Hilbert 90 for K_{3} for degree-two extensions

by Markus Rost

Regensburg, Mai 1986

In this paper^{∗} we consider Milnor K-theory of fields [Mi]. Let F be a field of characteristic different from 2 and let L be an extension of degree two with generator σ of Gal(L|F). The purpose of this paper is to prove

Theorem A

The sequence

K_{3}L ⟶^{1−σ} K_{3}L ⟶^{N_{L|F}} K_{3}F

is exact. It is a consequence of this theorem that the Galois symbol K_{3}F/2 → H^{3}(F, ZZ/2) is an isomorphism. This will be considered elsewhere. As in the proof of Hilbert 90 for K_{2} ([Me], [MS]), Theorem A follows from the exactness of a certain part of the localization sequence of a Severi-Brauer variety with respect to Milnor K-Theory. Let D be a quaternion algebra over F and let X be the (one-dimensional) Severi-Brauer variety associated to D. The basic results needed in the proof of Theorem A are the injectivity of the reduced norm Nrd : K_{2}D → K_{2}F ([R1]) and

Theorem B

The sequence

K_{3}F (X) → ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) →^{θ} K_{2}D → 0

is exact. (Here v runs through the closed points of X. The homomorphism θ is induced by the natural map θ_{K} : K_{2}K → K_{2}D for a splitting field K of D, finite over F; see [MS; § 1]).

Note that the corresponding statement for the K-Theory of Quillen follows from the computations K_{2}(X) = K_{2}D ⊕ K_{2}F and H^{0}(X, K_{2}) = K_{2}F ([MS]).

The philosophy of our proof is that Theorem A together with the injectivity of the Galois symbol is equivalent to Theorem B together with the injectivity of Nrd : K_{2}D → K_{2}F. Using Hilbert 90 for K_{2} it is not difficult to see that Theorem A holds for the universal Kummer extension of degree two of a pure transcendental extension F of a prime field (§ 1). We use this to show that Theorem B holds for a generic quaternion algebra D over F (§ 3). To prove Theorem B in general we make use of Rehmann’s description of K_{2}D in terms of generators and relations and do some specialization arguments using the results of § 3 (Since I hope that it is possible to shorten some arguments, the proof of Theorem B given here is rather sketchy). Finally, in § 5 we show that Theorem B and the injectivity of K_{2}D → K_{2}F imply Theorem A.

∗ This is a TEXed version (Sept. 1996) of the original preprint.

§ 1 Hilbert 90 in a special case

The reader is assumed to be familiar with Milnor K-Theory of fields as defined in [Mi]. For the product K_{1}K ⊗ K_{n}K → K_{n+1}K we use the notation (x, u) ↦ l(x)u = {x, u}. For the rational function field K(t) in one variable one has the exact sequence

(1.0.1)  0 → K_{n}K → K_{n}K(t) ⟶^{d} ⨁_{P^{∈}P_{K}} K_{n−1}K_{P} → 0

Here P runs over the set P_{K} of normed irreducible polynomials in t and K_{P} = K[t]/(P) (see [Mi; Theorem 2.3]). If H|K is finite extension (H may be a field or a direct sum of field extensions H_{i}|K; in the latter case K_{n}H = ⊕_{i}K_{n}H_{i} by definition), there is a restriction res_{H|K} : K_{n}K → K_{n}H and a corestriction or norm homomorphism cor_{H|K} = N_{H|K} : K_{n}H → K_{n}K (See [BT] for definition and [K] for uniqueness of the norm). One has the formulas

cor_{H|K} ∘ res_{H|K}= [H : K]
cor_{H|K}({u, v}) = {cor_{H|K}(u), v} for u ^{∈} K_{n}H, v ^{∈} K_{m}K
res_{H|K} ∘ cor_{H|K}= ∑_{σ^{∈}Gal(H|K)} if H|K is normal.

If P is irreducible over K and if Q_{1}, …, Q_{1} are the irreducible factors of P over H, one also has homomorphisms

�  ^{cor^{H|K}} �  _{res_{H|K}} K_{n}K_{P} ⨁_{i} K_{n}H_{Q_{i}}

satisfying the above formulas. They fit into a commutative diagram

�  ^{d} �_{res_{H(t)|K(t)}} �_{N_{H(t)|K(t)}} �  _{res_{H|K}} �  _{cor_{H|K}} �  ⨁_{P^{∈}P_{K}} K_{n−1}K_{P} K_{n}K(t)  ⨁_{Q^{∈}P_{H}} K_{n−1}H_{Q} K_{n}H(t)

(see [K]). Using this we construct an explicit section to d.

Lemma 1.1. Let u_{P} ^{∈} K_{n−1}K_{P} and let t_{P} be the residue class of t in K_{P}. Then

u_{P} = dN_{K_{P} (t)|K(t)}(t − t_{P} , u_{P})

Proof In order not to confuse the roles of K_{P} as residue class field and as base extension, let φ : K_{P} → H be an isomorphism over K. Then the statement reads as

u_{P} = dN_{H(t)|K(t)}({t − φ(t_{P}), φ(u_{P})})

Note that the composition K_{n−1}H → K_{n−1}H_{(t−φ(t_{P}))} \underrightarrow{cor_{H|K}} K_{n−1}K_{P} is the isomorphism induced by the inverse of φ. Hence

dN_{H(t)|K(t)}({t − φ(t_{P}), φ(u_{P})}) =cor_{H|K} ∘ d({t − φ(t_{P}), φ(u_{P})})
= cor_{H|K}(φ(u_{P}) mod (t − φ(t_{P}))) = u_{P} .

qed.

