CHAIN LEMMA FOR SPLITTING FIELDS OF SYMBOLS

MARKUS ROST

                      preliminary version

Contents

1. An invariant 1 2. Preliminaries, Conventions, and Notations 2 3. The forms A(α_{1}, …, α_{n}) (“algebras”) 4 4. The forms B(α_{1}, …, α_{n}) (“relative algebras”) 7 5. The forms C(α_{1}, …, α_{n}) (Chain lemma construction) 10 6. The forms K(α_{1}, …, α_{n}) (universal families of Kummer splitting fields) 15 7. Proof of the chain lemma 16 References 18

1. An invariant

We assume char k = 0 (to have available resolution of singularities). For a proper variety X we denote

I(X) = deg(CH_{0}(X)) ⊂ ℤ

Let X be an proper variety of dimension d, let X_{0} ⊂ X be closed subvariety, and let U = X \ X_{0}. We assume that U is smooth. Let W → U be a μ_{p}-torsor. We define

η(W, X, X_{0}) ∈ ℤ/I(X_{0})

as follows: Let L/U be the line bundle obtained as the image of [W/U] via

[W/U] ∈ H_{ét}^{1}(U, μ_{p}) → H_{ét}^{1}(U, 𝔾_{m}) = H_{Zar}^{1}(U, 𝔾_{m}) = Pic(U).

The degree map induces a map

deg: CH_{0}(U) → ℤ/I(X_{0}).

We define

η(W, X, X_{0}) = deg(c_{1}(L)^{d}).

Example 1.1. Let X be smooth, let F = k(X) and let K/F be a Kummer extension of degree p. Let W̄ → X be the normal closure of K/F. Then W̄ → X is etale over an open subset U = X \ X_{0} and we have an invariant

η′(K/F, X) = η(W̄ |U, X, X_{0}) ∈ ℤ/I(X_{0}).

Date: August 9, 1998.

Passing to the limit over all modells X of F one may define an invariant of K/F in ℤ/... where ... can be expressed in terms of valuations on F. But this is not important at the moment.

Proposition 1.2 (Degree formula). Let W → U = X \ X_{0} as above with X irreducible and let Y be proper and irreducible of dimension dim Y = dim X = d. Let f: Y → X be morphism, let Y_{0} = f^{−1}(X_{0}), let U′ = Y \ Y_{0} and W′ = W ×_{U} U′. Then

η(W′, Y, Y_{0}) = (deg f)η(W, X, X_{0}) mod I(X_{0}).

Note that

I(Y_{0}) ⊂ I(X_{0}).

Proof. This is pretty obvious: Let f̂: U′ → U be the restriction of f. Then the line bundle L′ given by W′/U′ is f̂^{∗}L, whence

c_{1}(L′)^{d} = f̂^{∗}c_{1}(L)^{d}

and

f̂_{∗}(c_{1}(L′)^{d}) = (deg f̂)c_{1}(L)^{d}.

Now apply the degree map.

Example 1.3. Suppose that in Proposition 1.2 one has I(X_{0}) ⊂ pZ and suppose further η(W′, Y, Y_{0}) ≢ 0 mod p. Then deg f is prime to p.

2. Preliminaries, Conventions, and Notations

• The ground field k has characteristic 0. We fix a prime p. We assume μ_{p} ⊂ k.

• By a scheme or a variety X (over k) we mean a separated scheme of finite type π_{X}: X → Spec k.

• If X is a smooth variety, then T X denotes the tangent bundle of X.

• Let V be vector bundle over X. We denote by π_{V} : ℙ(V) → X the projective bundle associated to V. Moreover

𝕃(V) → π_{V}^{∗} V

denotes the tautological line bundle on ℙ(V).

For the fiber tangent bundle T (ℙ(V)/X) one has

T (ℙ(V)/X) = π_{V}^{∗} V ⊗ 𝕃(V)^{∨}/O_{ℙ(V)}.

• Let V be vector (or an affine) bundle over X. We denote by 𝔸(V) → X the associated scheme V.

• By a form we understand a triple (T/S, L, α) where T → S are schemes, L is

line bundle on T and α ∈ H^{0}(T, L^{⊗−p}) is a form of degree p on L.

There is a natural homomorphism μ_{p} → Aut(T/S, L, α) induced from the standard action of 𝔾_{m} on L.

• Let (Spec k, L, α) be a nonzero form and let u ∈ L be a basis vector. Then the p-power class

{α} = {α(u)} ∈ K_{1}k/p = k^{∗}/(k^{∗})^{p}

is independent on the choice of u.

• Let (T/S, L, α) and let Γ be a finite group acting on (T/S, L, α) (i.e., there is given a homomorphism Γ → Aut(T/S, L, α)). We say that (T/S, L, α) is an admissable Γ-form if the following conditions hold:

– α is nonzero on an open dense subscheme of T.

– Γ has only finitely many fixed points on T (a fixed point is a point P ∈ T with gP = P for all g ∈ G).

– At each fixed point P the form α is nonzero.

• For vector bundles V, V′ on schemes X/S resp. X′/S we denote by V ⊞_{S} V′ the exterior direct sum, given by the sum of the pull backs to X ×_{S} X′. Similarly we denote by V ⊠_{S} V′ the exterior tensor product, given by the tensor product of the pull backs.

• For forms (T/S, L, α) and (T′/S, L′, α′) we denote by

(T/S, L, α) ⊠_{S} (T′/S, L′, α′) = ((T ×_{S} T′)/S, L ⊠_{S} L′, α ⊠_{S} α′)

their exterior product, with the form defined by

(α ⊠_{S} α′)(u ⊠_{S} u′) = α(u)α′(u′)

for sections u, u′ of L, L′, respectively.

If (T/S, L, α) and (T′/S, L′, α′) are admissable Γ-forms, then (T/S, L, α)⊠_{S} (T′/S, L′, α′) is an admissable Γ-form.

• Let (S, H_{i}, α_{i}), i = 1, …, n, be admissable Γ-forms and let P ∈ S be a k-rational fixed point. We say that P is twisting for the family (S, H_{i}, α_{i})_{i}, if the homomorphism

Γ → μ_{p}^{n} = ∏_{i=1}^{n} Aut(H_{i}|P, α_{i}|P)

is surjective.

• By a cellular variety we mean a variety which admits a stratification by affine spaces. The motive of a cellular variety is the direct sum of powers of the Tate motive L, with a summand L^{⊗i} for each i-cell. If X and Y are cellular, then X × Y is cellular and one has

CH_{∗}(X × Y) = CH_{∗}(X) ⊗_{ℤ} CH_{∗}(Y).

