ON TRIVIALITY OF THE FUNCTOR Coker(K_{1}(F) → K_{1}(D)) FOR DIVISION ALGEBRAS

ROOZBEH HAZRAT

Abstract. Let D be a cyclic division algebra over its centre F of index n. Consider the group CK_{1}(D) = D^{∗}/F^{∗}D′ where D^{∗} is the group of invertible elements of D and D′ is its commutator subgroup. In this note we shall show that the group CK_{1}(D) is trivial if and only if D is an ordinary quaternion division algebra over a real Pythagorean field F. We construct a division algebra D and a division subalgebra A ⊂ D such that CK_{1}(A) ≅ CK_{1}(D). Using valuation theory, the group CK_{1}(D) is computed for some valued division algebras.

1. Introduction

Let A be an Azumaya algebra over a commutative ring R. Consider the functor CK_{1}(A) = Coker(K_{1}(R) →^{i} K_{1}(A)) where i is the inclusion map. If R is semilocal, since the stable rank of R and A are one, CK_{1}(A) = A^{∗}/R^{∗}A′ where A^{∗} and R^{∗} are the group of invertible elements of A and R respectively and A′ is the derived subgroup of A^{∗}. A study of this group in the case of central simple algebras is initiated in [6] and further in [5]. It has been established that despite of a “different nature” of this group from the reduced Whitehead group SK_{1}, these two groups have the same functorial properties. In [6] this functor is determined for totally ramified division algebras and in particular for any finite abelian group G, a division algebra D such that CK_{1}(D) = G × G is constructed. Further in [5], this functor is studied in more cases and examples of cyclic CK_{1} (even over non local fields) constructed. It was conjectured in [6] that if the functor CK_{1} is trivial then the index of the division algebra is 2. In this note we characterise quaternion division algebras D such that CK_{1}(D) is trivial. In fact triviality of CK_{1} characterises the ground field. We shall show that if CK_{1}(D), for a quaternion division algebra D, is trivial then D is an ordinary quaternion and the centre of D is a real Pythagorean field. In fact, this is the only class of cyclic division algebras such that CK_{1} is trivial (Theorem 2.3). We shall show that some of the notions from the theory of quadratic forms, like rigidity of an element, can be formulated as a property of the group CK_{1}. It seems that this group is highly sensitive to the arithmetic property of the ground field. For example we will observe that CK_{1}(\frac{x,x}{ℝ((x))}) ≅ ℤ_{2} whereas CK_{1}(\frac{x,x}{𝔽_{3}((x))}) ≅ ℤ_{2} ⊕ ℤ_{2}.

Date: January 25, 2003.

2. Triviality of CK_{1}

Let F be a field with characteristic not 2. Let (\frac{a,b}{F}) be a quaternion division algebra over F. The elements of D = (\frac{a,b}{F}) have the form c_{0} + c_{1}i + c_{2}j+c_{3}ij and Nrd_{D}(c_{0}+c_{1}i+c_{2}j+c_{3}ij) = c_{0}^{2}−ac_{1}^{2}−bc_{2}^{2}+abc_{3}^{2} where Nrd_{D} is the reduced norm map for D. Note that CK_{1}(D) = D^{∗}/F^{∗}D′ where D^{∗} and F^{∗} are the group of invertible elements of D and F respectively and D′ is the commutator subgroup of D^{∗}. If CK_{1}(D) = 1 then Nrd_{D}(D) = F^{∗}^{2}. Thus Nrd_{D}(i) = −a ∈ F^{2} and Nrd_{D}(j) = −b ∈ F^{2}. Therefore (\frac{a,b}{F}) ≅ (\frac{−1,−1}{F}).

Our first theorem shows that triviality of (\frac{a,b}{F}) forces that F be a real Pythagorean field. In fact this property characterises this field. Recall that F is a real Pythagorean if −1 ∉ F^{∗2} and sum of any two square elements is a square in F. It follows immediately that F is an ordered field. F is called Euclidean if F^{∗2} is an ordering of F

Theorem 2.1. Let F be a field of characteristic not 2. Then the following are equivalent.

