a torsion group.

Corollary 2.6. Let 𝔊 be a reduced Dieudonne functor. Then for any division algebra D of index n, 𝔊(D) is a torsion group of bounded exponent n^{2} = [D : Z(D)].

Proof. Thanks to Proposition 2.5, for any finite field extension L of F = Z(D), we have the sequence of homomorphisms 𝔊(D) ⟶^{I} 𝔊(D ⊗_{F} L) ⟶^{P} 𝔊(D), such that PI(x) = x^{m}, where x ∈ 𝔊(D) and [L : F] = m. Now let L be a maximal subfield of D. Since L is a splitting field for D, we get the sequence of homomorphisms 𝔊(D) ⟶^{I} 𝔊(M_{n}(L)) ⟶^{P} 𝔊(D). Since 𝔊 is a reduced Dieudonne functor, it follows that 𝔊(M_{n}(L)) is a torsion group of bounded exponent n (See Definition 2.2). Now the fact that for any x ∈ 𝔊(D), PI(x) = x^{n}, shows that 𝔊(D) is a torsion group of bounded exponent n^{2} = [D : Z(D)]. □

Now it is immediate that if A is a central simple algebra, then 𝔊(A) is also torsion.

We later specialize a Dieudonne functor and show that we can reduce this bound for certain functors. The following corollary shows that the analog result of the behavior of SK_{1}(D) under extension of the ground field holds for a reduced Dieudonne functor. Namely, we show that 𝔊(D) embeds in 𝔊(D ⊗_{F} L) when the index of D and [L : F] are relatively prime.

Corollary 2.7. Let D be a division ring over its center F and L/F be a finite field extension such that [L : F] is relatively prime to the index of D. Then the canonical homomorphism 𝔊(D) ⟶^{I} 𝔊(D ⊗_{F} L) is injective.

Proof. Let i(D) = n and [L : F] = m. Suppose I(x) = 1 for some x ∈ 𝔊(D). By Proposition 2.5, PI(x) = x^{m} = 1. But 𝔊(D) is a torsion of bounded exponent n^{2}. Hence x^{n^{2}} = 1. Since m and n are relatively prime, x = 1 and the proof is complete. □

Computing a Dieudonne functor in general is a difficult task, as it covers groups of different nature like SK_{1}(D), or the group G(D) = D^{∗}/F^{∗}D′. In the next section we specialize a Dieudonne functor to 𝔊(D) = D^{∗}/F^{∗}D′ and compute it for certain division algebras. But before we continue with the functorial properties of 𝔊, let us consider a class of reduced Dieudonne functors which enjoy an additional property. Namely let τ be a natural transformation τ : K_{1} ⟶ 𝔊 such that,

(1) For any object A in C, τ_{A} : K_{1}(A) ⟶ 𝔊(A) is an epimorphism.

(2) For any division algebra D and a nonnegative integer n, the following diagram commutes,

\xymatrix{K_{1}(M_{n}(D)) \ar[r]^{τ} \ar[d]^{det} & 𝔊(M_{n}(D)) \ar[d]^{d} \\ K_{1}(D) \ar[r]^{τ} & 𝔊(D).}

For example, the functors of Example 2.3, or 𝔊(D) = Nrd_{D/F} (D^{∗})/F^{∗}^{i(D)} and 𝔊(A) = A^{∗}/Nrd_{A/F} (A^{∗})A′ enjoy the above property.

Even in this case, it is not clear when a reduced Dieudonne functor is trivial. The following theorem is almost the only known example where 𝔊(D) with above property is trivial.

Corollary 2.8. Let D be a division ring of quaternions over real-closed field. Let 𝔊 be a reduced Dieudonne functor with a natural epimorphism τ : K_{1} ⟶ 𝔊 as above. Then 𝔊(D) = 1.

Proof. For any finite field extension L of F = Z(D), the following diagram is commutative,

\xymatrix{K_{1}(D ⊗_{F} L) \ar[r]^{P} \ar[d]^{τ} & K_{1}(D) \ar[d]^{τ} \\ 𝔊(D ⊗_{F} L) \ar[r]^{P} & 𝔊(D).}

Now since D is algebraically closed (See [7], Section 16), thanks to Proposition 2.5 and above diagram, 𝔊(D ⊗_{F} L) ⟶^{P} 𝔊(D) is an epimorphism. Replace L by \overline{F}, the algebraic closure of F. Because \overline{F} is a splitting field for D, 𝔊(D⊗_{F} \overline{F}) = 𝔊(M_{2}(\overline{F})). We show that 𝔊(M_{2}(\overline{F})) is a trivial group and hence the corollary follows. Since τ : K_{1} ⟶ 𝔊 is a natural epimorphism, there is a sequence of homomorphism

ψ : K_{1}(\overline{F}) = \overline{F}^{∗} ⟶^{≃} K_{1}(M_{2}(\overline{F})) ⟶^{epi.} 𝔊(M_{2}(\overline{F})).

