CONSTRUCTION OF SPLITTING VARIETIES

MARKUS ROST

Contents

1. Preliminaries, Conventions, and Notations 2 2. Consequences of Voevodsky’s work 4 2.1. Voevodsky’s theorem 4 2.2. A degree formula for s_{d}(X) 4 2.3. On a higher degree formula 5 3. The Conner-Floyd theorem: computing s_{d}(X) 7 4. The forms A(α_{1}, …, α_{n}) (“algebras”) 8 5. The forms B(α_{1}, …, α_{n}) (“relative algebras”) 11 6. The forms C(α_{1}, …, α_{n}) (Chain lemma construction) 14 7. The forms K(α_{1}, …, α_{n}) (universal families of Kummer splitting fields) 19 8. Construction of a norm variety via chain lemma 21 9. Multiplicativity 25 10. On the norm principle 27 References 30

Date: April, 1998.

1. 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).

2. Consequences of Voevodsky’s work

In this paper p always is a prime, k is a field with char k = 0 and K_{n}^{M}k denotes Milnor’s n-th K-group of k. Let

h_{(n,p)} : K_{n}^{M}k/p → H_{ét}^{n}(k, μ_{p}^{⊗n}),
{a_{1}, …, a_{n}} ↦ (a_{1}, …, a_{n}).

be the norm residue homomorphism.

2.1. Voevodsky’s theorem. V. Voevodsky announced in October 1996 the following theorem:

Theorem (Voevodsky). Let p be a prime and let m be a natural number.

Suppose that for every subfield k ⊂ ℂ containing the p-th roots of unity and for every sequence of elements a_{1}, …, a_{n} ∈ k^{∗}, 2 ≤ n ≤ m, there exists a smooth projective variety X over k such that:

(V_{1}) {a_{1}, …, a_{n}}_{k(X)} = 0 in K_{n}^{M}k(X)/p.

(V_{2}) X has dimension d = p^{n−1} − 1.

(V_{3}) In the category of Chow motifs over k(X) with ℤ_{(p)}-coefficients there exist an effective object Y such that

X_{k(X)} = L^{⊗0} ⊕ (Y ⊗ L).

Here L denotes the Tate motive. (V_{4}) The characteristic number s_{d}(X(ℂ)) ∈ ℤ is not divisible by p^{2}. (V_{5}) The sequence

∐_{x∈X_{(1)}} K_{2}κ(x) ⟶^{d} ∐_{x∈X_{(0)}} K_{1}κ(x) ⟶^{N_{X}} K_{1}k

is exact. Here N_{X} = ∑ N_{κ(x)|k}.

Then one has: (BK) The Bloch-Kato conjecture holds in weight m and mod p, i.e., the norm

residue homomorphism h_{(m,p)} is bijective. (for all fields k with char k ≠ p )

(S) For n ≤ m, for elements a_{1}, …, a_{n} ∈ k^{∗}, and for a smooth projective variety X satisfying (V_{1})–(V_{5}), the sequence

∐_{x∈X_{(0)}} K_{1}κ(x) ⟶^{N} K_{1}k ⟶^{b ↦(a_{1},...,a_{n},b)} H_{ét}^{n+1}(k, μ_{p}^{⊗(n+1)})

is exact.

2.2. A degree formula for s_{d}(X). We fix a prime p be a prime and a number d of the form d = p^{n} − 1.

Let X, Y be irreducible smooth proper varieties over k with dim Y ≤ dim X = d and let f: X → Y be a morphism. Define deg f as follows: If dim f(X) < dim X, then deg f = 0. Otherwise deg f ∈ ℕ is the degree of the extension of the function fields:

f_{∗}([X]) = deg f · [Y].

Theorem 1 (“Degree formula”).

(s_{d}(X)/p) = (deg f)(s_{d}(Y)/p) mod I_{Y}

This is a consequence of algebraic cobordism theory. One uses the spectrum Φ considered in [4].

Corollary 2. The class

s_{d}(X)/p mod I_{X} ∈ ℤ/I_{X}

is a birational invariant

2.3. On a higher degree formula. All of what I am saying in the next lines are mainly guesses from my poor knowledge of Morava K-theories and algebraic cobordism. Everything has to be checked.