1) (\frac{−1,−1}{F}) is a division ring and CK_{1}(\frac{−1,−1}{F}) is trivial.

2) F is the real Pythagorean field.

3) (\frac{−1,−1}{F}) is a division ring and every maximal subfield of (\frac{−1,−1}{F}) is F -isomorphic to F (\sqrt{−1}).

Proof. We shall show that 1 and 2 are equivalent. The equivalency of 2 and 3 are known (see [2]).

1) ⇒ 2) Since D = (\frac{−1,−1}{F}) is a division ring, −1 ∉ F^{∗}^{2}. By induction, if −1 = f_{1}^{2} + ⋯ + f_{n}^{2}, then considering f_{1}^{2} + f_{2}^{2} = Nrd_{D}(f_{1} + f_{2}i) = f^{2}, a contradiction follows. This shows that −1 is not sum of squares plus sum of any two squares is a square

2) ⇒ 1) Suppose F is a real Pythagorean. It is easy to see that D = (\frac{−1,−1}{F}) is a division ring. Now for any x ∈ D^{∗}, Nrd_{D}(x) is a sum of four squares, thus Nrd_{D}(D^{∗}) = F^{∗}^{2}. Since SK_{1}(D) is trivial, this forces that CK_{1} be trivial. □

Over Euclidean fields, the only quaternion division algebra is the ordinary quaternion division algebra, and from the above Theorem it follows that its CK_{1} is trivial.

We shall show that an ordinary quaternion division algebra over a real Pythagorean field is in fact the only type of cyclic division algebras such that its CK_{1} is trivial. Before we state the theorem, note that there are examples of infinite dimensional division algebras D such that D^{∗} coincide with D′ [7]. In the finite dimensional case, it is not hard to see that D^{∗} ≠ D′, infact K_{1}(D) = D^{∗}/D′ is not torsion. In the case of algebraic division algebras, everything remains as a mystery.

We need the following theorem which says that the functor CK_{1} has a decomposition property similar to one for the reduced Whitehead group.

Theorem 2.2. [5] Let A and B be division algebras with centre F such that (i(A), i(B)) = 1. Then CK_{1}(A ⊗_{F} B) ≅ CK_{1}(A) × CK_{1}(B).

Theorem 2.3. Let D be a cyclic division algebra finite dimension over its centre F. Then CK_{1}(D) = 1 if and only if D is an ordinary quaternion division algebra over a real Pythagorean field F.

Proof. The only if part is already proved in Theorem 2.1. For the if part, we shall show that the index of D is 2. Then the claim follows again from Theorem 2.1. First suppose that i(D) = n is odd. Let α ∈ D such that F (α) is a maximal cyclic subfield of D with a mininal polynomial of the form x^{n} − f where f ∈ F. Since D^{∗} = F^{∗}D′, we can choose α ∈ D′. Now α^{n} = N_{F (α)/F} (α) = Nrd_{D}(α) = 1. From this it follows that the degree of the minimal polynomial of α is less than n which can not happen. Since CK_{1}(D) has a decomposition property (Theorem 2.2), the only case which remains, is n = 2^{k}, for some integer k. We shall show that k = 1. Again consider α as above. This time −α^{2^{k}} = N_{F (α)/F} (α) = Nrd_{D}(α) = 1. On the other hand since 1 + N_{F (α)/F} (α) = N_{F (α)/F} (α + 1) ∈ F^{∗2^{k}}, it follows that \sqrt{2} ∈ F. Thus if k > 1 then α^{2^{k}} + 1 = (α^{2^{k−1}} + 1)^{2} − 2α^{2^{k−1}} can be decomposed further which leads to a contradiction that the minimal polynomial of α has degree less than n. Therefore n = 2 and the claim follows from Theorem 2.1 □

Note that in the above theorem we could replace CK_{1} with a smaller group D^{∗}/F^{∗}D^{(1)}, where D^{(1)} is set of elements with reduced norm 1.

The above theorem states that if in the case of division algebra D of odd prime index, CK_{1}(D) happens to be trivial, then this forces the well-known conjecture that any division algebra of prime index is cyclic would be false!