• Let L be a line bundle L on a smooth and proper variety X over k of dimension d ≥ 0. We write

δ(L) = deg(c_{1}(L)^{d}) ∈ ℤ.

Here

deg: CH_{0}(X) → CH_{0}(Spec k) = ℤ

is the degree map. If d = 0 we understand by δ(L) the degree of X as a finite extension of k.

If V is a vector space of dimension n, then

δ(𝕃(V)) = deg(c_{1}(𝕃(V))^{n−1}) = (−1)^{n−1}.

• The index I_{X} of a proper variety is

I_{X} = deg(CH_{0}(X)) ⊂ ℤ

• If p is a prime, a field k is called p-special if char k ≠ p and if k has no finite field extensions of degree prime to p.

• Let (S, L, α) be a form. We consider the bundle of algebras

A = A(S, L, α) = T L/I

over R. Here T L is the tensor algebra of L and I is the ideal subsheaf generated by

λ^{⊗p} − α(λ)

for local sections λ of L. A a is bundle of commutative algebras of degree p. Note that

A = ⨁_{i=0}^{p−1} L^{⊗i}

as vector bundles. We denote by

N_{A}: A → O_{S}

the norm of the algebra A.

• We use the notation

Cyclic^{p}(Z) = (Z^{p})/(ℤ/p).
3. The forms A(α_{1}, …, α_{n}) (“algebras”)

Given a scheme S and forms (S, H_{i}, α_{i}), i = 1, …, m, we define forms

A(α_{1}, …, α_{n}) = (P_{n}/S, K_{n}, Φ_{n}), 0 ≤ n ≤ m.

For n = 0 we put

P_{0} = S,
K_{0} = O_{S},
Φ_{0}(t) = t^{p}.

Suppose (P_{n−1}/S, K_{n−1}, Φ_{n−1}) is defined. We consider the 2-dimensional vector bundle

V_{n} = O_{P_{n−1}} ⊕ H_{n} ⊠_{S} K_{n−1}

on P_{n−1}, and the form

φ_{n}: V_{n} → O_{P_{n−1}}

on V_{n} defined by

φ_{n}(t − u ⊗ v) = t^{p} − α_{n}(u)Φ_{n−1}(v)

for sections t, u, v of O_{P_{n−1}}, H_{n}, K_{n−1}, respectively.

Let (P_{n−1,j}, V_{n,j}, φ_{n,j}), j = 1, …, p − 1 be copies of (P_{n−1}, V_{n}, φ_{n}). We put

(P_{n}/S, K_{n}, Φ_{n}) = (P_{n−1}/S, K_{n−1}, Φ_{n−1}) ⊠_{S} _{⊠_{S}}_{j=1}^{p−1} (ℙ(V_{n,j}), 𝕃(V_{n,j}), φ_{n,j}).

We assume now that S = Spec k and list the most important properties of the forms (P_{n}, K_{n}, Φ_{n}).

Lemma 3.1. The variety P_{n} is smooth, proper, cellular, connected, and of dimension p^{n} − 1.

CHAIN LEMMA 5

Proof. Indeed, P_{n} is an iterated projective bundle. The computation of the dimension is clear for n = 0 and for n > 0 we find

dim P_{n} = dim P_{n−1} + (p − 1)(1 + dim P_{n−1})
= (p^{n−1} − 1) + (p − 1)p^{n−1} = p^{n} − 1

by induction on n. Lemma 3.2. δ(K_{n}) = (−1)^{n} mod p. Proof. This is clear for n = 0. Let

u_{n} = c_{1}(K_{n}) ∈ CH^{1}(P_{n}), n ≥ 0,
u_{n−1,j} = c_{1}(K_{n−1,j}) ∈ CH^{1}(P_{n−1,j}), n ≥ 1, j = 1, …, p − 1,
z_{n,j} = c_{1}(𝕃(V_{n,j})) ∈ CH^{1}(ℙ(V_{n,j})), n ≥ 1, j = 1, …, p − 1.

For n ≥ 1 let

P̂_{n} = P_{n−1} × ∏_{j=1}^{p−1} P_{n−1,j}

Then

CH^{∗}(P̂_{n}) = CH^{∗}(P_{n−1}) ⊗ ⨂_{j=1}^{p−1} CH^{∗}(P_{n−1,j})

and

CH^{∗}(P_{n}) = \frac{CH^{∗}(P̂_{n})[z_{n,j}; j = 1, …, p − 1]}{⟨z_{n,j}^{2} − z_{n,j}u_{n−1,j}; j = 1, …, p − 1⟩}.

Moreover

u_{n} = u_{n−1} + \overline{z}_{n}, with \overline{z}_{n} = ∑_{j=1}^{p−1} z_{n,j}.

Note that

u_{n−1}^{p^{n−1}} = u_{n−1,j}^{p^{n−1}} = 0, z_{n,j}^{p^{n−1}+1} = 0

by dimension reasons. Hence, calculating mod p,

u_{n}^{p^{n−1}} = (u_{n−1} + \overline{z}_{n})^{p^{n−1}} = u_{n−1}^{p^{n−1}} + \overline{z}_{n}^{p^{n−1}} = \overline{z}_{n}^{p^{n−1}}.

One finds (using Lemma 3.3 below)

u_{n}^{p^{n}−1} = u_{n}^{p^{n−1}−1}u_{n}^{p^{n−1}(p−1)} = u_{n}^{p^{n−1}−1}\overline{z}_{n}^{p^{n−1}(p−1)}

= u_{n}^{p^{n−1}−1}(z_{n,1}^{p^{n−1}} + z_{n,2}^{p^{n−1}} + ⋯ + z_{n,p−1}^{p^{n−1}})^{p−1}

= −u_{n−1}^{p^{n−1}−1}z_{n,1}^{p^{n−1}}z_{n,2}^{p^{n−1}} ⋯ z_{n,p−1}^{p^{n−1}}

= −u_{n−1}^{p^{n−1}−1}z_{n,1}u_{n,1}^{p^{n−1}−1}z_{n,2}u_{n,2}^{p^{n−1}−1} ⋯ z_{n,p−1}u_{n,p−1}^{p^{n−1}−1}.

It follows that

δ(K_{n}) = −δ(K_{n−1})(−δ(K_{n−1,1}))(−δ(K_{n−1,2})) ⋯ (−δ(K_{n−1,p−1}))
= −δ(K_{n−1}) mod p.

whence the claim.