Take x ∈ 𝔊(M_{2}(\overline{F})). Since \overline{F} is algebraically closed, there exist y ∈ K_{1}(\overline{F}) = \overline{F}^{∗} such that ψ(y^{2}) = x. But 𝔊(M_{2}(\overline{F})) is a torsion group of bounded exponent 2, hence x = 1. This shows that 𝔊(M_{2}(\overline{F})) is trivial and the proof is complete. □

Back to the functorial properties of 𝔊, the next step is to replace the field L in Proposition 2.5 by a division ring. The following proposition shows that the same result holds here too.

Proposition 2.9. Let 𝔊 be a Dieudonne functor. Let A and B be division rings with center F such that [B : F] is finite. Then there is a homomorphism P : 𝔊(A ⊗_{F} B) ⟶ 𝔊(A) such that PI = η_{[B:F]}.

Proof. Let [B : F] = m. We have the following sequence of F -algebra homomorphisms,

A ⟶ A ⊗_{F} B ⟶ A ⊗_{F} B ⊗_{F} B^{op} ⟶ A ⊗_{F} M_{m}(F) ⟶ M_{m}(A).

This implies the group homomorphism P : 𝔊(A ⊗_{F} B) ⟶ 𝔊(M_{m}(A)) ⟶^{d} 𝔊(A). The rest of the proof is similar to one in Proposition 2.5. □

Note that in the above proposition A could be of infinite dimension over its center F. A same statement as Corollary 2.7 could be obtained here too. In particular if (i(A) : i(B)) = 1 then 𝔊(A) embeds in 𝔊(A ⊗_{F} B) and similarly for B. Employing torsion theory of groups and sequences which are appeared in the above propositions, we can write the primary decomposition for 𝔊(D). The proof follows more or less the same pattern as for SK_{1}(D).

Theorem 2.10. Let 𝔊 be a reduced Dieudonne functor. Let A and B be division algebras with center F such that (i(A), i(B)) = 1. Then 𝔊(A ⊗_{F} B) = 𝔊(A) × 𝔊(B).

Proof. By Corollary 2.6, 𝔊(A ⊗_{F} B) is a torsion group of bounded exponent m^{2}n^{2} where m = i(A) and n = i(B). Therefore 𝔊(A ⊗_{F} B) ≃ G × H, where exp(G)|m^{2} and exp(H)|n^{2}. By Proposition 2.9, we have the sequence:

(2.2)  𝔊(A) ⟶^{ϕ} 𝔊(A ⊗_{F} B) ⟶^{ψ} 𝔊(A ⊗ B ⊗ B^{op}) ⟶^{θ} 𝔊(A)

such that θψϕ = η_{n^{2}}. Hence 𝔊(A) = η_{n^{2}}η_{n^{2}}(𝔊(A)) = η_{n^{2}}θψϕ(𝔊(A)) ⊆ θψη_{n^{2}}(G × H) = θψ(G) ⊆ 𝔊(A). This shows that θψ|_{G} : G ⟶ 𝔊(A) is surjective. Next we show that θψ|_{G} is injective. Considering the regular representation B^{op} ⟶ M_{n^{2}}(F), by Proposition 2.5, we have the following sequence

𝔊(A ⊗_{F} B) ⟶^{ψ} 𝔊(A ⊗ B ⊗ B^{op}) ⟶^{ψ′} 𝔊(A ⊗ B ⊗ M_{n^{2}}(F)) ⟶^{θ′} 𝔊(A ⊗_{F} B)

such that θ′ψ′ψ = η_{n^{2}}. Now if w ∈ G − 1, then θ′ψ′ψ(w) = η_{n^{2}}(w) = w^{n^{2}} ≠ 1. Therefore ψ|_{G} is injective. Rewrite the sequence (2.2) as follows:

𝔊(A ⊗_{F} B) ⟶^{ψ} 𝔊(A ⊗ B ⊗ B^{op}) ⟶^{iso.} 𝔊(M_{n^{2}}(A)) ⟶^{d} 𝔊(A).

Suppose x ∈ G such that θψ(x) = 1. The above sequence and the definition of reduced Dieudonne functor shows that ψ(x)^{n^{2}} = 1. Since ψ|_{G} is injective, x^{n^{2}} = 1. On the other hand because exp(G)|m^{2} then x^{m^{2}} = 1. Since m and n are relatively prime, x = 1. This shows that θψ is an isomorphism and so 𝔊(A) ≃ G. In the similar way it can be shown that 𝔊(B) ≃ H. Therefore the proof is complete. □

Now we specialize a Dieudonne functor to 𝔊(D) = D^{∗}/F^{∗}D′ where D is a division ring with center F. It turns out that the structure of this group carries significant information about the arithmetic of D. Let us start with the following definition.

Definition 2.11. Let A be a central simple algebra with center F. For each integer s ≥ 0 define G_{s}(A) = A^{∗}/F^{∗}^{s}A′ and Ĝ_{s}(A) = lim_{←_{k}} G_{s^{k}}(A) where k ≥ 1.