Let Φ_{r} be the Φ-construction of [4], iterated r-times, i.e., Φ_{r} is a tower consisting of Σ^{2id,id}H_{ℤ/p}, i = 0, …, r with all intermediate towers of length 2 being a suspension of Φ. Then the Thom class lifts to MU → Φ_{r} and for X of dimension rd we have a fundamental class

[X] ∈ π_{2rd,rd}(X ∧ Φ_{r}).

Define t(X) ∈ ℤ/p as the image of [X] in

π_{2rd,rd}(Spec k ∧ Φ_{r}) = π_{2rd,rd}(Spec k ∧ Σ^{2rd,rd}H_{ℤ/p}) = ℤ/p

From the known structure of Morava K-theories it follows that (perhaps up to multiplication with a number prime to p)

t(X_{1} × X_{2} × ⋯ × X_{r}) = (s_{d}(X_{1})/p)(s_{d}(X_{2})/p) ⋯ (s_{d}(X_{r})/p) mod p

if dim X_{i} = d.

Furthermore, let Ψ be the fibre of Φ_{r} → H_{ℤ/p} and define J(X) ⊂ ℤ as the image of

π_{2rd,rd}(X ∧ Ψ) → π_{2rd,rd}(Spec k ∧ Ψ) = π_{2rd,rd}(Spec k ∧ Σ^{2rd,rd}H_{ℤ/p}) = ℤ/p

Note that Ψ = Σ^{2d,d}Φ_{r−1}.

Then the “higher degree formula” is

t(X) = t(Y)(deg f) mod J(Y) (1)

for any f: X → Y with X, Y, smooth proper of dimension rd. It should be possible to show this in the same way as for the degree formula for s_{d}/p.

Moreover, one should have

(2)  J(X) = J(Y),  if deg f is prime to p

by a transfer argument.

I guess that the following is true:

Let X_{i}, i = 1, …, r be of dimension d and suppose that I_{(X_{i}_{F_{i}})} ⊂ pℤ for all i, where

F_{i} = k(X_{1} × ⋯ × X̂_{i} × ⋯ × X_{r}).

Let X = X_{1} × ⋯ × X_{r}. Then

J(X) ⊂ pℤ (3)

In the case of curves (d = 1 = 2^{1} − 1) one has (?)

J(X) = (π_{X})_{∗}(K_{0}(X)^{(1)})

where

K_{0}(X)^{(1)} = ker(K_{0}(X) → CH^{0}(X))

and π_{X}: X → Spec k is the structure map for X.

In this case (3) is not difficult to show:

Proof for d = 1 and r = 2: One has an exact sequence

∐_{x∈X_{1}_{(0)}} K_{0}(X_{2}_{κ(x)}) → K_{0}(X_{1} × X_{2})^{(1)} → K_{0}(X_{2}_{k(X_{1})})^{(1)} → 0

Push forward along π′: X_{1} × X_{2} → X_{1} maps this sequence into the sequence

CH_{0}(X_{1}) → K_{0}(X_{1}) → K_{0}(k(X_{1})) → 0

Since the index of X_{2}_{k(X_{1})} is 2-divisible, we see that

π′_{∗}(K_{0}(X_{1} × X_{2})^{(1)}) ⊂ CH_{0}(X_{1}) + 2K_{0}(k(X_{1}))

The claim (3) follows since I_{X_{1}} is 2-divisible.

It should be possible to extend this reasoning to the general case (?).

In my application one has r = p and the X_{i} are of the following type: Let a_{m} ∈ k_{0}^{∗} be such that {a_{1}, …, a_{n}} is a nontrivial symbol, let k = k_{0}(t_{1}, …, t_{p}) and let X_{i}/k be a norm variety for the symbol {a_{1}, …, a_{n}, t_{i}}. We may take X_{i} to be defined over k_{0}(t_{i}). Note that then each of the fields k_{0}(t_{i})(X_{i}) has a k_{0}-place, hence the field F_{i} has a k_{0}(t_{i})-place, whence {a_{1}, …, a_{n}, t_{i}} is nontrivial over F_{i}. Therefore the index of X_{i}_{F_{i}} is p.

3. The Conner-Floyd theorem: computing s_{d}(X)

In this section we indicate how one can get information about s_{d}(X) from a (ℤ/p)^{n−1}-action on X with isolated fixed points.