Lemma 3.3. Let R be a ring over 𝔽_{p} and let v_{1}, v_{2}, …, v_{p−1} ∈ R, be elements with v_{1}^{2} = v_{2}^{2} = ⋯ = v_{p−1}^{2} = 0. Then

(v_{1} + v_{2} + ⋯ + v_{p−1})^{p−1} = −v_{1}v_{2} ⋯ v_{p−1}.

Proof. Note that (p − 1)! = −1 mod p.

The construction of (P_{n}, K_{n}, Φ_{n}) is functorial in the forms (S, H_{i}, α_{i}). In particular the group

Γ_{n} = μ_{p}^{n} ⊂ ∏_{i=1}^{n} Aut(S, H_{i}, α_{i})

acts on (P_{n}, K_{n}, Φ_{n}).

From now on we suppose that α_{i} ≠ 0 for i = 1, …, n.

Lemma 3.4. The triple (P_{n}, K_{n}, Φ_{n}) is an admissable Γ_{n}-form. All fixed points are k-rational.

Proof. By induction on n. Suppose that (P_{n−1}, K_{n−1}, Φ_{n−1}) is an admissable Γ_{n−1}- form. It suffices to show that (ℙ(V_{n}), 𝕃(V_{n}), φ_{n}) is an admissable Γ_{n}-form. It is easy to see that φ_{n} is generically nonzero. Every Γ_{n}-fixed point on ℙ(V_{n}) lies over a Γ_{n−1}- fixed point P ∈ P_{n−1}. It suffices to show that the fibre (Spec κ(P), 𝕃(V_{n})|P, φ_{n}|P) is an admissable Γ-form where

Γ = Aut(S, H_{n}, α_{n}) = ker(Γ_{n} → Γ_{n−1}).

This is easy to see: If (Spec k, H, α) is a nonzero form over k, then

μ_{p} = Aut(Spec k, H, α)

has in ℙ(k ⊕ H) only the two fixed points ℙ(0 ⊕ H) and ℙ(k ⊕ 0). The form φ(t − u) = t^{p} − α(u) is nonzero on the lines t = 0 and u = 0.

Lemma 3.5. Let η_{n} ∈ P_{n} be the generic point. Then

{α_{1}, …, α_{n}, Φ_{n}(η_{n})} = 0 ∈ K_{n+1}^{M}k(P_{n})/p.

Proof. By induction on n. Suppose that

{α_{1}, …, α_{n−1}, Φ_{n−1}(η_{n−1})} = 0 ∈ K_{n}^{M}k(P_{n−1})/p.

One has

Φ_{n}(η_{n}) = Φ_{n−1}(η_{n−1}) · ∏_{j=1}^{p−1}(1 − α_{n}Φ_{n−1,j}(η_{n−1,j})).

Hence it suffices to show

{α_{1}, …, α_{n}, 1 − α_{n}Φ_{n−1,j}(η_{n−1,j})} ∈ K_{n+1}^{M}k(P_{n})/p

for each j = 1, …, p − 1. This follows from {a, 1 − ab} = −{b, 1 − ab}.

Remark 3.6. Given the forms (Spec k, H_{i}, α_{i}), form the vector space

A_{n} = ⨁_{j_{1},...,j_{n}=0}^{p−1} H_{1}^{⊗j_{1}} ⊗ ⋯ ⊗ H_{n}^{⊗j_{n}}.

One has dim A_{n} = p^{n}. On A_{n} there is the form

Θ_{n} = ⨁_{j_{1},...,j_{n}=0}^{p−1}(−α_{1})^{⊗j_{1}} ⊗ ⋯ ⊗ (−α_{n})^{⊗j_{n}}

Consider the form (ℙ(A_{n}), 𝕃(A_{n}), Θ_{n}). If p = 2, this form satisfies all the properties of (P_{n}, K_{n}, Φ_{n}) listed above (up to a sign in the computation of δ(𝕃(A_{n}))). If p > 2, all properties of (P_{n}, K_{n}, Φ_{n}) are also valid, except for the splitting of the symbol. If n = 1, n = 2, or n = p = 3, one may define on A_{n} an algebra structure with norm form Θ′_{n} in such a way that (ℙ(A_{n}), 𝕃(A_{n}), Θ′_{n}) satisfies all the properties. The (P_{n}, K_{n}, Φ_{n}) form an approximation to these algebras, with the advantage, that (P_{n}, K_{n}, Φ_{n}) can be constructed for all p and n.

4. The forms B(α_{1}, …, α_{n}) (“relative algebras”)

Let n ≥ 1. Given forms (S, H_{i}, α_{i}), i = 1, …, n − 1, and (S′/S, L, β), we define a form

B(α_{1}, …, α_{n−1}, β) = (P′_{n}/S′, K′_{n}, Φ′_{n})

as follows. Let (P_{n−1}/S, K_{n−1}, Φ_{n−1}) be as in section 3. Put

\overline{P}_{n−1} = S′ ×_{S} P_{n−1}

We consider the 2-dimensional vector bundle

\overline{V}_{n} = O_{\overline{P}_{n−1}} ⊕ L ⊠_{S} K_{n−1}

on \overline{P}_{n−1}, and the form

\overline{φ}_{n}: \overline{V}_{n} → O_{\overline{P}_{n−1}}

on \overline{V}_{n} defined by

\overline{φ}_{n}(t − u ⊗ v) = t^{p} − β(u)Φ_{n−1}(v)

for sections t, u, v of O_{\overline{P}_{n−1}}, L, K_{n−1}, respectively.

Let

(\overline{P}_{n−1,j}, \overline{V}_{n,j}, \overline{φ}_{n,j}, K_{n−1,j}, P_{n−1,j}), j = 1, …, p − 1

be copies of (\overline{P}_{n−1}, \overline{V}_{n}, \overline{φ}_{n}, K_{n−1}, P_{n−1}). We put

(P′_{n}/S′, K′_{n}, Φ′_{n}) = ⊠_{S′}_{j=1}^{p−1} (ℙ(\overline{V}_{n,j}), 𝕃(\overline{V}_{n,j}), \overline{φ}_{n,j}).

We assume now that S = Spec k and list the most important properties of the forms (P′_{n}, K′_{n}, Φ′_{n}).

Lemma 4.1. The variety P′_{n} is smooth and proper over S′, and of relative dimension p^{n} − p^{n−1}. If S′ is cellular, so is P′_{n}. The fibres of S/S′ are connected.

Proof. Note that P′_{n}/S′ is an iterated projective bundle. Moreover

dim P′_{n}/S′ = (p − 1)(dim P_{n−1} + 1) = p^{n} − p^{n−1}

by Lemma 3.1.