Clearly G_{0}(A) = K_{1}(A) and G_{1}(A) = A^{∗}/F^{∗}A′. It is easy to see that G_{s}(A) and Ĝ_{s}(A) are Dieudonne functors. Also 𝔊(A) = G_{1}(A) is a reduced Dieudonne functor. It is immediate from Corollary 2.6 that G_{1}(A) is a torsion group of bounded exponent m[D : Z(D)] where A = M_{m}(D). So it follows that G_{s}(A) are all torsions. Note that Ĝ_{s}(A) is not torsion in general. As we mentioned above we specialize 𝔊 to the group G(A) = A^{∗}/F^{∗}A′. In this case we can reduce the bound of the group G(A) as follows.

Let D be a division algebra over its center F. If a ∈ D is algebraic over F of degree m, then by Wedderburn’s factorization theorem, we can associate m conjugates to a such that the sum and the product of them are in F. The first observation is used, for instance, in [2] to compute the center of the group D^{∗}/1 + M_{D} (which we will use it in this note) where M_{D} is the maximal ideal of the valuation ring of D (See Section 3). The fact that the product of the conjugates is in F, shows that a^{[F (a):F]} is in F^{∗}D′. Therefore if D is an algebraic division algebra over its center F, then G(D) = D^{∗}/F^{∗}D′ is a torsion group. This is used in [10] to investigate the role of D′ in the structure of a division ring.

Now let D be division algebra over its center F with index n. Let N be a normal subgroup of D^{∗}. For x ∈ N, it follows that Nrd_{D/F} (x) is the product of n conjugates of x. This shows that Nrd|_{N} : N ⟶ Z(N) is well defined, where Z(N) = F^{∗} ∩ N is the center of the group N. We will use the following lemma in Section 3 for normal subgroups of D^{∗} which arise from a valuation on D.

Lemma 2.12. Let D be a division ring over its center F with index n. Let N be a normal subgroup of D^{∗}. Then N^{n} ⊆ Z(N)[D^{∗}, N].

Proof. Let x ∈ N. As stated above, Nrd_{D/F} (x) = d_{1}xd_{1}^{−1} ⋯ d_{n}xd_{n}^{−1}. This implies that x^{n} = Nrd_{D/F} (x)d_{x} where d_{x} ∈ [D^{∗}, N] and Nrd_{D/F} (x) ∈ F^{∗} ∩ N. This shows that N^{n} ⊆ (F^{∗} ∩ N)[D^{∗}, N]. □

If in the above lemma we take N = D^{∗}, then for any x ∈ D, x^{n} = Nrd_{D/F} (x)d_{x} where d_{x} ∈ D′. This in effect shows that G(D) = D^{∗}/F^{∗}D′ is a torsion group of bounded exponent n.

In the next section we will show yet another SK_{1}-like property for the group G(D). Namely G(D) satisfy the following stability, G(D) ≃ G(D((x))) where G(D((x))) is the division ring of Laurent series (Corollary 3.6). We close this section by the following theorem, which shows that the group G(D) = D^{∗}/F^{∗}D′ does not always follow the same pattern as the reduced Whitehead group SK_{1}(D).

Theorem 2.13. (J. -P. Tignol) Let D be a division algebra over its center F with index n. Then the following sequence where ℘ runs over the irreducible monic polynomials of F [x] and n_{℘} is the index of D ⊗_{F} F [x]/℘, is split exact.

1 ⟶ G(D) ⟶ G(D(x)) ⟶ ⨁_{℘} \frac{ℤ}{n/n_{℘}ℤ} ⟶ 1.

Proof. By Proposition 7 in [8], the sequence

1 ⟶ K_{1}(D) ⟶ K_{1}(D(x)) ⟶ ⨁_{℘} n_{℘}/nℤ ⟶ 1

which is obtained from the localization exact sequence of algebraic K-theory is split exact. Now since the group G(D) is the cokernel of the natural map K_{1}(F) ⟶ K_{1}(D), applying the snake lemma to the commutative diagram,

\xymatrix{1 \ar[r] & K_{1}(F) \ar[r] \ar[d] & K_{1}(F (x)) \ar[r] \ar[d] & ⨁_{℘} ℤ \ar[r] \ar[d] & 1 \\ 1 \ar[r] & K_{1}(D) \ar[r] & K_{1}(D(x)) \ar[r] & ⨁_{℘} n_{℘}/nℤ \ar[r] & 1}

the result follows. □

3. On The Group G(D) Over Henselian Division Algebra

In this section we assume that D is a finite dimensional division algebra over a henselian field F = Z(D). Recall that a valuation v on F is Henselian if and only if v has a unique extension to each field algebraic over F. Therefore v has a unique extension to D (see [6] and [15]). Denote by V_{D}, V_{F} the valuation rings, M_{D}, M_{F} their maximal ideals, and \overline{D}, \overline{F} the residue division algebra and the residue field of D and F respectively. We also take Γ_{D}, Γ_{F} for the value groups, U_{D}, U_{F} for the groups of units of V_{D}, V_{F} respectively. Furthermore, we assume that D is a tame division algebra, i.e., Z(\overline{D}) is separable over \overline{F} and Char\overline{F} does not divide i(D). The quotient group Γ_{D}/Γ_{F} is called the relative valued group of the valuation. In this setting it turns out that D is defectless, namely we have [\overline{D} : \overline{F}][Γ_{D} : Γ_{F}] = [D : F]. D is said to be unramified over F if [Γ_{D} : Γ_{F}] = 1. At the other extreme D is said to be totally ramified if [D : F] = [Γ_{D} : Γ_{F}]. D is called semiramified if \overline{D} is a field and [\overline{D} : \overline{F}] = [Γ_{D} : Γ_{F}] = i(D). For a recent account of the theory of henselian valued division algebras see [6].