Now let F_{0} ≠ ZZ/2 be a prime field, let F = F_{0}(a_{1}, …, a_{n}, a) be pure transcendental over F_{0} and let L = F (\sqrt{a}). The generator of Gal(L|F) is denoted by σ.

Proposition 1.2. The following sequences are exact

(1.2.1)  K_{3}L ⟶^{1−σ} K_{3}L ⟶^{N_{L|F}} K_{3}F
(1.2.2)  K_{2}F/2 ⟶^{l(a)} K_{3}F/2 \underrightarrow{res_{L|F}} K_{3}L/2
(1.2.3)  K_{2}F ⟶^{l(−1)} K_{3}F ⟶^{2} K_{3}F
(1.2.4)  K_{3}F ⊕ U_{F} ⟶^{(res_{L|F} ,l(\sqrt{a}))} K_{3}L ⟶^{1−σ} K_{3}L

where U_{F} = Ker(K_{2}F ⟶^{l(−1)} K_{3}F)

Proof (Sketch) (1.2.1): One uses Hilbert 90 for K_{2} and (1.0.1) with respect to the variables a_{i} to reduce to the case n = 0. Then, if α ^{∈} KerN_{F_{0}(\sqrt{a})|F_{0}(a)}, one uses again (1.0.1) for F_{0}(\sqrt{a})|F_{0} and F_{0}(a)|F_{0} to show that there exist β ^{∈} K_{3}F_{0}(\sqrt{a}) such that α − (1 − σ)(β) ^{∈} K_{3}F_{0}. However K_{3}F_{0} = 0 if F_{0} is finite and K_{3}Q = ZZ/2 generated by {−1, −1, −1}, see [Mi]. In the latter case one has \{−1, −1, −1\} = (1 − σ)(\{\sqrt{a}, −1, −1\}).

(1.2.2) follows from the fact that the Galois symbol K_{3}K/2 → H^{3}(K, ZZ/2) is an isomorphism for K = F, L ([Mi; Lemma 6.2; Theorem 6.3]) and the corresponding exact sequence for Galois cohomology.

(1.2.3) can be derived from (1.2.1) in the same way as the corresponding result for K_{2} (see [MS, Lemma 10.4] or [S; Lemma 3]); one also uses Lemma 1.1.

(1.2.4) will be proved in detail. Let α ^{∈} K_{3}L such that σ(α) = α. Since res_{L|F} ∘ N_{L|F} (α) = (1 + σ)(α) ^{∈} 2K_{3}L, (1.2.2) implies that

N_{L|F} (α) = {a, β} + 2γ

for some β ^{∈} K_{2}F, γ ^{∈} K_{3}F. Replacing α by α − res_{L|F} (γ) we may assume γ = 0. Put α′ = α − \{\sqrt{a}, β\}. Then

(1 − σ)(α′) = −\{\sqrt{a}, β\} + \{−\sqrt{a}, β\} = \{−1, β\}

On the other hand

(1 + σ)(α′) = res_{L|F} ∘ N_{L|F} (α′) = res_{L|F} ({a, β} − {−a, β}) = {−1, β}

Hence 2α′ = 0 and (1.2.3) implies α′ = {−1, δ} for some δ ^{∈} K_{2}L. Since

\{a, β\} = N_{L|F} (α) = N_{L|F} (α′ + \{\sqrt{a}, β\}) = \{−1, N_{L|F} (δ)\} + \{−a, β\}

we have β′ = β + N_{L|F} (δ) ^{∈} U_{F}. These facts yield

\begin{matrix}α = \{\sqrt{a}, β\} + \{−1, δ\} & = \{\sqrt{a}, β′\} − \{−\sqrt{a}, δ\} − \{\sqrt{a}, σ(δ)\} \\ = \{\sqrt{a}, β′\} − res_{L|F} ∘ N_{L|F} (\{−\sqrt{a}, δ\}). \end{matrix}

qed.

§ 2 Severi-Brauer Varieties

Let F be a field, CharF ≠ 2. For a, b ^{∈} F^{∗} let

D = D(a, b) =< A, B | A^{2} = a, B^{2} = b, AB = −AB > .

The Severi-Brauer variety to the quaternion algebra D is isomorphic to the quadric hypersurface X in IP^{3} defined by X_{1}^{2} − aX_{2}^{2} − bX_{3}^{2} = 0. It is well known that

D ≃ M_{2}(F) ⟺ X ≃ IP^{1} ⟺ b ^{∈} N_{F (\sqrt{a})|F} (F (\sqrt{a})^{∗} ⟺ \{a, b\} ^{∈} 2K_{2}F.