8 MARKUS ROST

Let

u′_{n} = c_{1}(K′_{n}) ∈ CH^{1}(P′_{n}),
u_{n−1,j} = c_{1}(K_{n−1,j}) ∈ CH^{1}(P_{n−1,j}),
v_{n} = c_{1}(L)  ∈ CH^{1}(S′).

Lemma 4.2. One has

u′_{n}^{p^{n}} = u′_{n}^{p^{n−1}}v_{n}^{p^{n}−p^{n−1}} mod p.

If S′ = Spec k, then

δ(K′_{n}) = deg(u′_{n}^{p^{n}−p^{n−1}}) = −1 mod p.

Proof. Let

ˆ\overline{P_{n}} = S′ × ∏_{j=1}^{p−1} P_{n−1,j}

Then

CH^{∗}(ˆ\overline{P_{n}}) = CH^{∗}(S′) ⊗ ⨂_{j=1}^{p−1} CH^{∗}(P_{n−1,j})

and

CH^{∗}(P′_{n}) = \frac{CH^{∗}(ˆ\overline{P_{n}})[z_{n,j}; j = 1, …, p − 1]}{⟨z_{n,j}^{2} − z_{n,j}(v_{n} + u_{n−1,j}); j = 1, …, p − 1⟩}.

Moreover

u′_{n} = \overline{z}_{n}, with \overline{z}_{n} = ∑_{j=1}^{p−1} z_{n,j}.

Recall that u_{n−1,j}^{p^{n−1}} = 0. Calculating mod p, one finds

u′_{n}^{p^{n}} = \overline{z}_{n}^{p^{n}}
= z_{n,1}^{p^{n}} + ⋯ + z_{n,p−1}^{p^{n}}

= z_{n,1}^{p^{n−1}}(v_{n} + u_{n−1,1})^{p^{n−1}(p−1)} + ⋯ + z_{n,p−1}^{p^{n−1}}(v_{n} + u_{n−1,p−1})^{p^{n−1}(p−1)} = z_{n,1}^{p^{n−1}}(v_{n}^{p^{n−1}} + u_{n−1,1}^{p^{n−1}})^{(p−1)} + ⋯ + z_{n,p−1}^{p^{n−1}}(v_{n}^{p^{n−1}} + u_{n−1,p−1}^{p^{n−1}})^{(p−1)}

= z_{n,1}^{p^{n−1}}v_{n}^{p^{n−1}(p−1)} + ⋯ + z_{n,p−1}^{p^{n−1}}v_{n}^{p^{n−1}(p−1)}
= \overline{z}_{n}^{p^{n−1}}v_{n}^{p^{n−1}(p−1)} = u′_{n}^{p^{n−1}}v_{n}^{p^{n−1}(p−1)}.

This proves the first claim.

Suppose v_{n} = 0. Then z_{n,j}^{p^{n−1}+1} = 0. One finds mod p (using Lemma 3.3)

u′_{n}^{p^{n−1}(p−1)} = (z_{n,1}^{p^{n−1}} + z_{n,2}^{p^{n−1}} + ⋯ + z_{n,p−1}^{p^{n−1}})^{p−1}
= −z_{n,1}^{p^{n−1}}z_{n,2}^{p^{n−1}} ⋯ z_{n,p−1}^{p^{n−1}}

= −z_{n,1}u_{n−1,1}^{p^{n−1}−1}z_{n,2}u_{n−1,2}^{p^{n−1}−1} ⋯ z_{n,p−1}u_{n−1,p−1}^{p^{n−1}−1} Since δ(K_{n−1}) ≠ 0 mod p, it follows that

δ(K′_{n}) = −(−δ(K_{n−1,1}))(−δ(K_{n−1,2})) ⋯ (−δ(K_{n−1,p−1}))
= −1 mod p,

whence the second claim.

From now on we suppose that α_{i} ≠ 0 for i = 1, …, n−1. Let Γ be a finite group, let Γ → Γ_{n−1} be an epimorphism and let Γ → Aut(S′, L, β) be a homomorphism. Thus Γ acts on all the forms (Spec k, H_{i}, α_{i}), i = 0, …, n − 1, and (S′, L, β).

Lemma 4.3. Suppose that (S′, L, β) is an admissable Γ-form with all fixed points k-rational. Moreover suppose that each fixed point is twisting for the forms

(S, H_{i}, α_{i}), i = 1, …, n − 1, and (S′, L, β).

Then (P′_{n}, K′_{n}, Φ′_{n}) is an admissable Γ-form with all fixed points k-rational.

Proof. This follows as for Lemma 3.4.

Lemma 4.4. Suppose that S′ is irreducible. Let η_{n} ∈ P_{n} be the generic point. Then

{α_{1}, …, α_{n−1}, β(η_{n}), Φ_{n}(η_{n})} = 0 ∈ K_{n+1}^{M}k(P_{n})/p.

Proof. This follows as for Lemma 3.5.

Remark 4.5. Given the form (S′, L, β) one may define the “Kummer algebra”

A = A(S′, L, β) = L^{⊗0} ⊕ L^{⊗1} ⊕ ⋯ ⊕ L^{⊗p−1}

with the product given by the natural multiplication in the tensor algebra using the form β: L^{⊗p} → L^{⊗0} to reduce the degree mod p. One finds

CH^{∗}(ℙ(A)) ⊗ 𝔽_{p} = CH^{∗}(S′) ⊗ 𝔽_{p}[x]/⟨x^{p} − x^{p−1}y⟩

with x = c_{1}(𝕃(A)) and y = c_{1}(L).

Hence we have a homomorphism

R = 𝔽_{p}[x]/⟨x^{p} − x^{p−1}y⟩ → CH^{∗}(ℙ(A)) ⊗ 𝔽_{p}

Lemma 4.2 shows that there is a homomorphism

R → CH^{∗}(P′_{n}) ⊗ 𝔽_{p}, x ↦ u′_{n}^{p^{n−1}}, y ↦ v_{n}^{p^{n−1}}.

If one thinks in terms of the (in general nonexisting) algebras

A_{n} = A(α_{1}, …, α_{n−1}, β)

with “subalgebras”

A_{n−1} = A(α_{1}, …, α_{n−1}),

and one imagines to form something like the projective space ℙ_{A_{n−1}}(A_{n}), then one may think of P′_{n} as an approximation P′_{n} → ℙ_{A_{n−1}}(A_{n}) with the homomorphism R → CH^{∗}(P′_{n}) ⊗ 𝔽_{p} being the pull back on the Chow rings (if say S′ = ℙ^{∞} and with L the universal line bundle).