We start with the following theorem which describes a fundemental connection between the group G(D) and its residue version.

Theorem 3.1. Let D be a tame division algebra over a henselian field F = Z(D) with index n. Let L/F be a subfield of D. Then the following sequence is exact.

(3.1)  1 ⟶ \overline{D}^{∗}/\overline{L}^{∗}\overline{D′} ⟶ D^{∗}/L^{∗}D′ ⟶ Γ_{D}/Γ_{L} ⟶ 1.

Proof. Consider the normal subgroup 1 + M_{D} of D^{∗}. Thanks to Lemma 2.12, we have (1+M_{D})^{n} ⊆ ((1+M_{D})∩F^{∗})[D^{∗}, 1+M_{D}]. But applying the Hensel lemma to the elements of the group 1 + M_{D}, shows that, this group is n-divisible. Therefore 1 + M_{D} = (1 + M_{D})^{n}. Hence 1 + M_{D} ⊆ (1 + M_{F})D′. Now consider the reduction map U_{D} ⟶ \overline{D}^{∗}. We have the following sequence:

\begin{aligned}\overline{D}^{∗} ⟶^{≃} U_{D}/1 + M_{D} ⟶^{nat.} U_{D}/(1 + M_{F})D′ & ⟶^{nat.} U_{D}/(1 + M_{L})D′ ⟶^{nat.} \\ & ⟶^{nat.} U_{D}/U_{L}D′ ⟶^{≃} L^{∗}U_{D}/L^{∗}D′. \end{aligned}

Therefore ψ : \overline{D}^{∗}/(\overline{L}^{∗})\overline{D′} ⟶ L^{∗}U_{D}/L^{∗}D′ is an isomorphism. Considering the fact that D^{∗}/L^{∗}U_{D} ≃ Γ_{D}/Γ_{L}, the theorem follows. □

Now we are ready to compute G(D) for some certain cases. The statements i. and ii. of the following theorem were first appeared in [5] using results from reduced K-theory.

Theorem 3.2. Let D be a henselian division algebra tame over its center F with index n. Then

i. If D is unramified over F then G(D) ≃ G(\overline{D}).

ii. If D is totally ramified over F then G(D) = Γ_{D}/Γ_{F}.

iii. If D is semiramified and \overline{D} is cyclic over \overline{F} then the following sequence where N_{\overline{D}/\overline{F}} is the norm function, is exact.

1 ⟶ N_{\overline{D}/\overline{F}} (\overline{D})/\overline{F}^{n} ⟶ G(D) ⟶ Γ_{D}/Γ_{F} ⟶ 1.

iv. If D is unramified and s is an integer relatively prime to Char(\overline{F}), then the sequence 1 ⟶ Ĝ_{s}(\overline{D}) ⟶ Ĝ_{s}(D) ⟶ Γ̂_{D_{(s)}} is exact.

Proof. i. Writing (3.1) for L = F, we have:

1 ⟶ \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} ⟶ G(D) ⟶ Γ_{D}/Γ_{F} ⟶ 1.

Now if (D, v) is unramified, namely [Γ_{D} : Γ_{F}] = 1, then \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} ≃ D^{∗}/F^{∗}D′. On the other hand Z(\overline{D}) = \overline{F} and D^{∗} = F^{∗}U_{D}. Therefore, for a, b ∈ D^{∗}, the element c = aba^{−1}b^{−1} may be written in the form c = αβα^{−1}β^{−1}, where α and β ∈ U_{D}. This shows \overline{D′} = \overline{D}′, so G(D) ≃ G(\overline{D}).

ii. If D is totally ramified over F then \overline{D} = \overline{F}. Writing (3.1) for L = F, since the group \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} is trivial then G(D) = Γ_{D}/Γ_{F}.

iii. Let D be semiramified and \overline{D} be cyclic over \overline{F}. Consider the norm function N_{\overline{D}/\overline{F}} : \overline{D}^{∗} ⟶ \overline{F}^{∗}. Moreover for any x ∈ U_{D} we have \overline{Nrd_{D/F} (x)} = N_{\overline{D}/\overline{F}} (\overline{x}) (See [4]). This shows that \overline{D′} ⊆ KerN_{\overline{D}/\overline{F}}. But if x ∈ KerN_{\overline{D}/\overline{F}} then by Hilbert theorem 90, there is a \overline{a} such that x = \overline{a}σ(\overline{a})^{−1}, where σ is the generator of Gal(\overline{D}/\overline{F}). It is well known that the fundemental homomorphism D^{∗} ⟶ Gal(Z(\overline{D})/\overline{F}) is surjective. Therefore σ : \overline{D} ⟶ \overline{D} is of the form σ(\overline{a}) = \overline{cac^{−1}}, for some c ∈ D^{∗}. This shows that x ∈ \overline{D′}. Therefore KerN_{\overline{D}/\overline{F}} = \overline{D′}. Now it is easy to see

that \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} ≃ N_{\overline{D}/\overline{F}} (\overline{D}^{∗})/\overline{F}^{∗}^{n}. So thanks to (3.1), 1 ⟶ N_{\overline{D}/\overline{F}} (\overline{D}^{∗})/\overline{F}^{∗}^{n} ⟶ G(D) ⟶ Γ_{D}/Γ_{F} ⟶ 1 is exact.