Now suppose a /^{∈} (F^{∗})^{2} and let L = F (\sqrt{a}). An explicit isomorphism X_{L} → IP_{L}^{1} is given by

[X_{1} : X_{2} : X_{3}]⟶[(X_{1} + \sqrt{a}X_{2}) : X_{3}] = [bX_{3} : (X_{1} − \sqrt{a}X_{2})] with inverse
[S_{1} : S_{2}]  ⟶[\sqrt{a}(S_{1}^{2} + bS_{2}^{2}) : (S_{1}^{2} − bS_{2}) : 2\sqrt{a}S_{1}S_{2}]

The function t = S_{1}/S_{2} is a generator of the function field of IP_{L}^{1}. In this paper we identify the function field L(X) of X_{L} with L(t) by means of the above isomorphism. Note that the action of Gal(L|F) is given by t → b/t; in particular N_{L(X)|F (X)}(t) = b.

We have to use the following result of Merkur’ev and Suslin.

Proposition 2.1.

i) The sequence

0 ⟶ K_{2}F ⟶^{d} K_{2}F (X) ⟶ ⨁_{v^{∈}X^{(1)}} K_{1}κ(v) ⟶^{θ} K_{1}D ⟶ 0

is exact. ii) For every α ^{∈} ⨁_{v^{∈}X^{(1)}} K_{1}κ(v) there exist v_{0} ^{∈} X^{(1)} of degree two and α_{0} ^{∈} K_{1}κ(v_{0})

such that α − α_{0} ^{∈} Im d. For i) see [MS] or [S; Proposition 3]. To prove ii) represent θ(α) by x ^{∈} D^{∗}. Let F_{x} be a maximal commutative subfield of D containing x. Now choose v_{0} such that κ(v_{0}) ≃ F_{x} and take for α_{0} the element corresponding to x. qed.

§ 3 Theorem B in a special case

Let F_{0} ≠ ZZ/2 be a prime field and let F/F_{0}(a_{1}, …, a_{n}, a, b) be pure transcendental over F_{0}. Put D = D(a, b) and let X be the Severi-Brauer variety corresponding to D. L = F (\sqrt{a}) is a splitting field of D.

Theorem 3.1. ^{∗} The sequence

K_{3}F (X) ⟶^{d} ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) ⟶^{N} K_{2}F

is exact. Here N is induced by the norm for finite extensions. Since N = Nrd ∘ θ, Theorem 3.1 implies Theorem B in this case. Note that, over L, the sequence of Theorem 3.1 reads as

(3.1.1)  K_{3}L(t) ⟶^{d} ⨁_{p^{∈}P_{L}} K_{2}L_{P} ⊕ K_{2}κ(w_{∞}) ⟶^{N} K_{2}L

under the identification L(X) = L(t) of § 2 (w_{∞} denotes the point of X_{L} defined by t = ∞). Since κ(w_{∞}) = L, the exactness of (3.1.1) is clear by the exactness of (1.0.1).

∗ There is a simpler proof of Theorem 3.1 than the one given here: One has to compare the sequence of Theorem 3.1 via Galois symbol (which is an isomorphism for F (X), see the proof of 3.1) with the spectral sequence for Galois cohomology associated to the field extension F (X)|F.

We have to consider the following commutative diagram

� � �  _{d_{L}} � �  _{d_{L}} �  _{N} �  _{d_{F}} �  _{N} �_{N_{L|F}} � �_{1−σ} �  _{1−σ} �  _{1−σ} � � �  K_{2}F ⨁_{v} K_{2}F_{v} K_{3}F (X)  K_{2}L ⨁_{v} K_{2}L_{v} K_{3}L(X) K_{3}L  ⨁_{v} K_{2}L_{v} K_{3}L(X) K_{3}L  K_{3}F (X) K_{3}F

Here I have changed notation a little bit. v runs everywhere (!) over the closed points of X, F_{v} = κ(v) is the residue field of v and L_{v} = F_{v} ⊗_{F} L.

Lemma 3.2 Let α ^{∈} ⨁_{v} K_{2}F_{v} such that N (α) = 0. Then there exist β ^{∈} K_{3}L(X) and γ ^{∈} K_{3}L such that

i) d_{L}(β) = res_{L|F} (α). ii) res_{L(X)|L}(γ) = (1 − σ)(β). iii) N_{L|F} (γ) = 0.

Proof The exactness of (3.1.1) implies the existence of β such that i) holds. Since d_{L}∘(1−σ)(β) = (1−σ)∘d_{L}(β) = (1−σ)∘res_{L|F} (σ) = 0, there exist a (unique) γ ^{∈} K_{3}L such that ii) holds. Note that N_{L|F} (γ) depends only on α. However, to prove that indeed N_{L|F} (γ) = 0 one constructs β more explicitly (I don’t know a direct argument, because K_{3}F → K_{3}F (X) is not injective; e.g. {−1, a, b} = 0 in K_{3}F (X)). I use the identification X_{L} ≃ IP_{L}^{1} of § 2. Let v_{∞} be the closed point of X which splits over L into the points w_{0}, w_{∞} given by t = 0, t = ∞ respectively. Let us first assume that α = (α_{v})_{v} ^{∈} ⨁_{v≠v_{∞}} K_{2}F_{v}. Denote by t_{v} the residue class of t in L_{v}, v ≠ v_{∞}. Put

β = ∑_{v≠v_{∞}} N_{L_{v}(t)|L(t)}({t − t_{v}, res_{L_{v}|F_{v}}(α_{v})})

It is clear from Lemma 1.1 that i) holds. Since σ(t) = b/t we also have σ(t_{v}) = b/t_{v}.