We assume that p is odd and k ⊂ ℂ. For odd p the Chern number s_{d}(X) = s_{d}(T X) of a complex variety is also an Pontrjagin number of the underlying differentiable manifold M = X(ℂ). Therefore the number s_{d}(X) can be computed in terms of the class [M] of M in the oriented cobordism ring.

In order to compute this number for certain norm varieties we use the following theorem of Conner and Floyd: ( [2]).

Theorem 3. Let d = p^{n}−1, let G = (ℤ/p)^{n}, and let M be an oriented differentiable manifold. Suppose that there exist a fixed point free G-action on M. Then the class of M in the oriented cobordism ring Ω_{∗} lies in the ideal I_{n−1,p} generated by Milnor the base elements M_{0,p} = p · Point, M_{1,p}, …, M_{n−1,p} (dim M_{i,p} = p^{i} − 1).

Corollary 4. Let d = p^{n} − 1, let G = (ℤ/p)^{n}, and let M be an oriented differentiable manifold of (real) dimension 2d. Suppose that there exist an fixed point free G-action on M. Then s_{d}(M) is divisible by p^{2}.

Proof. This follows from s_{d}(M) ∈ pℤ for all M of dimension 2d and s_{d}(M_{1} ×M_{2}) = 0 if d > dim X_{i} > 0.

Using canonical desingularization [1] and the Conner-Floyd theorem one finds:

Corollary 5. Let p be odd, let X, Y be proper varieties with Y smooth. Suppose that G = (ℤ/p)^{n} acts on X and Y such that the fixed point schemes F_{X} and F_{Y} are of dimension 0 and suppose that F_{X} ⊂ X_{reg}. Suppose further that the families of G-representations (T_{P} X)_{P ∈F_{X}(ℂ)}, (T_{P} Y)_{P ∈F_{Y} (ℂ)} are isomorphic. Then there exist a smooth proper variety X̃ together with a birational isomorphism X̃ → X such that X̃(ℂ) and Y (ℂ) represent the same element in Ω_{∗}/I_{n−1,p}.

In particular, if dim X = dim Y = d = p^{n} − 1, then

s_{d}(X̃) = s_{d}(Y) mod p^{2}

This consequence is extremely useful to compute the birational invariant of Corollary 2.

Proof. By canonical desingularization [1] we may assume that X is smooth. Let Z be the multifold connected sum of the differentiable manifolds X(ℂ) and −Y (ℂ), build by glueing together pairs of fixed points with isomorphic G-normal structures. Since S^{2d} and S^{1} × S^{2d−1} are bordant, one has [Z] = [X] − [Y] for the cobordism classes. On Z we have a fixed point free G-action, and the Conner-Floyd theorem shows [Z] ∈ I_{n−1,p}.

If the families of G-representations (T_{P} X)_{P ∈F_{X}(ℂ)}, (T_{P} Y)_{P ∈F_{Y} (ℂ)} are isomorphic, we say that X and Y are G-fixed point equivalent.

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

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 7. 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 8 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 8. 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 9. (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 10. 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 1. 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.

5. 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 4. 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 11. 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 6.

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 12. 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 8)

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 13. 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 9.

Lemma 14. 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 10.

Remark 2. 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 12 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).

6. 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 5, 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 15. 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 11. 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 11.

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 16. 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 17. 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 18. 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 19. u^{p^{2}} = u^{p} for all u ∈ R_{p}(0, 1).

Corollary 20. 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 17. Moreover, by Corollary 19, 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 19 and 20 yield the following two corollaries:

Corollary 21. u^{p^{2}} = u^{p}x^{p^{2}−p} for all u ∈ R′.

Corollary 22. 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 16: 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 12.

We have a homomorphism

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

It follows from Corollary 22 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 12.

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 23. 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 13.

Lemma 24. 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 14.

We will need the following special case:

Corollary 25.

{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 3. 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}.

7. 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 6. 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 26. The variety R_{i} is smooth, proper, cellular, and of dimension p^{n} − p^{i}.

Proof. This follows from Lemma 15. For the dimension note

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

by Lemma 15.

Lemma 27. δ(J_{i}) = 1 mod p.