iv. We can write the similar statement as Theorem 3.1 for L = F^{s}, where s is an integer relatively prime to Char\overline{F}. In this case we only have to change Γ_{L} to sΓ_{F}. Hence we obtain the exact sequence:

1 ⟶ \overline{D}^{∗}/(\overline{F}^{∗})^{s}\overline{D′} ⟶ D^{∗}/F^{∗}^{s}D′ ⟶ Γ_{D}/sΓ_{F} ⟶ 1.

Now because inverse limit is a left exact functor, we obtain the following exact sequence:

1 ⟶ lim_{←_{k}} \overline{D}^{∗}/(\overline{F}^{∗})^{s^{k}} \overline{D′} ⟶ Ĝ_{s}(D) ⟶ lim_{←_{k}} Γ_{D}/s^{k}Γ_{F} .

If D is unramified, thanks to i. \overline{F} = Z(\overline{D}) and \overline{D′} = \overline{D}′. therefore the result follows. □

Remark 3.3. If \overline{D} is a cyclic field extension of \overline{F}, a modification of the proof of iii. above shows that N_{\overline{D}/\overline{F}} (\overline{D}^{∗})/\overline{F}^{∗}^{f} ⟶ \overline{D}^{∗}/\overline{F}^{∗}\overline{D′}, where [\overline{D} : \overline{F}] = f, is always

surjective. Therefore if N_{\overline{D}/\overline{F}} (\overline{D}^{∗})/\overline{F}^{∗}^{f} = 1 then G(D) = Γ_{D}/Γ_{F}.

Example 3.4. Let ℂ be a field of complex numbers. Let 1 ≠ σ ∈ Gal(ℂ/ℝ) where ℝ is real numbers. Then by Hilbert construction, D = ℂ((x, σ)) is a division ring with center F = ℝ((x^{2})). We show that G(D) = ℤ_{2}. D has a natural valuation such that Γ_{D}/Γ_{F} = ℤ/2ℤ = ℤ_{2}. Clearly \overline{D} = ℂ and \overline{F} = ℝ. Since N_{ℂ/ℝ}(ℂ) = ℝ^{2} by Theorem 3.2 iii., G(D) = Γ_{D}/Γ_{F} = ℤ_{2}.

There have been significant results on the structure of relative valued group in the case of totally ramified algebra. Using Theorem 3.2 we can write interesting statements relate group structure of G(D) to algebraic structure of D. Recall that the group G(D) is torsion of bounded exponent n.

Theorem 3.5. Let D be a valued division algebra tame and totally ramified over henselian field F = Z(D) of index n. Then,

i. There is a one to one correspondence between isomorphism classes of F - subalgebra of D and the subgroups of G(D).

ii. exp(G(D)) divides the exponent of D, i.e., the order of [D] in Br(F), the Brauer group of F.

iii. D is cyclic division algebra if and only if exp(G(D)) = n.

Proof. The theorem follows by comparing Theorem 3.2 ii., with the results on relative valued group in the case of totally ramified valuation (See for example [15]). □

Corollary 3.6. Let D be a finite dimensional division algebra over its center F such that CharF ∤ i(D)s. Then there is a monomorphism Ĝ_{s}(D) ⟶ Ĝ_{s}(D((x))). In particular if CharF ∤ i(D) then G(D) ⋍ G(D((x))). □

Now we are in a position to use the group G(D) to compute SK_{1}(D). The following theorem enables us to compute SK_{1}(D) when, roughly speaking, G(\overline{D}) is trivial. Note that we do not use any results from reduced K-theory.

Theorem 3.7. Let D be a tame division algebra over henselian field F = Z(D) of index n.

i. If \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} = 1 then SK_{1}(D) = μ_{n}(F)/Z(D′).

ii. If D is a cyclic division algebra with a maximal cyclic extension L/F such that \overline{D}^{∗}/\overline{L}^{∗}\overline{D′} = 1 then SK_{1}(D) = 1.

Proof. i. As the proof of Theorem 3.1 shows, we have a natural isomorphism,

ψ : \overline{D}^{∗}/(\overline{F}^{∗})\overline{D′} ⟶ U_{D}/U_{F} D′.