It is a celeberated result of Wedderburn that a division algebra of index 3 is cyclic. This together with Theorem 2.2 and Theorem 2.3 implies that

Theorem 2.4. If D be a division algebra of index 3n where 3 ∤ n, then CK_{1}(D) ≠ 1.

Remark 2.5. Let D be a finite dimensional division algebra with centre F, of index n. Consider the sequence

(1)  K_{1}(D) ⟶^{Nrd} K_{1}(F) ⟶^{i} K_{1}(D)

Where Nrd_{D} is the reduced norm map and i is the inclusion map. One can see that the composition i Nrd_{D/F} = η_{n} where η_{n}(a) = a^{n}. (See for example the proof of Lemma 4, p. 157 [1]). From this the formula a^{n} = Nrd_{D/F} (a)c_{a} where a ∈ D^{∗} and c_{a} ∈ D′ follows. In particular this implies that the exponent of the abelian group CK_{1}(D) divides n, the index of D.

It is not known if exp(CK_{1}(D)) < i(D) what would be imposed on the algebraic structure of D. We mention that if D is a totally ramified division algebra then exp(CK_{1}(D)) = i(D) if and only if D is cyclic [5]. In fact from the above theorem it follows that if D is a cyclic division algebra of index p, an odd prime, then the exponent of CK_{1}(D) is exactly p. The converse is not true, as the following example shows. This is an example of a cyclic decomposable F -division algebra D of index 2p, p an odd prime, such that it has a proper F -division subalgebra A ⊂ D, where CK_{1}(A) ≅ CK_{1}(D). This in particular shows that the exponent of CK_{1}(D) is less than the index of D, but D is cyclic, thats is, the phenomena of cyclicity of D and exp(CK_{1}(D)) does not follow the same pattern as in the case of totally ramified. For this we need the Fein-Schacher-Wadsworth example of a division algebra of index 2p over a Pythagorean field F [3]. We briefly recall the construction. Let p be an odd prime and K/F be a pair of real Pythagorean fields such that K is cyclic over F with the generating automorphism σ. Now consider

D = (\frac{−1, −1}{F ((x))}) ⊗_{F ((x))} (K((x))/F ((x)), σ, x).

K((x))/F ((x)) is a cyclic extension where K((x)) and F ((x)) are Laurant power series fields of K and F respectively. D is a division algebra of index 2p. Since F is real Pythagorean, so does F ((x)). Now by Theorem 2.2 the primary decomposition of the division algebra D induces a corresponding decomposition for CK_{1}(D). Thus CK_{1}(D) ≅ CK_{1}(\frac{−1,−1}{F ((x))}) × CK_{1}(A) where A = (K((x))/F ((x)), σ, x). By Theorem 2.1, CK_{1}(\frac{−1,−1}{F ((x))}) = 1, thus CK_{1}(D) ≅ CK_{1}(A).

Question. If exp(CK_{1}(D)) is less than the index i(D), what can be said about the algebraic structure of D.

Recall that a ∈ F is called rigid if a ∉ ±F^{∗2} and F^{∗2}+aF^{∗2} = F^{∗2}∪aF^{∗2}. The rigidity concept plays a role in the study of the extension of Pythagorean fields. In particular if K = F (\sqrt{a}) is a quadratic extension of F, then K is real Pythagorean if and only if F is real Pythagorean and a is rigid [8].

Theorem 2.6. Let F be a real Pythagorean field. Let a ∉ ±F^{∗2}. Then a is rigid if and only if CK_{1}(\frac{−1,−a}{F}) = ℤ_{2}.

Proof. Suppose a is rigid. Since −a ∉ F^{∗2}, then D = (\frac{−1,−a}{F}) is a division ring. Since F is Pythagorean, Nrd_{D}(D^{∗}) = c_{0}^{2}+ac_{1}^{2}, where c_{0}, c_{1} ∈ F^{∗}. From rigidity of a it follows that Nrd_{D}(D^{∗}) = F^{∗2} ∪ aF^{∗2}. Because a ∉ F^{∗2}, then F^{∗2} and F^{∗2}a are two distinct elements in the group Nrd_{D}(D^{∗})/F^{∗2}. Since SK_{1}(D) is trivial, CK_{1}(D) ≅ Nrd_{D}(D^{∗})/F^{∗2} ≅ ℤ_{2}. The if part follows the same way. □