Proof. By Theorem 16 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 28. 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 23.

20 MARKUS ROST

Lemma 29. 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 25. In particular we have

(4)  {α_{1}, …, α_{n}} = {α_{1}, γ_{2}, γ′_{3}, …, γ′_{n}},
(5)  {α_{1}, γ_{2}} = {γ_{1}, γ′_{2}},
(6)  {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 30. The symbol {α_{1}, …, α_{n}} vanishes in the generic point of R̃. Proof. This follows from Lemma 29 (see (6)).

8. Construction of a norm variety via chain lemma

We fix a p-th root of unity ζ ≠ 1.

Let (R, J, γ) be the form of defined at the end of section 7 and let G = Γ_{n}. Moreover let A = A(R, J, γ) be the associated algebra bundle, with norm

N_{A}: A → O_{R}.

Let b ∈ k^{∗}. We define the variety

X = X_{b} = { [x, t] ∈ ℙ(A ⊕ O_{R}) | N_{A}(x) = bt^{p} }.

The G-action extends to a G-action on A, A ⊕ O_{R}, ℙ(A ⊕ O_{R}), and X.

Proposition 31. The variety X has the following properties:

(1) X is proper of dimension d = dim X = p^{n} − 1.

(2) One has

{α_{1}, …, α_{n}, b}_{k(X)} = 0 in K_{n+1}^{M}k(X)/p

(3) The fixed point scheme F_{X} of the G-action on X is a smooth 0-dimensional subscheme of X contained in the smooth part of X.

(4) There exist a proper smooth G-variety Y such that a) X and Y are G-fixed point equivalent.

b) s_{d}(Y) ≢ 0 mod p^{2}.

Proof. (1) follows from Lemma 26 and (2) follows from Corrollary 30. For the variety Y we take Y = ℙ(A) with the natural G-action.

Proof of 4a): We have the map

X ⟶^{π} Y, π([x, t]) = [x]

The map π is a branched covering of degree p. It should be noted that the map π seems to have no real significance for the applications of the proposition, however it turns out to be useful to compare the fixed points of X and Y.

To compute the fixed point sets F_{X} and F_{Y} we first note that both lie over F_{R}. For each P ∈ F_{R}(k_{sep}) let G_{P} = μ_{p} ⊂ Aut(J|P) and let X_{P}, Y_{P} be the fibres of X resp. Y over P.

Recall that P is twisting by Lemma 28. Therefore the homomorphism

G → G_{P}

is surjective. From this one sees that all fixed points in X are contained in the smooth locus of X.

For x ∈ F_{X} ∩ X_{P}, y ∈ F_{Y} ∩ Y_{P} one has G-equivariant decompositions

T_{x}X = T_{x}X_{r} ⊕ T_{P} R
T_{y}Y = T_{y}Y_{r} ⊕ T_{P} R.

Therefore, in order to prove 4a), it suffices to show that for each r ∈ F_{R}(k_{sep}) the fibres X_{r} and Y_{r} are G_{P} -fixed point equivalent. Moreover, we may assume that k = k_{sep}. Hence we are reduced to the case n = 1, G = μ_{p}, R = Spec k, J = k, γ(λ) = λ^{p}, and b = 1.

In this case the G-fixed points of X resp. Y are

[ζ^{i}, 1, …, 1]  0 ≤ i ≤ p − 1  (for X)
[0, …, 0, 1_{i}, 0, …, 0]  0 ≤ i ≤ p − 1  (for Y)

with respect to the coordinates

A ⊕ k = k ⊕ L ⊕ ⋯ ⊕ L^{⊗p−1} ⊕ k = k^{p+1}

resp.

A = k ⊕ L ⊕ ⋯ ⊕ L^{⊗p−1} = k^{p}.

The fixed points of Y have all the same tangential G-structure, since the cyclic permutation of the coordinates on Y = ℙ(A) commutes with the G-action. Moreover the map π: X → Y induces isomorphisms between the tangent spaces at the fixed points of X and the fixed point [1, 0, …, 0] of Y. Hence X and Y have both p fixed points, with all having the same tangential G-structure. (Of course this can be verified also directly by computing the tangential G-structures: they are all isomorphic to the sum of the p − 1 irreducible representations of G.)