Now if \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} = 1 then U_{D} = U_{F} D′. But D^{(1)} ⊆ U_{D}. This shows that D^{(1)} = μ_{n}(F)D′. Using the fact that μ_{n}(F) ∩ D′ = Z(D′) the theorem follows.

ii. The same proof as i. shows that if \overline{D}^{∗}/\overline{L}^{∗}\overline{D′} = 1 then U_{D} = U_{L}D′. Therefore D^{(1)} ⊆ U_{L}D′. Let x ∈ D^{(1)}. Then x = ld where l ∈ L and d ∈ D′. So Nrd_{D/F} (x) = N_{L/F} (l) = 1. Hilbert theorem 90 for the cyclic extension L/F guarantee that l = aσ(a)^{−1}, where σ is a generator of Gal(L/F). Now Skolem-Noether theorem implies that σ(a) = cac^{−1} where c ∈ D^{∗}. Therefore l = aca^{−1}c^{−1}. This shows that D^{(1)} = D′. □

The part i. of the above theorem shows that if D is totally ramified, then SK_{1}(D) = μ_{n}(F)/Z(D′).

We conclude both theorems of [9] which are obtained by using a heavy machinery of reduced K- theory, as natural examples of the above theorem.

Example 3.8. For any division algebra D with center F = ℝ((x_{1}, ⋯, x_{m})) where ℝ is the real numbers, SK_{1}(D) is trivial.

Proof. From number theory, it is well known that [D : F] = 2^{s} where s ≤ m. Since the complete field F = ℝ((x_{1}, ⋯, x_{m})) has a natural valuation, then D enjoys a valuation which is obviously tame. It is clear that \overline{F} = ℝ. Because the only division algebras over real numbers are either the quaternion ℍ_{ℝ} or the field ℂ of complex numbers, therefore \overline{D} = ℍ_{ℝ} or \overline{D} = ℂ. Now Corollary 2.6 and Remark 3.3 show that in either case \overline{D}^{∗}/\overline{F}^{∗}\overline{D′} = 1. Now by Theorem 3.7, SK_{1}(D) ≃ μ_{i(D)}(F)/Z(D′). But clearly μ_{i(D)}(F) = {1, −1} and because the index of D is even, −1 ∈ D′. This implies that SK_{1}(D) = 1.

Example 3.9. For any division algebra with center F = C((x_{1}, ⋯, x_{m})) where C is an algebraically closed field, SK_{1}(D) is cyclic.

Example 3.10. Hilbert classical construction of division algebras. Let L be a field and σ ∈ Aut(L) such that o(σ) = n. Let F = F ix(σ) be the fixed field of σ. Hence Gal(L/F) is a cyclic group with the generator σ. Let D = L((x, σ)) be the division ring of formal Laurent series. It follows that Z(D) = F ((x^{n})) and i(D) = n. D has a natural valuation, and it is easy to see that with this valuation D is semiramified and L((x^{n})) is a maximal subfield of D. Now by Theorem 2.7 ii., SK_{1}(D) is trivial.

Example 3.11. From Theorem 3.7 ii., it is immediate that reduced Whitehead group of division algebra over a local field is trivial.

Because most of the interesting valued division algebras arise from the iterated formal power series fields, we may consider r-iterated Henselian division algebras. Following Platonov in [13], we define inductively an r-iterated henselian field F if its residue field \overline{F} is an (r − 1)-iterated henselian field.^{1} Let (D_{i}, v_{i}), 0 ≤ i ≤ r − 1 be an r iterated henselian division algebra (\overline{D_{i}} = D_{i+1}). Let Φ_{i} : U_{D_{i−1}} ⟶ D_{i} be the i-th natural reduction map. Then Φ_{i}(Φ_{i−1}(⋯ (Φ_{1}(a)) ⋯)) is called i-iterated reduction, if it is defined. Denote the r iterated henselian division algebra by D, (D = D_{0}, Z(D) = F = F_{0}). We also need the following notations in order to state the following lemma. By [U_{D}]_{i} and [U_{F}]_{i} we denote the set of all elements of D and F respectively, such that i iterated reduction defines. Also by [1 + M_{D}]_{i} and [1 + M_{F}]_{i}, we denote the subsets of [U_{D}]_{i} and [U_{F}]_{i} such that i iterated reduction equals one. Clearly [1 + M_{F}]_{1} = 1 + M_{F}. we can write the main lemma of [5] in this setting.

Lemma 3.12. Let D be a i iterated tame division algebra of finite dimension over a henselian field F = Z(D) with index n.

i. For each a ∈ [1 + M_{D}]_{i} there is b ∈ [1 + M_{F}]_{i} such that ab ∈ D′.

ii. [1 + M_{D}]_{i} ⫋ [1 + M_{F}]_{i}D′.

