WEDDERBURN’S FACTORIZATION THEOREM

APPLICATION TO REDUCED K-THEORY

R. Hazrat

Abstract. This article provides a short and elementary proof of the key theorem of reduced K-theory, namely Platonov’s Congruence theorem. Our proof is based on Wedderburn’s factorization theorem.

Let D be a division algebra with center F. If a ∈ D is algebraic over F of degree m, then by Wedderburn’s factorization theorem, one can find m conjugates of a such that the sum and the product of them are in F. This observation has been used in many different circumstances to give a short proof of known theorems of central simple algebras. (See [8] for a list of these theorems.) Here we will use this fact to prove Platonov’s congruence theorem.

The non-triviality of the reduced Whitehead group SK_{1}(D) was first shown by V. P. Platonov who developed a so-called reduced K-theory to compute SK_{1}(D) for certain division algebras. The key step in his theory is the “congruence theorem” which is used to connect SK_{1}(\overline{D}) where \overline{D} is a residue division algebra of D to SK_{1}(D). This in effect enables one to compute the group SK_{1}(D) for certain division algebras. (See [5] and [6].)

Before we describe the congruence theorem, we employ Wedderburn’s factorization theorem to obtain a result regarding normal subgroups of division algebras.

Suppose that D has index n. Let N be a normal subgroup of the group of units D^{∗} of D. Let a ∈ N with the minimal polynomial f(x) ∈ F [x] of degree m. Then from the theory of central simple algebras we have the following equality,

(1)  f(x)^{n/m} = x^{n} − T rd_{D/F} (a)x^{n−1} + ⋯ + (−1)^{n}Nrd_{D/F} (a),

where Nrd_{D/F} : D^{∗} → F^{∗} is the reduced norm, T rd_{D/F} : D → F is the reduced trace and the right hand side of the equality is the reduced characteristic polynomial of a. (See [7], §9.)

Typeset by AMS-TEX

Key words and phrases. Division algebra, Reduced K-theory, Congruence theorem, Reduced Whitehead group.

Using Wedderburn’s factorization theorem for the minimal polynomial f(x) of a, one obtains f(x) = (x − d_{1}ad_{1}^{−1}) ⋯ (x − d_{m}ad_{m}^{−1}) for certain d_{i} ∈ D. Now from the equality (1), it follows that Nrd_{D/F} (a) is the product of n conjugates of a. Since N is a normal subgroup of D^{∗}, it follows that Nrd_{D/F} (a) ∈ N. Therefore Nrd|_{N} : N ⟶ Z(N) is well defined, where Z(N) = F^{∗} ∩ N is the center of the group N.

Before stating the main Lemma, we fix some notation.

Let μ_{n}(F) denote the group of n − th roots of unity in F and let Z(D′) denote the center of the commutator subgroup D′ of D^{∗}. Let D^{(1)} stand for the kernel of the reduced norm. Observe that μ_{n}(F) = F^{∗} ∩ D^{(1)} and Z(D′) = F^{∗} ∩ D′. If G is a group, denote by G^{n} the subgroup of G generated by all n − th powers of elements of G. If H and K are subgroups of G, denote by [H, K] the subgroup of G generated by all mixed-commutators [h, k] = hkh^{−1}k^{−1}, where h ∈ H and k ∈ K.

We are now in a position to state our main lemma which is interesting in its own right.

Lemma 1. Let D be a division algebra with center F, of index n. Let N be a normal subgroup of D^{∗}. Then N^{n} ⊆ (F^{∗} ∩ N)[D^{∗}, N].