Hence

(1 − σ)(β)= ∑_{v≠v_{∞}} N_{L_{v}(t)|L(t)}(\{\frac{t−t_{v}}{σ(t)−σ(t_{v})}, res_{L_{v}|F_{v}}(α_{v})\})
= ∑_{v≠v_{∞}} N_{L_{v}(t)|L(t)}(\{\frac{tt_{v}}{−b}, res_{L_{v}|F_{v}}(α_{v})\})
= ∑_{v≠v_{∞}}\{\frac{−t}{b}, N_{L_{v}|F_{v}}(α_{v})\} + res_{L(t)|L}(γ)
= \{\frac{t}{b}, res_{L|F} ∘ N (α)\} + res_{L(t)|L}(γ)

where γ = ∑_{v≠v_{∞}} N_{L_{v}|L}({t_{v}, res_{L_{v}|F_{v}}(α_{v})}). With this choice of γ ii) and iii) hold, since N (α) = 0 and N_{L|F} (γ) = ∑_{v} N_{F_{v}|F} {b, α_{v}} = {b, N (α)}. For the general case it suffices to show K_{2}F_{v_{∞}} ⊂ Im d_{F} ⊕ ⨁_{v≠v_{∞}} K_{2}F_{v}. Note that cor_{L|F} : K_{2}L_{v_{∞}} → K_{2}F_{v_{∞}} induces an isomorphism K_{2}κ(w_{∞}) → K_{2}F_{v_{∞}}. For α ^{∈} K_{2}F_{v_{∞}} let α′ ^{∈} K_{2}κ(w_{∞}) = K_{2}L such that cor_{L|F} (α′) = α. Now, if f ^{∈} L(X) is any function having a zero at w_{∞} and no zero or pole at w_{0}, then α − d_{F} cor_{L(X)|F (X)}{f, α} ^{∈} ⨁_{v≠v_{∞}} K_{2}F_{v}. qed.

Proof of Theorem 3.1. For α ^{∈} Ker N we have to show α ^{∈} Im d_{F}. Let β and γ be as in Lemma 3.2. By (1.2.1) there exist β′ ^{∈} K_{3}L such that (1 − σ)(β′) = γ. Replacing β by β − β′ we may assume (1 − σ)(β) = 0. Hence, by (1.2.4), β = res_{L(X)|F (X)}(β′′) + \{\sqrt{a}, δ\} for some β′′ ^{∈} K_{3}F (X), δ ^{∈} U_{F (X)}. (We can apply (1.2.4), since F (X) = qf F_{0}[a_{1}, …, a_{n}, a, b, X_{1}, X_{2}]/(X_{1}^{2} − aX_{2}^{2} − b) = F_{0}(a_{1}, …, a_{n}, a, X_{1}, X_{2})). After replacing α by α − d_{F} (β′′) and β by β − res_{L(X)|F (X)}(β′′), we have the following situation i) d_{L}(β) = res_{L|F} (α). ii) β = \{\sqrt{a}, δ\}, δ ^{∈} K_{2}F (Y). iii) {−1, δ} = 0 in K_{3}F (Y).

Claim There exist ρ ^{∈} K_{2}L(X) such that cor_{L|F} ∘ d_{L}(ρ) = d_{F} (δ).

Proof Since

{a, d_{F} (δ)}= d_{F} ({a, δ}) = d_{F} ({−a, δ}) = d_{F} ∘ N_{L(X)|F (X)}(β)
= cor_{L|F} ∘ d_{L}(β) = cor_{L|F} ∘ res_{L|F} (α) = 2α

there exist μ ^{∈} ⨁_{v} K_{1}L_{v} such that cor_{L|F} (μ) = d_{F} (δ). (use the general fact: \{a, b\} ^{∈} 2K_{2}K ⟺ b ^{∈} N_{K(\sqrt{a})|K}(K(\sqrt{a})^{∗})). We now alter μ such that N (μ) = 0, i.e., μ ^{∈} Im d_{L}. Since N_{L|F} ∘ N (μ) = N ∘ d_{F} (δ) = 0, there exist by Hilbert 90 λ ^{∈} K_{1}L such that (1 − σ)(λ) = N (μ). Let w be a rational point of X_{L}; the residue class field κ(w) is a direct factor of L_{v} for some closed point v of X. Let φ : L → κ(w) be the natural isomorphism and put μ′ = μ − (1 − σ)(φ(λ)) ^{∈} ⨁_{v} K_{1}L_{v}. Then N (μ′) = N (μ) − (1 − σ)(λ) = 0, so there exist ρ ^{∈} K_{2}L(X) such that d_{L}(ρ) = μ′. The claim follows by

cor_{L|F} ∘ d_{L}(ρ) = cor_{L|F} (μ − (1 − σ)(φ(λ))) = cor_{L|F} (μ) = d_{F} (δ).