Proof. i. Let a ∈ [1+M_{D}]_{i}. Then a is contained in a maximal subfield of D, say L. Therefore a ∈ [1+M_{L}]_{i}. By lemma 3 [13], we have N_{L/F} ([1+M_{L}]_{i}) = [1+M_{F}]_{i}. So Nrd_{D/F} (a) = N_{L/F} (a) ∈ [1+M_{F}]_{i}. Let t = Nrd_{D/F} (a). Using inductive argument for Hensel lemma, we will show that there exist c ∈ [1 + M_{F}]_{i} such that c^{n} = t. Let s ∈ 1+M_{F} = [1+M_{F}]_{1}. Applying Hensel lemma for f(x) = x^{n}−s gives c ∈ 1+M_{F} such that c^{n} = s. Now it is not hard to see that Φ_{1}([1 + M_{F}]_{i}) = [1 + M\overline{F}]_{i−1}. Therefore [1 + M_{F}]_{i}/[1 + M_{F}]_{1} ≃ [1 + M\overline{F}]_{i−1}. Now by induction, we conclude that [1 + M_{F}]_{i} is n-divisible. Therefore exist c ∈ [1 + M_{F}]_{i} such that c^{n} = t. Now Nrd_{D/F} (a) = c^{n}. So Nrd_{D/F} (ac^{−1}) = 1. Hence ac^{−1} ∈ D^{(1)} ∩ [1 + M_{D}]_{i}. Applying the Platonov’s generalized congruence theorem (cf. [13] and [4]) , we obtain ac^{−1} ∈ D′. Take b = c^{−1} and the proof is complete.

ii. Applying the first part of the lemma for i = 1, in each step of reduction we have, 1 + M_{D_{i}} ⊆ (1 + M_{K_{i}})D_{i}′ where K_{i} = Z(D_{i}). First we show that in each step of reduction, D_{i}′ ⊈ 1 + M_{D_{i}}. Consider the groups Δ = D_{i}^{∗}/1 + M_{D_{i}} and P (D_{i}) = (1 + M_{K_{i}})D_{i}′/(1 + M_{D_{i}}). One can easily observe that P (D_{i}) = Δ′ and as Theorem 2.11 of [2] shows, the center of Δ is K_{i}^{∗}(1 + M_{D_{i}})/(1 + M_{D_{i}}). We claim that Δ is not an abelian group, for otherwise U_{D_{i}} = U_{K_{i}}(1 + M_{D_{i}}) which implies that D_{i} is totally ramified. Thus \overline{D_{i}′} = μ_{e}(\overline{K_{i}}), where e = exp(Γ_{D_{i}}/Γ_{K_{i}}), (cf. the proof of Theorem 3.1 of [15]) which leads us to a contradiction. Therefore D_{i}′ is not in 1 + M_{D_{i}} and Δ is not abelian. But Φ^{−1}(1 + M_{D_{i}}) = [1 + M_{D}]_{i+1}. If D′ ⊆ [1 + M_{D}]_{i+1} then Φ(D′) ⊆ Φ([1 + M_{D}]_{i+1}) = 1 + M_{D_{i}}. But D_{i}′ ⊆ Φ(D′) so D_{i}′ ⊆ 1 + M_{D_{i}} a contradiction. □

1See Ershov’s comment in [3] on iterated valued field. Among other things, Considering iterated valued field, enables us to have more insight in each step of reduction.

Remark 3.13. In the proof of i. above, we could use Lemma 2.12 and avoid the Platonov congruence theorem.

Theorem 3.14. Let D be a tame division algebra of r iterated over henselian field F of index n. If there is an 0 ≤ ℓ ≤ r − 1 such that \overline{D}_{ℓ}/\overline{F}_{ℓ}\overline{D′_{ℓ}} = 1 then SK_{1}(D) ≃ μ_{n}(F)/Z(D′).

Proof. For any 0 ≤ k ≤ r − 1, consider the k + 1 − th reduction map

[U_{D}]_{k+1} ⟶^{Φ_{k+1}Φ_{k}···Φ_{1}} \overline{D_{k}}^{∗}.

Thanks to Lemma 3.12 i., we have:

\overline{D_{k}}^{∗} →^{≃} [U_{D}]_{k+1}/[1 + M_{D}]_{k+1} →^{nat.} [U_{D}]_{k+1}/[1 + M_{F}]_{k+1}D′ →^{nat.} [U_{D}]_{k+1}/[U_{F}]_{k+1}D′.

Therefore,

\overline{D_{k}}^{∗}/\overline{F_{k}}^{∗}\overline{D′_{k}} ⟶^{≃} [U_{D}]_{k+1}/[U_{F}]_{k+1}D′.

Hence if there is a ℓ such that \overline{D_{ℓ}}^{∗}/\overline{F_{ℓ}}^{∗}\overline{D′_{ℓ}}^{∗} = 1 then [U_{F}]_{ℓ+1}D′ = [U_{D}]_{ℓ+1}. By lemma 1 in [13] D^{(1)} ⊆ [U_{D}]_{ℓ+1} so D^{(1)} = μ_{n}(F)D′. Using the fact that μ_{n}(F) ∩ D′ = Z(D′) the theorem follows. □

Considering the fact that each henselian division algebra is 1 iterated division algebra, we recover Theorem 3.7 from the above theorem.

4. On The Unitary Setting

In this section we introduce the unitary version of the group G(D) and obtain the similar results in the unitary setting. Let D be a division ring with an involution τ over its center F with index n. Let S_{τ} (D) = {a ∈ D|a^{τ} = a} be the subspace of symmetric elements and Σ_{τ} (D) the subgroup of D^{∗} generated by nonzero symmetric elements. Here we concentrate on the involution of the first kind, i.e. Σ_{τ} (D)∩F^{∗} = F^{∗}.