This proves 4a) and along the way we have also seen (3).

Proof of 4b): The tangent bundle of Y decomposes (in K_{0}(Y)) as the sum of the tangent bundle T R of R and the fibre tangent bundle T (Y/R) of the projection π_{A}: Y = ℙ(A) → R. Hence

s_{d}(T Y) = s_{d}(π_{A}^{∗}(T R)) + s_{d}(T (Y/R)).

We have

s_{d}(π_{A}^{∗}(T R)) = π_{A}^{∗}(s_{d}(T R)) = 0

since dim R < d. Moreover

s_{d}(T (Y/R)) = s_{d}(π_{A}^{∗}(A) ⊗ 𝕃(A)^{∨} − O_{R})

= s_{d}(⨁_{i=0}^{p−1} J^{⊗i} ⊗ 𝕃(A)^{∨})

= ∑_{i=0}^{p−1} s_{d}(J^{⊗i} ⊗ 𝕃(A)^{∨})

= ∑_{i=0}^{p−1}(c_{1}(J^{⊗i} ⊗ 𝕃(A)^{∨}))^{d}.

We put

x = −c_{1}(J) ∈ CH^{1}(R)
y = −c_{1}(𝕃(A)) ∈ CH^{1}(Y).

Then

CH^{∗}(Y) = CH^{∗}(R)[y]/(∏_{i=0}^{p−1}(y − ix))

= ⨁_{i=0}^{p−1} y^{i} CH^{∗}(R)

(by the computation of the Chow ring of projective bundles) and

s_{d}(T Y) = ∑_{i=0}^{p−1}(y − ix)^{d}.

In the ring

ℤ[x, y]/(∏_{i=0}^{p−1}(y − ix))

write

∑_{i=0}^{p−1}(y − ix)^{d} = ∑_{i=0}^{p−1} a_{i}y^{i}x^{d−i}, a_{i} ∈ ℤ.

Since d − i = dim R + p − 1 − i, we have

s_{d}(T Y) = a_{p−1}y^{p−1}x^{dim R}

Moreover (π_{A})_{∗}(y^{p−1}) = [R] ∈ CH^{0}(R). Hence

s_{d}(Y) = deg(s_{d}(T Y)) = a_{p−1} deg((−c_{1}(J))^{dim R}).

The claim follows now from δ(J) = 1 mod p (Lemma 27) and from the following Lemma 32.

Lemma 32. Let p be a prime, Z = ℤ/p^{2}, and let

S = Z[y]/(∏_{i=0}^{p−1}(y − i)).

For u ∈ S define a_{i}(u) ∈ Z by

u = ∑_{i=0}^{p−1} a_{i}(u)y^{i}.

For n ≥ 1 let

u_{n} = ∑_{i=0}^{p−1}(y − i)^{p^{n}−1} ∈ S.

Then a_{p−1}(u_{n}) = p.

Proof. One easily sees a_{p−1}(u_{1}) = p. We show that u_{n} does not depend on n. The homomorphism

Φ: S → ∏_{i=0}^{p−1} Z

y ↦ (0, 1, 2, …, p − 1)

is an isomorphism of rings. Hence it suffices to show that Φ(u_{n}) does not depend on n.

This means that for each j = 0, …, p − 1 the residue class

∑_{i=0}^{p−1}(j − i)^{p^{n}−1} mod p^{2}

is independent of n. In fact, for any integer h one has h^{p^{n}−1} = h^{(p−1)} mod p^{2}. This is obvious if h = 0 mod p (if p ≠ 2). Otherwise h^{p−1} = 1 mod p and h^{(p−1)p} = 1 mod p^{2}. Then

h^{p^{n}−1} = h^{(p−1)(1+p+···+p^{n−1})} = h^{(p−1)} mod p^{2}.

24 MARKUS ROST

If p = 2, then

S = Z[y]/(y^{2} − y)

and the claim is easy to check.

9. Multiplicativity

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 7.