We continue the proof of Theorem 3.1. Let α′ = α − d_{F} ∘ N_{L(X)|F (X)}(\sqrt{a}, ρ). Since

\begin{matrix}2α = \{a, d_{F} (δ)\} & = \{a, cor_{L|F} ∘ d_{L}(ρ)\} = d_{F} ∘ N_{L(X)|F (X)}\{a, ρ\} \\ = 2d_{F} ∘ N_{L(X)|F (X)}(\{\sqrt{a}, ρ\}) \end{matrix}

we have 2α′ = 0. The analogue to (1.2.3) for K_{2} ([MS, Lemma 10.4]) implies α′ = {−1, ξ} for some ξ ⊕_{v} K_{1}F_{v}. By Proposition 2.1 ii) there exist a closed point v ^{∈} X of degree two, ξ_{0} ^{∈} K_{1}F_{v} and η ^{∈} K_{2}F (X) such that d_{F} η = ξ − ξ_{0}. Then

α = \{−1, ξ_{0}\} + d_{F} (N_{L(X)|F (X)}(\{\sqrt{a}, ρ\}) + \{−1, η\}).

Hence we may assume that α is concentrated in some point v of X of degree two, i.e., α ^{∈} K_{2}F_{v}. Let ε be the generator of Gal(F_{v}|F). Since N_{F_{v}|F} (α) = N (α) = 0, Hilbert 90 for K_{2} implies α = (1 − ε)(λ) for some λ ^{∈} K_{2}F_{v}. We consider the base extension F → F′, where F′|F is isomorphic to F_{v}|F. Let ε′ be the generator of Gal(F′|F) and let v_{0} and v_{1} = ε′(v_{0}) be the points over v. Moreover let φ_{i} : F_{v} = F′ → κ(v_{i}) be the natural identification. If we put x = φ_{0}(λ) − φ_{1}(λ) ^{∈} ⊕_{v}K_{2}F′_{v}. then cor_{F′|F} (x) = (1 − ε)(λ) = α. Now take y ^{∈} K_{3}F′(X) such that d_{F′}(y) = x; this is possible since D is split over F′ and N (x) = 0. Then α = cor_{F′|F} (x) = d_{F} ∘ N_{F′(X)|F (X)}(y) ^{∈} Im d_{F} qed.

§ 4 Proof of Theorem B (Sketch)

The hard point in the proof of Theorem B is

Theorem 4.1. If α ^{∈} _{⨁_{v^{∈}X^{(1)}} ; deg v=2}K_{2}κ(v) and θ(α) = 0, then α ^{∈} Im d.

The general case is covered by the following two lemmas

Lemma 4.2 Every element of Ker θ/Im d is of order 2.

This follows by adjoining a splitting field of D of degree two and the usual transfer arguments. So we may assume that F has no extension of odd degree.

Lemma 4.3 If F has no extension of odd degree, then

⨁_{v^{∈}X^{(1)}} ; deg v>2K_{2}κ(v) ⊂ Im d +_{⨁_{v^{∈}X^{(1)}} ; deg v=2}K_{2}κ(v).

The proof is similar to that of the K_{1}-case in [R2].

In the following we use the notation of [Re]. One has an exact sequence

1 → K_{2}D → U_{D} →^{π}[D^{∗}, D^{∗}] → 1

where U_{D} is generated by elements c(x, y), x, y ^{∈} D^{∗} and π(c(x, y)) = [x, y]. Now choose maps ψ_{0}, ψ_{1} : [D^{∗}, D^{∗}] → D^{∗} such that [ψ_{0}(x), ψ_{1}(x)] = x and ψ_{i}(1) = 1. The defining relations for the c(u, v) in [Re] and the Reidemeister-Schreier method [MKS] yield the following representation of K_{2}D.

Lemma 4.4 K_{2}D is generated by the elements

d(u; x, y) = c(ψ_{0}(u), ψ_{1}(u)) · c(x, y) · c(ψ_{0}(u[x, y]), ψ_{1}(u[x, y]))^{−1}
u ^{∈} [D^{∗}, D^{∗}], x, y ^{∈} D^{∗}

with the following set of defining relations:

\begin{matrix}R_{0}(u, x) & d(u; x, 1 − x) = 1 \\ R_{1}(u, x, y, z) & d(u; xy, z) = d(u; xyx^{−1}, xzx^{−1}) · d(ux[y, z]x^{−1}; x, z) \\ R_{2}(u, x, y, z) & d(u; x, yz) · d(u[x, yz]; y, zx) · d(u[xy, z]; z, xy) = 1 \\ R_{3}(u) & d(1; ψ_{0}(u), ψ_{1}(u)) = 1 \end{matrix}

Let

\begin{aligned}H_{D} = ⟨ h(x, y); x, y ^{∈} D^{∗}, [x, y] = 1 & | h(x, 1 − x) = 1; \\ & h(x, y)h(x, z) = h(x, yz); [h(x, y), h(x′, y′)] = 1 ⟩ \end{aligned}