5. The forms C(α_{1}, …, α_{n}) (Chain lemma construction)

Let n ≥ 2. Given forms (S, H_{i}, α_{i}), i = 1, …, n − 1, and (S′/S, L, β), we define forms

C_{r} = C_{r}(α_{1}, …, α_{n−1}, β) = (S_{r}/S_{r−1}, L_{r}, β_{r}), r ≥ −1.

For r = −1, 0 we put

(S_{−1}/S_{−2}, L_{−1}, β_{−1}) = (S/S, H_{n−1}, α_{n−1}),
(S_{0}/S_{−1}, L_{0}, β_{0}) = (S′/S, L, β).

Let r > 0 and suppose C_{r−2} and C_{r−1} are defined.

Let

(P′_{n−1,r}/S_{r−1}, K′_{n−1,r}, Φ′_{n−1,r}) = B(α_{1}, …, α_{n−1}, β_{r−1})

be the form constructed in section 4, starting from (S, H_{i}, α_{i}), i = 1, …, n − 2, and (S_{r−1}/S_{r−2}, L_{r−1}, β_{r−1}). Put

(S_{r}/S_{r−1}, L_{r}, β_{r}) = (S_{r−2}/S_{r−3}, L_{r−2}, β_{r−2}) ⊠_{S_{r−2}} (P′_{n−1,r}/S_{r−1}, K′_{n−1,r}, Φ′_{n−1,r}).

We assume now that S = Spec k and list the most important properties of the forms (S_{r}/S_{r−1}, L_{r}, β_{r}).

Lemma 5.1. The variety S_{r} is smooth and proper over S′, and of relative dimension r(p^{n−1} − p^{n−2}). If S′ is cellular, so is S_{r}. The fibres of S/S′ are connected.

Proof. This follows from Lemma 4.1. For the dimension note

dim S_{r}/S_{r−1} = dim P′_{n−1,r}/S_{r−1} = p^{n−1} − p^{n−2}

by Lemma 4.1.

Thus if dim S′ = (p^{l} − 1)p^{n} for some ℓ ≥ 0, then dim S_{p} = (p^{l+1} − 1)p^{n−1}.

Theorem 5.2. Let ℓ ≥ 0 and suppose that S′ is smooth and proper of dimension (p^{l} − 1)p^{n}. Then

δ(L_{p}) = δ(L) mod p.

The proof requires some calculations.

Let a, b ∈ 𝔽_{p}, and let r ≥ 0 be an integer. In the ring 𝔽_{p}[z_{1}, …, z_{r}] let

x_{−1} = a,
x_{0} = b,
x_{m} = z_{m} + x_{m−2}, 1 ≤ m ≤ r.

Then

x_{2k} = z_{2k} + z_{2k−2} + ⋯ + z_{4} + z_{2} + b,
x_{2k+1} = z_{2k−1} + z_{2k−3} + ⋯ + z_{3} + z_{1} + a.

We denote by I the ideal generated by

z_{m}^{p} − z_{m}x_{m−1}^{p−1}, 1 ≤ m ≤ r

and put

R_{r}(a, b) = 𝔽_{p}[z_{1}, …, z_{r}]/I.

The elements

z^{J} = z_{1}^{i_{1}} ⋯ z_{r}^{i_{r}}, J = (i_{1}, …, i_{r}), 0 ≤ i_{j} ≤ p − 1

form an 𝔽_{p}-basis of R_{r}(a, b). For u ∈ R_{r}(a, b) let c_{m}(u) be the coefficient of z_{1}^{p−1} ⋯ z_{m}^{p−1}.

Lemma 5.3. If 1 ≤ r ≤ p one has c_{r}(x_{r}^{r(p−1)}) = 1 in R_{r}(a, b).

Proof. One has for 1 ≤ m ≤ p:

x_{m}^{m(p−1)} = x_{m}^{p(m−1)+(p−m)}
= (z_{m} + x_{m−2})^{p(m−1)+(p−m)}
= (z_{m}^{p} + x_{m−2}^{p})^{(m−1)}(z_{m} + x_{m−2})^{(p−m)}
= (z_{m}x_{m−1}^{p−1} + x_{m−2}^{p})^{(m−1)}(z_{m} + x_{m−2})^{(p−m)}.

Hence for m ≤ p one has

c_{m}(x_{m}^{m(p−1)}) = c_{m−1}(x_{m−1}^{(m−1)(p−1)}).

The claim follows by induction.

Proposition 5.4. If (a, b) ≠ (0, 0), then R_{r}(a, b) is isomorphic to a product of rings of the form

𝔽_{p}[v_{1}, …, v_{k}]/(v_{1}^{p}, …, v_{k}^{p}), k ≥ 0.

Proof. By induction on r ≥ 0. The case r = 0 is obvious.

Suppose b ≠ 0. Then the polynomial

z_{1}^{p} − z_{1}x_{0}^{p−1}

is separable with roots z_{1} = ib, i ∈ 𝔽_{p}. It follows that we have isomorphism

R_{r}(a, b) ⟶^{∼} ∏_{i∈𝔽_{p}} R_{r}(a, b)/(z_{1} − ib).

The ring R_{r}(a, b)/(z_{1} − ib) is the quotient of 𝔽_{p}[z_{2}, …, z_{r}] by the ideal generated by

z_{m}^{p} − z_{m}x_{m−1}^{p−1}, 2 ≤ m ≤ r

with

x_{0} = b,
x_{1} = ib + a,
x_{m} = z_{m} + x_{m−2}, 2 ≤ m ≤ r.

Hence R_{r}(a, b)/(z_{1} − ib) ≃ R_{r−1}(b, ib + a). The claim follows from the induction hypothesis.

Suppose b = 0. Then a ≠ 0. In this case we consider the homomorphism

φ: 𝔽_{p}[z_{1}, …, z_{r}] → 𝔽_{p}[z_{1}]/(z_{1}^{p}) ⊗ R_{r−1}(0, 1),
z_{m} ↦ (a + z_{1}) ⊗ z_{m−1}, 2 ≤ m ≤ r,
z_{1} ↦ z_{1} ⊗ 1.

We claim that φ(I) = 0. For this it suffices to show

φ(z_{m}^{p} − z_{m}x_{m−1}^{p−1}) = 0, 1 ≤ m ≤ r.