We have diagram of varieties

𝔸^{1}  𝔸^{1}  𝔸^{1}
N_{A}↑  ^{N_{A′}} ↑  ↑^{mult}
𝔸(A) ⟵^{ḡ} 𝔸(A′) ⟶^{f̄} Cyclic^{p}(Z × 𝔸^{1})
↓  ↓  ↓
R ⟵^{g} R′ ⟶^{f} Cyclic^{p}(Z)

Here g is of degree prime to p and f is a morphism. The maps f, g come from the fact 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. This point defines the map f. The maps f, g are covered by the cyclic extensions of degree p:

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

with R̃′ = R̃_{R}R′.

They are also covered by line bundles:

J ⟵^{ĝ} J′ ⟶^{f̂} (Z^{p} × \overline{𝔸^{1}})/(ℤ/p)
↓  ↓  ↓
R ⟵^{g} R′ ⟶^{f} Cyclic^{p}(Z).

with \overline{𝔸^{1}} ⊂ 𝔸^{p} the image of 𝔸^{1} → 𝔸^{p}, t ↦ t(1, ζ, ζ^{2}, …, ζ^{p−1}) with 1 ≠ ζ ∈ μ_{p}. This diagram induces the maps f̄, ḡ.

Note that deg ḡ = deg g and deg f̄ = deg f.

The fibre X_{t} of N_{A} over the generic point Spec k(t) is a splitting variety of the symbol

{α_{1}, …, α_{n}, t}.

One has I_{X_{t}} = pℤ, since {α_{1}, …, α_{n}, t} ≠ 0.

We assume now p ≠ 2. By Proposition 31 (4), Corollary 5, and Corollary 2, one finds that the birational invariant of X_{t} in

ℤ/I_{X_{t}} = ℤ/p

is nonzero. Since deg ḡ is prime to p, the degree formula implies that the fibre X′_{t} of N_{A′} has nontrivial invariant. The degree formula shows then that deg f̄ is prime to p, hence deg f is prime to 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)  {α_{1}, …, α_{n}} = {α_{1}, γ_{2}(P), γ′_{3}(P), …, γ′_{n}(P)},
(8)  {α_{1}, γ_{2}(P)} = {b, γ′_{2}(P)},
(9)  {α_{1}, …, α_{n}} = {b, γ′_{2}(P), …, γ′_{n}(P)}.

(see (4)–(6) after Lemma 29).

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 33. 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 [3].

Corollary 34. If k is p-special, then the set V(N_{A}) of values ≠ 0, ∞ of the map

N_{A}: 𝔸(A)(k) → k

is a multiplicative subgroup of k^{∗}.

10. On the norm principle

We fix a p-th root of unity ζ ≠ 1.

Let (Spec k, I, ϵ) be a nonzero form and let K = A(Spec k, I, ϵ) be the associated algebra.

Let (R, J, γ) be the form defined at the end of section 7, let G = Γ_{n}, and let A = A(R, J, γ) be the associated algebra bundle, with norm

N_{A}: A → O_{R}.

We denote by

N_{A_{K}}: A ⊗ K → O_{R} ⊗ K

the induced map of degree p.

Let B = O_{R} ⊕ A ⊗ I. We have a natural inclusion B → A ⊗ K. Let

M: B → K ⊗_{k} O_{R}

be the restriction of N_{A_{K}} to B.

Let (B_{i}, R_{i}, M_{i}), i = 1, …, p − 1 be copies of (B, R, M). We put

U = ℙ(A) × ∏_{i=1}^{p−1} ℙ(B_{i}),

L = 𝕃(A) ⊠ ⊠_{i=1}^{p−1} 𝕃(B_{i}),
V = L ⊕ O_{U}.

Then dim U = p^{n} − 1 + (p − 1)p^{n} = p^{n+1} − 1 and L is a line bundle on U and V is a 2-dimensional vector bundle on U.

Let ω ∈ K^{∗}. On V we define a K-valued form

Θ: V → O_{U} ⊗ K

of degree p by

Θ_{ω}(u ⊗ ⊗_{i=1}^{p−1}u_{i} + t) = N_{A}(u)M(u_{1}) ⋯ M(u_{p−1}) − ωt^{p},

where u, u_{i}, t are sections of 𝕃(A), 𝕃(B_{i}), O_{U}, respectively.

We define the variety

\overline{X}_{ω} = \{ [x] ∈ ℙ(V) | Θ_{ω}(x) = 0 \}.