Definition 4.1. Let D be a division ring with an involution τ. Then the group KU_{1}(D) = D^{∗}/Σ_{τ} (D)D′ is called unitary Whitehead group and the GU(D) = Σ_{τ} (D)D′/F^{∗}D′ the unitary version of G(D).

We will prove that there is a stability theorem for GU(D) similar to one in Corollary 3.6. The first part of the following theorem was first proved by Platonov and Yanchevskii [14].

Theorem 4.2. Let D be a finite dimensional tame and unramified division algebra with an involution of the first kind over a henselian field Z(D)=F. Then KU_{1}(\overline{D}) ⋍ KU_{1}(D) and GU(\overline{D}) ⋍ GU(D) .

Proof. Consider the following sequence:

\overline{D}^{∗} ⟶ U_{D}/1 + M_{D} ⟶ F^{∗}U_{D}/F^{∗}(1 + M_{D}) ⟶ D^{∗}/Σ_{τ} (D)D′.

Because the valuation is unramified, we have Σ\overline{τ} (\overline{D}) = \overline{Σ_{τ} (D) ∩ U_{D}} (See [14]), and \overline{D}′ = \overline{D′} (See Theorem 3.2 i.). Therefore we have the following isomorphism:

\overline{D}^{∗}/Σ\overline{τ} (\overline{D})\overline{D}′ ⟶^{⋍} D^{∗}/Στ(D)D′.

For the second part, consider the following commutative diagram with exact rows:

\xymatrix{1 \ar[r] & GU(\overline{D}) \ar[r] \ar[d] & G(\overline{D}) \ar[d]^{iso.} \ar[r] & KU_{1}(\overline{D}) \ar[d]^{iso.} \ar[r] & 1 \\ 1 \ar[r] & GU(D) \ar[r] & G(D) \ar[r] & KU_{1}(D) \ar[r] & 1.}

The two of the vertical arrows are isomorphisms, thanks to the first part of this theorem and Theorem 3.2 i.. Therefore the third one is also isomorphism which completes the proof. □

If D has an involution of the first kind, then D((x)) enjoys a natural involution which is induced by the one from D. Therefore if CharF ∤ i(D) then thanks to the above theorem, we have GU(D) ⋍ GU(D((x))) which is a stability for GU(D).

Acknowledgments. I thank Oleg Izhboldin for some helpful conversations. I am also grateful to J. -P. Tignol for his attention to my work.

References

1. S. Akbari, M. Mahdavi-Hezavehi, M. G. Mahmudi, Maximal subgroups of GL_{1}(D), J. Algebra 217 (1999), no. 2, 422-433.

2. M. Chacron, c-valuations and normal c-ordering, Canad. J. Math. 41 (1989), 14-67.

3. P. Draxl, Skew Field, London Math. Soc. Lecture Note Ser. Vol 81, Cambridge, Univ. Press. Cambridge, 1983.

4. Y. L. Ershov, Henselian valuation of division rings and the group SK_{1}(D), Math. USSR Sb. 45 (1983), 63-71.

5. R. Hazrat, M. Mahdavi-Hezavehi, B. Mirzaii, Reduced K-theory and the group G(D) = D^{∗}/F^{∗}D′, Algebraic K-theory and its Application, H. Bass, A.O.Kuku, and C. Pedrini.(Ed). World Sci. Publishing, River Edge, NJ. (1999), 403-409.

6. B. Jacob, A. R. Wadsworth, Division algebras over Henselian fields, J. Algebra 128 (1990), no. 1, 126-179.

7. T. Y. Lam, A first course in noncommutative rings, Springer-Verlag, New York, 1991.

8. D. W. Lewis, J. -P. Tignol, Square class groups and Witt rings of central simple algebras, J. Algebra 154 (1993), no. 2, 360-376.

9. V. A. Lipnitskii, On the Tannaka-Artin problem over special fields, Soviet Math. Dokl. 17 (1976), no. 3, 639-642.

10. M. Mahdavi-Hezavehi, On derived groups of division rings, Comm. Algebra 23 (1995), no. 3, 913-926.

11. R. S. Pierce, Associative Algebras, Springer-Verlag, New York, 1982.

12. V. P. Platonov, The Tannaka-Artin problem and reduced K-theory, Math USSR Izv. 10 (1976), 211-243.

13. V. P. Platonov, On reduced K-theory for n-fold Hensel fields, Soviet Math. Dokl. 20 (1979), no. 6, 1436-1439.

14. V. P. Platonov, V. I. Yanchevskii, Dieudonne’s conjecture on the structure of unitary groups over a division ring, and hermitian K-theory, Math. USSR Izv. 25 (1985), no. 3, 573-598.

15. J. -P. Tignol, A. R. Wadsworth, Totally ramified valuations on finite-dimensional division algebras, Tran. Amer. Math. Soc. 302 (1987), no. 1, 223-250.

Department of Mathematics, University of Bielefeld, P. O. Box 100131, 33501 Bielefeld, Germany.

   E-mail address: rhazrat@mathematik.uni-bielefeld.de