Proof. Let a ∈ N. As stated above, using Wedderburn’s factorization theorem, Nrd_{D/F} (a) = d_{1}ad_{1}^{−1} ⋯ d_{n}ad_{n}^{−1}. But

d_{1}ad_{1}^{−1} ⋯ d_{n}ad_{n}^{−1} = [d_{1}, a]a[d_{2}, a]a ⋯ [d_{n}, a]a = a^{n}d_{a}

for some d_{a} ∈ [D^{∗}, N]. This implies that a^{n} = Nrd_{D/F} (a)d_{a}^{−1}. Therefore N^{n} ⊆ (F^{∗} ∩ N)[D^{∗}, N]. □

Let N = D^{∗}. Then by above Lemma, for any x ∈ D^{∗}, x^{n} = Nrd_{D/F} (x)d_{x} where d_{x} ∈ D′. This shows that the group G(D) = D^{∗}/F^{∗}D′ is a torsion group of bounded exponent n. Some algebraic properties of this group are studied in [4].

In order to describe Platonov’s congruence theorem, we need to recall some concepts from valued division algebras. Let D be a finite dimensional division algebra with center a Henselian field F. Recall that a valuation v on a field F is called Henselian if and only if v has a unique extension to each field algebraic over F. Therefore v has a unique extension denoted also by v to D ([11]). Denote by V_{D}, V_{F} the valuation rings of v on D and F respectively and let M_{D}, M_{F} denote their maximal ideals and \overline{D}, \overline{F} their residue division algebra and residue field, respectively. We let Γ_{D}, Γ_{F} denote the value groups of v on D and F, respectively and U_{D}, U_{F} the groups of units of V_{D}, V_{F} respectively. Furthermore, we assume that D is a tame division algebra, i.e., Char\overline{F} does not divide i(D), the index of D.

Platonov’s congruence theorem asserts that if D is a tame division algebra over a Henselian field F then (1 + M_{D}) ∩ D^{(1)} ⊆ D′. This is the crucial theorem of reduced K-theory which is proved in [5] (Note that [5] provides a lengthy and complicated proof for the special case of a complete discrete valuation of rank 1, and [3] notes that the same proof works for general case of tame Henselian valued division algebras). Here we give a short and elementary proof of this fact.

Theorem 2 (Congruence Theorem). Let D be a tame division algebra over a Henselian field F = Z(D), of index n. Then (1 + M_{D}) ∩ D^{(1)} = [D^{∗}, 1 + M_{D}].

Proof. First we show that (1+M_{F})∩D^{(1)} = 1. Let 1+f ∈ 1+M_{F}. If 1+f ∈ D^{(1)}, then (1 + f)^{n} = 1. But v((1 + f)^{n} − 1) = v(f). This shows that f = 0 and so our claim. Now take N = 1 + M_{D}. By Lemma 1,

(1 + M_{D})^{n} ⊆ ((1 + M_{D}) ∩ F^{∗})[D^{∗}, (1 + M_{D})].

Since the valuation is tame and Henselian, Hensel’s lemma shows that (1+M_{D})^{n} = 1 + M_{D}. Therefore 1 + M_{D} = (1 + M_{F})[D^{∗}, (1 + M_{D})]. Now using the fact that

(1 + M_{F}) ∩ D^{(1)} = 1, the theorem follows. □

Remark. There is an elegant proof of the congruence theorem by A. Suslin in [9], in the case of a discrete valuation of rank 1. This proof uses substantial results from valuation theory and the fact that the group SK_{1}(D) is torsion of bounded exponent n = i(D). Using results of Ershov in [3], Suslin’s proof can be written for arbitrary tame Henselian division algebras.

Having the congruence theorem, it is easy to see, in the case of discrete valuation of rank 1, that the sequence,

SK_{1}(\overline{D}) → SK_{1}(D) → L_{1}/L_{σ^{−}1} → 1,

is exact where L = Nrd(\overline{D}), L_{1} = L ∩ N_{Z(\overline{D})/\overline{F}}^{−1} (1) and L_{σ^{−}1} = the image of L

under the homomorphism a ↦ σ(a)a^{−1}, where ⟨σ⟩ = Gal(Z(\overline{D})/\overline{F}). This leads to computations of SK_{1}(D) for certain division algebras. (See [5], [6] and [9].)