There is a natural map μ : H_{D} → K_{2}D, sending h(x, y) to c(x, y) = d(1; x, y). By [RS; § 4] μ is surjective. Note that [x, y] = 1 implies that x and y are contained in a maximal commutative subfield of D which is unique if x /^{∈} F^{∗} or y /^{∈} F^{∗}. There is a bijection

v : {maximal commutative subfields of D} →^{≈} {closed points of X of degree 2},

such that κ(v(L)) ≃ L. Let Ω_{D} = ⊕_{deg v=2}K_{2}κ(v)/Im d. One defines an homomorphism ϕ : H_{D} → Ω_{D} by ϕ(h(x, y)) = {x, y} ^{∈} K_{2}κ(v(L)) mod Im d, where L ⊂ D is a maximal commutative subfield containing x and y. It turns out that ϕ is well defined, surjective and that θ ∘ϕ = μ. (I can show that ϕ is also injective, at least if F has no extension of odd degree. Theorem B then implies H_{D} ≅ K_{2}D). So we have a commutative diagram

�  ^{ϕ} �  _{θ} �  _{μ} �  _{s} K_{2}D  Ω_{D} H_{D}

To prove Theorem 4.1 we construct a surjective section s as follows. For every generator d(u; x, y) choose a preimage g(u; x, y) ^{∈} H_{D}. Now put s(d(u; x, y)) = ϕ(g(u; x, y)). The problem is of course to show that s is well defined. (To guarantee surjectivity of s one takes g(1; x, y) = h(x, y) if [x, y] = 1). In any case one gets an homomorphism s′ : G_{D} → Ω_{D}, where G_{D} is the free group generated by the d(u; x, y). To show that s′ vanishes on the relations of Lemma 4.4. one has to be very careful in the choice of ψ_{0}, ψ_{1} and the g(u, x, y). For a certain specific choice of s′ (I don’t see another way than to give explicit formulas using the method of proof of [RS; Proposition 4.1]) one shows:

Lemma For every u ^{∈} [D^{∗}, D^{∗}]; x, y, z ^{∈} D^{∗} there exist

i) a rational function field F_{0}(ā_{1}, … ā_{n}, ā, b̄) over the prime field F_{0} of F. ii) elements ū ^{∈} [D̄^{∗}, D̄^{∗}]; x̄, ȳ, z̄^{∈} D̄^{∗}, where D̄ = D(ā, b̄) iii) maps ψ̄_{0}, ψ̄_{1} : [D̄^{∗}, D̄^{∗}] → D̄^{∗} and an homomorphism s̄ : GD̄ → ΩD̄ with the corre-

sponding properties as ψ_{0}, ψ_{1} and s′. iv) a specialization ρ : F [ā_{1}, …, ā_{n}, ā, b̄] → F such that ρ(ā) = a, ρ(b̄) = b, ρ(ū) = u,

ρ(x̄) = x, ρ(ȳ) = y, ρ(z̄) = z.

v) a diagram of homomorphisms

�  ^{s̄′} �  _{s′} �_{ρ} �  _{ρ} Ω_{D} G_{D}  ΩD̄ GD̄

which is commutative at least on the elements d(.; ., .) which occur in the relations R_{0}(ū, x̄), R_{1}(ū, x̄, ȳ, z̄) etc., that is d(ū, x̄, 1 − x̄), d(ū, x̄ȳ, z̄) etc.

Using this lemma one argues as follows: By Theorem 3.1 s̄′ vanishes on the relations for K_{2}D̄; in particular s̄′(R_{0}(ū; x̄)) = 0, s̄′(R_{1}(ū; x̄, ȳ, z̄) = 0, etc. Then v) shows s′(R_{0}(u, x)) = 0, s′(R_{1}(u, x, y, z)) = 0 etc., which is the desired conclusion.

§ 5 Proof of Hilbert 90 (Theorem A)

Theorem B and the injectivity of the reduced norm Nrd : K_{2}D → K_{2}F (see [R2]) yield:

Theorem 5.1 The sequence

K_{3}F (X) →^{d} ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) →^{N} K_{2}F

is exact.

We have to generalize this theorem to a product of Severi-Brauer varieties. Let X_{1}, …, X_{n} be a family of Severi-Brauer varieties over F of dimension 1. Put X = X_{1} × … × X_{n} and X̂_{i} = X_{1} × … × X_{i−1} × X_{i+1} × … × X_{n}. Let X̄_{i} be the fiber over the generic point of X̂_{i} with respect to the natural projection. X̄_{i} is a Severi-Brauer variety over the function field F (X̂_{i}); thus we have an exact sequence as in Theorem 5.1:

K_{3}F (X̄_{i}) ⟶^{d} ⨁_{v^{∈}X̄_{i}^{(1)}} K_{2}κ(v) ⟶^{N_{i}} K_{2}F (X̂_{i}).

We have X̄_{i}^{(1)} ⊂ X^{(1)} and a bijection X^{(1)} \ X̄_{i}^{(1)} → X̂_{i}^{(1)} induced by projection. Therefore

⨁_{v^{∈}X^{(1)}} K_{2}κ(v) = ⨁_{v^{∈}X̄_{i}^{(1)}} K_{2}κ(v) ⊕ ⨁_{vX̂_{i}^{(1)}} K_{2}(κ(X_{i} ×_{F} κ(v))).