Remark 2.7. Let F be a real Pythagorean field and a ∈ F^{∗} such that −a ∉ F^{∗2}. Then CK_{1}(\frac{−1,−a}{F}) ≅ L^{∗}/F^{∗}L^{∗2} where L = F (\sqrt{−a}). For consider the square class exact sequence

1 ⟶ {F^{∗2}, F^{∗2}a} ⟶ F^{∗}/F^{∗2} ⟶ L^{∗}/L^{∗2} ⟶^{N} N_{L/F} (L^{∗})/F^{∗2} ⟶ 1.

In this setting, it is easy to see that N_{L/F} (L^{∗}) = Nrd_{D/F} (D^{∗}) thus the claim.

3. CK_{1} for valued quaternion division algebras

In this section we study the valued quaternion division algebras. Valuation theory for division algebras is the main tool for computation of CK_{1}. Recall that if D is equipped with a valuation, then D̄ and F̄ denote the residue division algebra and the residue field of D and F respectively. Γ_{D} and Γ_{F} are the value groups of the valuation.

A tame valued quaternion division algebra could be unramified, semiramified or totally ramified. This follows from the fact that D is defectless. Recall that D is tame if char(F̄) does not divide the index of D. Thus D is defectless, namely [D̄ : F̄][Γ_{D} : Γ_{F}] = [D : F]. D is called unramified if [D̄ : F̄] = [D : F] and totally ramified if [Γ_{D} : Γ_{F}] = [D : F]. D is called semiramified if D̄ is a field and [D̄ : F̄] = [Γ_{D} : Γ_{F}] = i(D). For an excellent survey of the valuation theory of division algebras see [10].

We will see that there are quaternion division algebras of any type mentioned above.

• D = (\frac{−1,−1}{ℝ((x))}) is unramified, and CK_{1}(D) = 1 (see Theorem 2.1).

• D = (\frac{−1,x}{ℝ((x))}) is semiramified, and CK_{1}(D) = ℤ_{2} (see Theorem 3.2).

• D = (\frac{x,y}{ℝ((x,y))}) is totally ramified, and CK_{1}(D) = ℤ_{2} ⊕ ℤ_{2} (see Remark 3.3).

Although all the examples that we are dealing with here have finite CK_{1}, let us mention that CK_{1} of a quaternion division algebra could be infinite. It is known that if p is a rational prime such that p ≡ 3(mod4) then (\frac{−1,p}{ℚ}) is a division algebra. It is not hard to see that CK_{1}(\frac{−1,p}{ℚ}) = ⨁ ℤ_{2} is infinite.

In order to compute the functor CK_{1} in the case of a valued division algebra, we need the following Theorem.

Theorem 3.1. [5] Let D be a Henselian division algebra tame over its centre with index n. Then

1) If D is unramified, then CK_{1}(D) ≅ CK_{1}(D̄).

2) If D is totally ramified, then CK_{1}(D) = Γ_{D}/Γ_{F} .

3) If D is semiramified, and D̄ is cyclic over F̄, then there is an exact sequence

1 ⟶ N_{D̄/F̄}(D̄^{∗})/F̄^{∗n} ⟶ CK_{1}(D) ⟶ Γ_{D}/Γ_{F} ⟶ 1.

It is known that if Z(D̄) is separable over F̄, then the fundamental homomorphism Φ : Γ_{D}/Γ_{F} → Gal(Z(D̄)/F̄) is epimorphism [10]. Since in the above theorem, part 3) D is semiramified over F, Φ is an isomorphism. On the other hand D̄ is cyclic over F̄, Thus Γ_{D}/Γ_{F} ≅ ℤ_{n}. So we can rewrite the above exact sequence as

3′) If D is semiramified, and D̄ is cyclic over F̄, then there is an exact sequence

1 ⟶ N_{D̄/F̄}(D̄^{∗})/F̄^{∗n} ⟶ CK_{1}(D) ⟶ ℤ_{n} ⟶ 1.