This is obvious for m = 1. If m = 2, then

φ(z_{2}^{p} − z_{2}x_{1}^{p−1}) = φ(z_{2}^{p} − z_{2}(z_{1} + a)^{p−1})
= (a + z_{1})^{p} ⊗ z_{1}^{p} − ((a + z_{1}) ⊗ z_{1})(z_{1} ⊗ 1 + 1 ⊗ a)^{p−1}
= (a + z_{1})^{p} ⊗ z_{1}^{p} − ((a + z_{1}) ⊗ z_{1})((z_{1} + a) ⊗ 1)^{p−1}
= (a + z_{1})^{p} ⊗ (z_{1}^{p} − z_{1}) = 0.

If m = 2k ≥ 2, then

φ(z_{2k}^{p} − z_{2k}x_{2k−1}^{p−1}) = φ(z_{2k}^{p} − z_{2k}(z_{2k−1} + ⋯ + z_{3} + z_{1} + a)^{p−1})
= (a + z_{1})^{p} ⊗ z_{2k−1}^{p} −
−((a + z_{1}) ⊗ z_{2k−1})((a + z_{1}) ⊗ z_{2k−2} + ⋯ + (a + z_{1}) ⊗ z_{2} + z_{1} ⊗ 1 + 1 ⊗ a)^{p−1}
= (a + z_{1})^{p} ⊗ z_{2k−1}^{p} −
−((a + z_{1}) ⊗ z_{2k−1})((a + z_{1}) ⊗ z_{2k−2} + ⋯ + (a + z_{1}) ⊗ z_{2} + (a + z_{1}) ⊗ 1)^{p−1}
= (a + z_{1})^{p} ⊗ (z_{2k−1}^{p} − z_{2k−1}(z_{2k−2} + ⋯ + z_{2} + 1))^{p−1}
= (a + z_{1})^{p} ⊗ (z_{2k−1}^{p} − z_{2k−1}x_{2k−2}^{p−1}) = 0.

If m = 2k − 1 ≥ 3, then

φ(z_{2k−1}^{p} − z_{2k−1}x_{2k−2}^{p−1}) = φ(z_{2k−1}^{p} − z_{2k−1}(z_{2k−2} + ⋯ + z_{2})^{p−1})
= (a + z_{1})^{p} ⊗ (z_{2k−2}^{p} − z_{2k−2}(z_{2k−3} + ⋯ + z_{1})^{p−1})
= (a + z_{1})^{p} ⊗ (z_{2k−2}^{p} − z_{2k−2}x_{2k−3}^{p−1}) = 0.

It follows that φ induces a homomorphism

\overline{φ}: R_{r}(a, b) → 𝔽_{p}[z_{1}]/(z_{1}^{p}) ⊗ R_{r−1}(0, 1),
z_{m} ↦ (a + z_{1}) ⊗ z_{m−1}, 2 ≤ m ≤ r,
z_{1} ↦ z_{1} ⊗ 1.

\overline{φ} is obviously surjective. By dimension reasons, \overline{φ} must be an isomorphism. Again the claim follows from the induction hypothesis.

Corollary 5.5. u^{p^{2}} = u^{p} for all u ∈ R_{p}(0, 1).

Corollary 5.6. Let n ≥ 2, and let u_{n} = x_{p}^{p^{n}−p} ∈ R_{p}(0, 1). Then c_{p}(u_{n}) = 1.

Proof. For n = 2 this is Lemma 5.3. Moreover, by Corollary 5.5, the element u_{n} does not depend on n.

We rewrite things in a homogenous form. Let x be a variable and let

R′ = 𝔽_{p}[x, z_{1}, …, z_{p}]/I′

where I′ is the homogenous ideal generated by

z_{m}^{p} − z_{m}x_{m−1}^{p−1}, 1 ≤ m ≤ p

with

x_{−1} = 0,
x_{0} = x,
x_{m} = z_{m} + x_{m−2}, 1 ≤ m ≤ p.

Then R′/(x − 1) = R_{p}(0, 1). Corollaries 5.5 and 5.6 yield the following two corollaries:

Corollary 5.7. u^{p^{2}} = u^{p}x^{p^{2}−p} for all u ∈ R′.

Corollary 5.8. Let n ≥ 2. Then

x_{p}^{p^{n}−p} = z_{1}^{p−1}z_{2}^{p−1} ⋯ z_{p}^{p−1}x^{p^{n}−p^{2}} mod x^{p^{n}−p^{2}+1}R′

Proof. Recall the basis elements (z^{J})_{J} of R_{p}(0, 1) considered above. The elements (z^{J}x^{p^{n}−p−|J|})_{J} form a basis of the homogenous subspace of R′ of degree p^{n} − p. It follows that

x_{p}^{p^{n}−p} = c_{p}(x_{p}^{p^{n}−p})z_{1}^{p−1}z_{2}^{p−1} ⋯ z_{p}^{p−1}x^{p^{n}−p^{2}} mod ⟨z^{J}x^{p^{n}−p−|J|}; |J| < p^{2} − p⟩.

But if |J| < p^{2} − p then z^{J}x^{p^{n}−p−|J|} ∈ x^{p^{n}−p^{2}+1}R′.

Proof of Theorem 5.2: Let

x_{r} = c_{1}(L_{r})^{p^{n−2}}  ∈ CH^{p^{n−2}}(S_{r}),  r ≥ −1,
z_{r} = c_{1}(K′_{n−1,r})^{p^{n−2}} ∈ CH^{p^{n−2}}(P′_{n−1,r}),  r ≥ 1.

Then, calculating mod p,

x_{−1} = 0,
x_{0} = c_{1}(L)^{p^{n−2}} ∈ CH^{p^{n−2}}(S′) ⊗ 𝔽_{p},
x_{r} = x_{r−2} + z_{r},  r ≥ 1,

since

c_{1}(L_{r}) = c_{1}(L_{r−2}) + c_{1}(K′_{n−1,r}).

Moreover

z_{r}^{p} = z_{r}x_{r−1}^{p−1}

by Lemma 4.2.

We have a homomorphism

R′(x) → CH^{∗}(S_{p}) ⊗ 𝔽_{p}, z_{m} ↦ z_{m}, x ↦ x_{0}.

It follows from Corollary 5.8 that (mod p)

x_{p}^{p^{ℓ+2}−p} = z_{1}^{p−1}z_{2}^{p−1} ⋯ z_{p}^{p−1}x_{0}^{p^{ℓ+2}−p^{2}} mod ⟨x^{p^{ℓ+2}−p^{2}+1}⟩

Now if dim S′ = (p^{l} − 1)p^{n}, then x_{0}^{p^{l+2}−p^{2}+1} = 0. Hence

x_{p}^{p^{ℓ+2}−p} = δ(K′_{n−1,1})δ(K′_{n−1,2}) ⋯ δ(K′_{n−1,p−1})δ(L) = δ(L) mod p,

where the last equation follows from Lemma 4.2.