Let π_{i} : ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) → ⨁_{v^{∈}X̄_{i}^{(1)}} K_{2}κ(v) be the corresponding projection. Put N_{i} = N_{i} ∘ π_{i}.

Corollary 5.2. The sequence

K_{3}F (X) ⟶^{d} ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) ⟶^{(d′,⊕N_{i})} ⨁_{v^{∈}X^{(2)}} K_{1}κ(v) ⊕ ⨁_{i=1}^{n} K_{2}F (X̂_{i})

is exact.

Proof If n = 1 this is Theorem 5.1. For an induction proof let us denote X = X^{n}, X̄_{i} = X̄_{i}^{n} and N_{i} = N_{i}^{n} to make clear the dependency on n (i ≤ n). Consider the commutative diagram

�  ^{d} �  ^{(d′,⊕N_{i}^{n−1})} �  _{d} �  _{(d′,⊕N_{i}^{n})} �  _{d} �  _{N_{n}} � �  _{f} �  _{d} �  _{g} � �  _{π_{n}} �  K_{2}F (X̂_{n}^{n}) ⨁_{v^{∈}(X̄_{n}^{n})^{(1)}} K_{2}κ(v) K_{3}F (X̄_{n}^{n})  ⨁_{v^{∈}(X^{n})^{(2)}} K_{1}κ(v) ⊕ ⨁_{i=1}^{n} K_{2}F (X̂_{i}^{n}) ⨁_{v^{∈}(X^{n})^{(1)}} K_{2}κ(v) K_{3}F (X^{n})  ⨁_{v^{∈}(X^{n−1})^{(2)}} K_{1}κ(v) ⊕ ⨁_{i=1}^{n−1} K_{2}F (X̂_{i}^{n−1}) ⨁_{v^{∈}(X^{n−1})^{(1)}} K_{2}κ(v) K_{3}F (X^{n−1})

Note that X^{n−1} = X̂_{n}. The homomorphisms denoted by f and g are injective by Proposition 2.1. i). Now let α ^{∈} ⨁_{v^{∈}(X^{n})^{(1)}} K_{2}κ(v) such that d′(α) = 0 and N_{i}^{n}(α) = 0; we have to show α ^{∈} Im d + Im f. Since N_{n} ∘ π_{n}(α) = 0 and the lower sequence is exact, there is a β ^{∈} K_{3}F (X^{n}) such that π_{n}(α − d(β)) = 0. So we may assume π_{n}(α) = 0, that is

α ^{∈} ⨁_{v^{∈}(X^{n−1})^{(1)}} K_{2}κ(X_{n} ×_{F} κ(v)).

The homomorphism d′ in the middle row can be written as

? ⊕ ⨁_{v^{∈}(X^{n−1})^{(1)}} K_{2}κ(X_{n} ×_{F} κ(v)) ⟶^{d′} ⨁_{v′^{∈}X_{n}^{(1)},v^{∈}(X^{n−1})^{(1)}} K_{1}κ(v′ × v) ⊕ ?

Hence d′(α) = 0 and Proposition 2.1. i) imply

α ^{∈} ⨁_{v^{∈}(X^{n−1})^{(1)}} K_{2}κ(v) = Im f.

qed.

Now we are ready to start the proof of Hilbert 90.

Lemma 5.3 Let α ^{∈} K_{3}L such that N_{L|F} (α) = 0. Then there exist r, n, m, p_{ij} ^{∈} IN, b_{i} ^{∈} F^{∗}, α_{i} ^{∈} K_{2}L, c_{j} ^{∈} F^{∗} (0 ≤ i ≤ n, 1 ≤ j ≤ m) and ρ ^{∈} K_{2}F such that

i) α = ∑_{i}{b_{i}, α_{i}} ii) b_{0}^{r} = 1 iii) N_{L|F} (α_{0}) = ∑_{j} p_{0j}{1 − d_{j}, c_{j}} + rρ,

N_{L|F} (α_{i}) = ∑_{j} p_{ij}{1 − d_{j}, c_{j}}, i ≥ 1 where d_{j} = π_{i}b_{i}^{p_{ij}}.

The proof is completely analogous to that of [MS; Lemma 13.3]. □ Let X_{i} be the Severi-Brauer variety associated to D(a, b_{i}) and let X = X_{1} × … × X_{n}. L(X) denotes the function field of X_{L}.

Lemma 5.4 There exist β ^{∈} K_{3}L(X) and γ ^{∈} ⨁_{v^{∈}X^{(1)}} K_{2}κ(v) such that

i) res_{L(X)|L}(α) = (1 − σ)(β). ii) dβ = res_{L|F} (γ) iii) (d′, ⊕N_{i})(γ) = 0.

Suppose the lemma holds. Then, by iii) and Corollary 5.2, we have γ = d(δ) for some δ ^{∈} K_{3}F (X). Put β′ = β − res_{L(X)|F (X)}(δ). Then ii) implies dβ′ = 0, i.e., β′ ^{∈} K_{3}L and i) yields α = (1 − σ)(β′) ^{∈} (1 − σ)(K_{3}L), which was to be shown.