Theorem 3.1 is our main tool to compute the functor CK_{1} for a valued quaternion division algebra.

Let F be a real Pythagorean field. Let N be the set of non rigid elements of F. Recall that if N = ±F^{∗2} then F is called super-Pythagorean [8]. The following theorem is a demonstration of how valuation theory enables us to compute CK_{1}.

Theorem 3.2. Let F be a real Pythagorean field and N the set of non rigid elements of F. Then for any t ∈ F ((x))\ ± F ((x))^{2} ∪ N, CK_{1}(\frac{−1,−t}{F ((x))}) = ℤ_{2}.

Proof. The valuation of F ((x)) extends to D = (\frac{−1,−1}{F ((x))}) as follows; v(x) = \frac{1}{2}v(Nrd_{D}(x)) where v denotes the valuation map for both F and D. Suppose first that t ∈ F. It is easy to see that

(\frac{−1, −t}{F ((x))}) ≅ (\frac{−1, −t}{F}) ⊗_{F} F ((x))

is unramified. Thus by 3.1 1), CK_{1}(\frac{−1,−t}{F ((x))}) ≅ CK_{1}(\frac{−1,−t}{F}). Since t is rigid in F by Theorem 2.6, CK_{1}(\frac{−1,−t}{F}) ≅ ℤ_{2} and we are done.

Now if t ∈ F ((x))\F, then it is not hard to see that D is semiramified, D̄ = F (\sqrt{−}1) and ND̄(F (\sqrt{−}1)) = F^{∗2}. Now from Theorem 3.1 3′) it follows that CK_{1}(D) = ℤ_{2}. □

Theorem above is another way to observe that if F is super-Pythagorean then so is F ((x)).

Remark 3.3. Consider the division algebra D = (\frac{x,y}{ℝ((x,y))}). Since Nrd_{D}(i) = −x and Nrd_{D}(j) = −y, it can be seen that the extension of the valuation from ℝ((x, y)) to D is totally ramified. From Theorem 3.1 2) follows that CK_{1}(D) = ℤ_{2} ⊕ ℤ_{2}.

3.1. CK_{1} for local quaternion division algebras. Let F be a local field and D a quaternion division algebra over D. We shall use Theorem 3.1 to show that CK_{1}(D) ≅ ℤ_{2} ⊕ ℤ_{2}. Recall that characteristic of F is not 2. We further assume that char(F̄) is not 2.

Theorem 3.4. Let D be a quaternion division algebra over a local field F. Then CK_{1}(D) ≅ ℤ_{2} ⊕ ℤ_{2}.

Proof. The valuation of F extends to D. Furthermore one can see that with this valuation D is semiramified over F and D̄ is cyclic over F̄. Since D̄ is finite, ND̄(D̄^{∗}) = F̄^{∗} and the exact sequence of Theorem 3.1 3′ has the look

1 ⟶ F̄^{∗}/F̄^{2∗} ⟶ CK_{1}(D) ⟶ ℤ_{2} ⟶ 1.

Since char(F̄) is not 2, the quotient F̄^{∗}/F̄^{2∗} is not trivial. Because F̄^{∗} is a cyclic group therefore F̄^{∗}/F̄^{2∗} ≅ ℤ_{2}. Since exp(CK_{1}(D)) divides 2, from the above exact sequence it follows that CK_{1}(D) ≅ ℤ_{2} ⊕ ℤ_{2}. □

The above theorem shows that CK_{1}(\frac{−1,x}{𝔽_{3}((x))}) = ℤ_{2} ⊕ ℤ_{2} as mentioned before.

We shall mention that this result was not unknown. In fact, since SK_{1}(D) is trivial and the reduced norm is surjective for division algebras over local fields, CK_{1}(D) ≅ F^{∗}/F^{∗2}. From the theory of quadratic forms for local fields we can thus obtain the same result. Also if charF̄ = 2, it is known that F^{∗}/F^{∗2} ≅ ⨁_{2^{n+1}} ℤ_{2} (See P.217 [9])

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Centre for Mathematics and its Applications, John Dedman Building (# 27), Australian National University, Canberra ACT 0200, Australia

  E-mail address: hazrat@maths.anu.edu.au