From now on we suppose that α_{i} ≠ 0 for i = 1, …, n−1. Let Γ be a finite group, let Γ → Γ_{n−1} be an epimorphism and let Γ → Aut(S′, L, β) be a homomorphism. Thus Γ acts on all the forms (Spec k, H_{i}, α_{i}), i = 0, …, n − 1, and (S′, L, β).

Lemma 5.9. Suppose that (S′, L, β) is an admissable Γ-form, that all fixed points are k-rational and that each fixed point P ∈ S′ is twisting for the forms

(S′, H_{i}, α_{i}), i = 1, …, n − 1, and (S′, L, β).

Then for all r ≥ 0, (S_{r}, L_{r}, β_{r}) is an admissable Γ-form, all fixed points are k-rational, and each fixed point P ∈ S_{r} is twisting for the forms

(S_{r}, H_{i}, α_{i}), i = 1, …, n − 2, (S_{r}, L_{r−1}, β_{r−1}), and (S_{r}, L_{r}, β_{r}).

Proof. Let P ∈ S_{r} be a fixed point. By induction we may assume that P is krational and that

Γ → Aut(L_{r−2}|P, β_{r−2}|P) × Aut(L_{r−1}|P, β_{r−1}|P) × ∏_{i=1}^{n−2} Aut(H_{i}|P, α_{i}|P)

is surjective. We claim that

Γ → Aut(L_{r}|P, β_{r}|P) × Aut(L_{r−1}|P, β_{r−1}|P) × ∏_{i=1}^{n−2} Aut(H_{i}|P, α_{i}|P)

is surjective. Note that L_{r}|P = L_{r−2}|P ⊗ K_{n−1,r}|P. The claim follows now from the fact that Aut(L_{r−2}|P, α_{r−2}|P) acts trivially on K_{n−1,r}|P.

The remaining parts of the statement follow from Lemma 4.3.

Lemma 5.10. Suppose that S′ is irreducible. Let η_{r} ∈ S_{r} be the generic point. Then

{α_{1}, …, α_{n−2}, β_{r−1}(η_{r−1}), β_{r}(η_{r})} = (−1)^{r}{α_{1}, …, α_{n−2}, α_{n−1}, β(η_{0})}

in K_{n}^{M}k(S_{r})/p.

Proof. We show

{α_{1}, …, α_{n−2}, β_{r−1}(η_{r−1}), β_{r}(η_{r})} = {α_{1}, …, α_{n−2}, β_{r−1}(η_{r−1}), β_{r}(η_{r−2})}.

We have

β_{r}(η_{r}) = β_{r}(η_{r−2})Φ′_{n−1,r}.

The claim follows now from Lemma 4.4.

We will need the following special case:

Corollary 5.11.

{α_{1}, …, α_{n−2}, β_{p}(η_{p}), β_{p−1}(η_{p−1})} = {α_{1}, …, α_{n−2}, α_{n−1}, β(η_{0})}

in K_{n}^{M}k(S_{p})/p.

Remark 5.12. Let S′ = Spec k. We think of the symbol

{α_{1}, …, α_{n−2}, β_{p}(η_{p})}

as a family of symbols of weight n − 1 “between”

{α_{1}, …, α_{n−2}} and {α_{1}, …, α_{n−2}, α_{n−1}, β},

with S_{p} as parameter space.

Our later considerations indicate that this family is universal over p-special fields. For n = 2 we will make this precise, and for p = 2 this can be done using Pfister forms. I have no idea how to show this in general. In the case n = p = 3 the universality would have important consequences for the classification of groups of type F_{4}.

6. The forms K(α_{1}, …, α_{n}) (universal families of Kummer splitting fields)

Let n ≥ 1. Given forms (S, H_{i}, α_{i}), i = 1, …, n, we define forms

K_{i} = K_{i}(α_{1}, …, α_{n}) = (R_{i}/R_{i+1}, J_{i}, γ_{i}),  1 ≤ i ≤ n,
K′_{i} = K′_{i}(α_{1}, …, α_{n}) = (R_{i}/R_{i+1}, J′_{i}, γ′_{i}),  1 ≤ i ≤ n.

We put

(R_{n}/R_{n+1}, J_{n}, γ_{n}) = (S/S, H_{n}, α_{n})

and

(R_{n}/R_{n+1}, J′_{n}, γ′_{n}) = (S/S, O_{S}, τ)

with τ(t) = t^{p}.

Let i < n and suppose that K_{i+1} is defined.

Recall the forms

C_{r} = C_{r}(α_{1}, …, α_{i}, γ_{i+1}) = (S_{r}/S_{r−1}, L_{r}, β_{r})

defined in section 5. Let π: S_{p} → S_{p−1} be the projection.

We put

K_{i} =C_{p}(α_{1}, …, α_{i}, γ_{i+1}),
K′_{i} =π^{∗}C_{p−1}(α_{1}, …, α_{i}, γ_{i+1}).

We assume now that S = Spec k and list the most important properties of the forms (R_{i}/R_{i+1}, J_{i}, γ_{i}) and (R_{i}/R_{i+1}, J′_{i}, γ′_{i}).

Lemma 6.1. The variety R_{i} is smooth, proper, cellular, and of dimension p^{n} − p^{i}.

Proof. This follows from Lemma 5.1. For the dimension note

dim R_{i}/R_{i+1} = p^{i+1} − p^{i}, i < n

by Lemma 5.1.

Lemma 6.2. δ(J_{i}) = 1 mod p.

Proof. By Theorem 5.2 we have

δ(J_{i}) = δ(J_{i+1}) mod p.

Hence δ(J_{i}) = δ(J_{n}) = 1 mod p.

The construction of (R_{i}/R_{i+1}, J_{i}, γ_{i}) is functorial in the forms (S, H_{i}, α_{i}). In particular the group

Γ_{n} = μ_{p}^{n} ⊂ ∏_{i=1}^{n} Aut(S, H_{i}, α_{i})

acts on (R_{i}/R_{i+1}, J_{i}, γ_{i}) .

From now on we suppose that α_{i} ≠ 0 for i = 1, …, n.

Lemma 6.3. The forms (R_{i}/R_{i+1}, J_{i}, γ_{i}) are admissable Γ_{n}-forms, all fixed points are k-rational, and each fixed point P ∈ R_{i} is twisting for the forms

(R_{i}, H_{m}, α_{m}), m = 1, …, i − 1, and (R_{i}, J_{i}, γ_{i}).