Proof of Lemma 5.4 We identify L(X_{i}) with L(t_{i}) as in § 2; then L(X) = L(t_{1}, …, t_{n}). Moreover σ(t_{i}) = b_{i}/t_{i}, hence N_{L(X)|F (X)}(t_{i}) = b_{i}. Put s_{j} = ∏_{i} t^{p_{ij}}; then N_{L(X)|F (X)}(s_{j}) = d_{j}. Let F_{j} = F [x_{j}]/ (x_{j}^{2} − d_{j}) and L_{j} = F_{j} ⊗_{F} L. We have

\begin{aligned}α & = ∑_{i}\{b_{i}, α_{i}\} = ∑_{i}\{t_{i}, N_{L|F} (α_{i})\} − (1 − σ) ∑_{i}\{t_{i}, σ(α_{i})\} \\ & = ∑_{j}\{s_{j}, 1 − d_{j}, c_{j}\} + \{t_{0}^{r}, ρ\} − (1 − σ) ∑_{i}\{t_{i}, σ(α_{i})\} \end{aligned}

by Lemma 5.3. Put

β = ∑_{j} N_{L_{j}(X)|L(X)}{x_{j} + s_{j}, 1 − x_{j}, c_{j}} + {1 + t_{0}^{r}, ρ} − ∑_{i}{t_{i}, σ(α_{i})}.

Then α = (1 − σ)(β), since

N_{L_{j}(X)|L(X)} ∘ (1 − σ)(\{x_{j} + s_{j}, 1 − x_{j}\}) = N_{L_{j}(X)|L(X)}(\{\frac{s_{j}}{x_{j}} , 1 − x_{j}\}) =

= {s_{j}, N_{F_{j}|F} (1 − x_{j})} = {s_{j}, 1 − d_{j}} and N_{L(X)|F (X)}(t_{0}^{r}) = b_{0}^{r} = 1.

Denote by P_{i}, P′_{i} ^{∈} ⨁_{w^{∈}X_{L}^{(1)}} K_{0}κ(w) the canonical generators of K_{0}κ({t_{i} = 0}), K_{0}κ({t_{i} = ∞}), respectively; in particular d(t_{i}) = P_{i} − P′_{i}. Define R_{0} ^{∈} ⨁_{w^{∈}X_{L}^{(1)}} K_{0}κ(w) and Q_{j} ^{∈} ⨁_{w^{∈}X_{L_{j}}^{(1)}} K_{0}κ(w) by

\begin{aligned}d(1 + t_{0}^{r}) & = R_{0} − rP′_{0} \\ d(x_{j} + s_{j}) & = Q_{j} − ∑_{i} p_{ij}res_{L_{j}|L}(P′_{j}) \end{aligned}

A little calculation shows

dβ = ∑_{j} cor_{L_{j}|L}({1 − x_{j}, c_{j}} · Q_{j}) + ρR_{0} − ∑_{i}(σ(α_{i})P_{i} + α_{i}P′_{i})

Note that σ(P_{i}) = P′_{i}, σ(R_{0}) = R_{0} and σ(Q_{j}) = Q_{j}. In particular R_{0} ^{∈} ⨁_{v^{∈}X^{(1)}} K_{0}κ(w) and Q_{j} ^{∈} ⨁_{v^{∈}X_{F_{j}}^{(1)}} K_{0}κ(w). Therefore dβ = res_{L|F} (γ), where

γ = ∑_{j} cor_{F_{j}|F} ({1 − x_{j}, c_{j}} · Q_{j}) + ρR_{0} − cor_{L|F} (∑_{i} α_{i}P′_{i}) ^{∈} ⨁_{v^{∈}X^{(1)}} K_{2}κ(v).

It is straight forward to verify iii) for this choice of γ. qed.

Bibliography

[BT] Bass, H.; Tate, J.: The Milnor ring of a global field, Algebraic K-theory II, Springer, Lecture Notes in Mathematics 342 (1973), 349-446.

[K] Kato, K.: The norm homomorphism of Milnor’s K-group.

[Mi] Milnor, J.: Algebraic K-Theory and Quadratic Forms, Invent. math. 9, 318-344, 1970.

[Me] Merkur’ev, A.S.: K_{2} of fields and the Brauer group. Proceedings of the Boulder Con-

ference on K-Theory, 1983.

[MKS] Magnus, W.; Karras, A.; Solitar, D.: Combinatorial group theory, Dover Publications Inc., New York, 1976.

[MS] Merkur’ev, A.S.; Suslin, A.A.: K-Cohomology of Severi-Brauer varieties and the norm residue homomorphism. Izv. Akad. Nauk SSSR 46, 1982, 1011-1046.

[R1] Rost, M.: Injectivity of K_{2}D → K_{2}F for quaternion algebras; in preparation.

[R2] Rost, M.: SK_{1} of some nonsingular quadrics; in preparation.

[S] Soulè, C.: K_{2} et le Groupe de Brauer, Sem. Bourbaki, 1982/83, no. 601, 79-93.

[Re] Rehmann, U.: Zentrale Erweiterungen der speziellen linearen Gruppe eines Schiefkörpers, J. Reine Angew. Mathematik, 301, 77-104 (1978).

[RS] Rehmann, U:; Stuhler, U: On K_{2} of finite dimensional division algebras over arithmetical fields, Inv. Math. 50, 75-90, (1978).

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