Another look at the proof of Theorem 2 shows that 1 + M_{D} ⊆ (1 + M_{F})D′ and therefore 1 + M_{D} ⊆ U_{F} D′. Put G(\overline{D}) = \overline{D}^{∗}/\overline{F}^{∗}\overline{D′}. In many applications, it is easy to obtain information about the residue data of division algebras. The following theorem gives an explicit formula for the group SK_{1}(D) when the group G(\overline{D}) is trivial.

Theorem 3. Let D be a tame division algebra over a Henselian field F = Z(D), of index n. If G(\overline{D}) = 1 then SK_{1}(D) = μ_{n}(F)/Z(D′).

Proof. The reduction map U_{D} ⟶ \overline{D}^{∗} induces an isomorphism \overline{D}^{∗} ⟶ U_{D}/1+M_{D}, \overline{a} ↦ (1 + M_{D})a. Since 1 + M_{D} ⊆ U_{F} D′, it follows that

\overline{D}^{∗}/\overline{F}^{∗}\overline{D′} ⟶^{≃} U_{D}/U_{F} D′.

Now if G(\overline{D}) = \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. □

Note that Hensel’s lemma implies that μ_{n}(F) ≃ μ_{n}(\overline{F}). In particular if D is a totally ramified division algebra, i.e. \overline{D} = \overline{F}, then G(\overline{D}) = 1.

Example 4. Let ℂ be the field of complex numbers and r be a nonnegative integer. Let D_{1} = ℂ((x_{1})) and define σ_{1} : D_{1} → D_{1} by the rule σ_{1}(x_{1}) = −x_{1}. Now let D_{2} = D_{1}((x_{2}, σ_{1})) and set D_{3} = D_{2}((x_{3})). Again define σ_{3} : D_{3} → D_{3} by σ_{3}(x_{3}) = −x_{3}. In general, if i is even, set D_{i+1} = D_{i}((x_{i+1})) and if i is odd define σ_{i} : D_{i} → D_{i} by σ_{i}(x_{i}) = −x_{i} and D_{i+1} = D_{i}((x_{i+1}, σ_{i})). By Hilbert’s construction (see [1], §1 and §24), D = D_{2r} = ℂ((x_{1}, ⋯, x_{2r}, σ_{1}, ⋯, σ_{2r−1})) is a division algebra with center F = ℂ((x_{1}^{2}, x_{2}^{2}, ⋯, x_{2r−1}^{2}, x_{2r}^{2})) and n = i(D) = 2^{r}. Finally define v : D^{∗} → Γ_{D} = Z^{2r} by the rule v(∑ c_{i}x_{1}^{i_{1}} ⋯ x_{2r}^{i_{2r}}) = (i_{1}, ⋯, i_{2r}) where i_{1}, ⋯, i_{2r} are the smallest powers of the x_{i}’s in the lexicographic order. It can be observed that v is a tame valuation and \overline{D} = ℂ and \overline{F} = ℂ. Therefore G(\overline{D}) = 1. Theorem 3 implies that SK_{1}(D) = μ_{n}(F)/Z(D′). From the multiplication rule in D, it follows that

D′ ⊆ { ± 1 + ∑_{i>0} c_{i}x_{1}^{i_{1}} ⋯ x_{2r}^{i_{2r}}}.

Since Z(D′) ⊆ μ_{n}(F), it follows that Z(D′) = \{1, −1\}. But μ_{n}(F) = μ_{n}(\overline{F}) = ℤ_{2^{r}}, hence SK_{1}(D) = ℤ_{2^{r−1}}.

In [4], as another application of Lemma 1, we obtain theorems of reduced Ktheory which previously required heavy machinery, as simple consequence of this approach.

Acknowledgments. I wish to thank Anthony Bak who read the note and made numerous corrections.

References

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

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