Proof. This follows form Lemma 5.9.

Lemma 6.4. Let η_{i} ∈ R_{i} be the generic point. Then, for 1 ≤ i < n,

{α_{1}, …, α_{i−1}, γ_{i}(η_{i}), γ′_{i}(η_{i})} = {α_{1}, …, α_{i}, γ_{i+1}(η_{i+1})}

in K_{i+1}^{M}k(R_{i})/p.

Proof. This follows from Lemma 5.11.

In particular we have

(6.1)  {α_{1}, …, α_{n}} = {α_{1}, γ_{2}, γ′_{3}, …, γ′_{n}},
(6.2)  {α_{1}, γ_{2}} = {γ_{1}, γ′_{2}},
(6.3)  {α_{1}, …, α_{n}} = {γ_{1}, γ′_{2}, …, γ′_{n}}.

We write

(R, J, γ) = (R_{1}, J_{1}, γ_{1})

We denote by R̃ → R be the degree p “Kummer extension” corresponding to γ, defined locally by OR̃ = O_{R}[t]/(t^{p} − γ(λ)) where λ is a local nonzero section of J.

Corollary 6.5. The symbol {α_{1}, …, α_{n}} vanishes in the generic point of R̃.

Proof. This follows from Lemma 6.4 (see (6.3)).

7. Proof of the chain lemma

A splitting variety of a symbol is called p-generic, if it is a generic splitting variety over any p-special field.

Let Z be a p-generic splitting variety of {α_{1}, …, α_{n}} of dimension p^{n−1} − 1. We assume {α_{1}, …, α_{n}} ≠ 0. It follows that I_{Z} ⊂ pℤ.

Let (R, J, γ) be the form of defined at the end of section 6.

Note that Z has point of degree prime to p over k(R̃), hence has a k(R̃′)-rational point where R′/R is of degree prime to p. We have diagram of varieties covered by cyclic extensions of degree p:

R̃ ⟵^{ĝ} R̃′ ⟶^{f̂}  Z^{p}
↓  ↓  ↓
R ⟵^{g} R′ ⟶^{f} Cyclic^{p}(Z).

Let

R_{0} ⊂ R

be the zero locus of γ. Inspection shows that I(R_{0}) ⊂ pℤ. We have

η(R̃/R, R, R_{0}) = c_{1}(J)^{d} mod p = 1 mod p ≠ 0 mod p

by Lemma 6.2.

Let

R′_{0} =⊂ R′

be the subscheme of ramification of R̃′/R′. Then g(R′_{0}) ⊂ R_{0} and therefore I(R′_{0}) ⊂ pℤ. The degree formula tells that

η(R̃′/R′, R′, R′_{0}) = (deg g)^{−1} mod p ≠ 0 mod p.

Moreover let

Cyclic^{p}(Z)_{0} = Z ⊂ Cyclic^{p}(Z)

be the image of the diagonal. One has I(Cyclic^{p}(Z)_{0}) = pℤ. Further, Cyclic^{p}(Z)_{0} contains the subscheme of ramification of Z^{p}/ Cyclic^{p}(Z). Therefore f(R′_{0}) ⊂ Cyclic^{p}(Z)_{0}. The degree formula tells that

deg f ≠ 0 mod p.

Now let K = k(\sqrt[p]{b}) be a cyclic extension of degree p which splits \{α_{1}, …, α_{n}\}. We assume that k is p-special. It follows that there is a point Spec K → R̃ lying over a rational point P: Spec k → R. Then b = γ(P) in k^{∗}/(k^{∗})^{p}. It follows that

(7.1)  {α_{1}, …, α_{n}} = {α_{1}, γ_{2}(P), γ′_{3}(P), …, γ′_{n}(P)},
(7.2)  {α_{1}, γ_{2}(P)} = {b, γ′_{2}(P)},
(7.3)  {α_{1}, …, α_{n}} = {b, γ′_{2}(P), …, γ′_{n}(P)}.

(see (6.1)–(6.3) after Lemma 6.4).

We have proved:

Corollary 7.1. The chain lemma for cyclic algebras of degree p over p-special fields.

Corollary 7.2. The chain lemma for symbols (a, b, c) mod p over p-special fields.

Now let k(\sqrt[p]{b}), k(\sqrt[p]{c}) be two cyclic extensions of degree p which split the symbol {α_{1}, …, α_{n}}. Applying the last arguments twice, one finds first b_{i} ∈ k^{∗} such that

{α_{1}, …, α_{n}} = {b, b_{1}, b_{2}, …, b_{n}},

and then c_{i}, c′_{i} ∈ k^{∗} such that

{b, b_{1}, b_{2}, …, b_{n}} = {b, c_{1}, c_{2}, …, c_{n}},
{b, c_{1}} = {c, c′_{2}}.

Let X(b, c_{1}) be the Brauer-Severi variety associated to the symbol {b, c_{1}}. It has rational points over k(\sqrt[p]{b}) and over k(\sqrt[p]{c}). Morover, since Z is a p-generic spliting field, we have a correspondence X(b, c_{1}) → Z lying over ℤ → ℤ of degree prime to p.

Corollary 7.3. Let x, y ∈ Z be points of degree p and let α ∈ κ(x)^{∗}, β ∈ κ(y)^{∗}. Then there exist z ∈ Z of degree p and γ ∈ κ(z)^{∗}, such that

[α] + [β] = [γ] in A_{0}(Z, K_{1}).

Proof. By the previous considerations, and using that CH_{0}(Z_{K}) = ℤ whenever Z(K) ≠ ∅, we may reduce to the case of Brauer-Severi variety. In this case the statement is known [1].

Remark 7.4. In the last proof we assumed CH_{0}(Z_{K}) = ℤ whenever Z(K) ≠ ∅. This can be shown for n = 3 for Z the usual SL(p)-form.

Without this assumption, we get at least the last corollary with A_{0}(Z, K_{1}) replaced by

coker A_{0}(Z^{2}, K_{1}) → A_{0}(Z, K_{1}),

the group considered in my MSRI-talk.

References

[1] A. S. Merkurjev and A. A. Suslin, The group of K_{1}-zero-cycles on Severi-Brauer varieties, Nova J. Algebra Geom. 1 (1992), no. 3, 297–315.

NWF I - Mathematik, Universität Regensburg, D-93040 Regensburg, Germany

E-mail address: markus.rost@mathematik.uni-regensburg.de
URL: http://www.physik.uni-regensburg.de/~rom03516