Proposition 35. There exist an open dense subset Ω ⊂ 𝔸(K) such that the variety \overline{X} = \overline{X}_{ω} has the following properties:

(1) d = dim X = p(p^{n} − 1).

(2) One has

\{α_{1}, …, α_{n}, ω\}_{k(\overline{X})} = 0 in K_{n+1}^{M}K(\overline{X})/p

(3) The fixed point scheme F\overline{X} of the G-action on \overline{X} is a smooth 0-dimensional subscheme of \overline{X}.

(4) There exist a proper smooth G-variety \overline{Y} such that a) \overline{X} and \overline{Y} are G-fixed point equivalent.

b) \overline{Y} is the union of (p−1)! copies of Y^{p}, where Y is a variety of dimension d = p^{n} − 1 with s_{d}(Y) ≢ 0 mod p^{2}.

Proof. (1) and (2) are obvious from former considerations (for all ω ∈ K^{∗}).

We have to determine the fixed points on X. Any fixed point lies over a fixed point P ∈ R^{p}. There are only finitely many such P, and any P is twisting for the bundles (Spec k, H_{i}, α_{i}), i = 1, …, n. On ℙ(A|P) there are exactly p fixed points. The fixed point scheme of ℙ(B|P) ≃ ℙ^{p} consists of a 1-dimensional component

P = ℙ(k ⊕ I ⊕ 0 ⊕ ⋯ ⊕ 0) ≃ ℙ^{1}

and p − 1 isolated fixed points

ℙ(0 ⊕ 0 ⊕ 0 ⊕ ⋯ ⊕ J^{⊗i} ⊗ I ⊕ ⋯ ⊕ 0)

Hence any of the fixed point components C in U|P is a product

C = C_{0} × C_{1} × ⋯ × C_{p−1}

with C_{0} a point, and, for i > 0, C_{i} a point or ≃ ℙ^{1}.

Let s = dim C. If s < p − 1, then the equation Θ_{ω}(x) = 0 has no solution at all, provided ω is generic. If s = p − 1 and

C_{0} = [0, …, 𝕃(A)^{⊗i}, …, 0]

for some i > 0, then ℙ(V)|C has only two fixed points, which do not lie in X. Hence fixed points appear only if

C = [1, 0, …, 0] × C_{1} × ⋯ × C_{p−1}

with C_{i} ≃ ℙ^{1}. One finds that there exactly (p − 1)!p^{p} fixed points (for generic ω). For this use the following facts:

Let

Φ: (ℙ^{1})^{p−1} → ℙ^{p−1}
Φ(([x_{i} + ty_{i}])_{i=1,...,p−1}) = [∏_{i}(x_{i} + ty_{i})]

be the “morphism of the fundamental theorem of algebra”. Φ is a finite morphism of degree (p − 1)!. Let

Φ̂: (ℙ^{1})^{p−1} → ℙ^{p−1}
Φ̂(([x_{i} + ty_{i}])_{i=1,...,p−1}) = [∏_{i}(x_{i} + ty_{i})^{p}].

Φ̂ is a finite morphism of degree (p − 1)!p^{p−1}.

Hence there are exactly (p − 1)!p^{p} fixed points over P. These are given by just (p − 1)! copies of the fixed point set of ℙ(A|P)^{p}. The characteristic number of Y = ℙ(A) has been computed in section 8.

We assume {α_{1}, …, α_{n}}_{K} ≠ 0. Let ω be the generic element of 𝔸(K). Then {α_{1}, …, α_{n}, ω} ≠ 0.

Let further T = N_{A}^{−1}(ω) (the splitting variety over K for {α_{1}, …, α_{n}, ω}, constructed from the family (R, J, γ) of Kummer splitting fields of {α_{1}, …, α_{n}}).

Let R_{K/k}(T) be the transfer of T. We claim the R_{K/k}(T) has a point of degree prime to p over k(X_{ω}).

In fact, by applying Corollary 34 to the ground field K(X_{ω}), we see that there exist an extension H/K(X_{ω}) of degree prime to p, such that ω is a value of N_{A} over H. But then there exist an extension H′/k(X_{ω}) of degree prime to p, such that ω is a value of N_{A} over H′ ⊗ K.

Writing H′ = k(W) we get maps

X_{ω} ⟵^{g} W ⟶^{f} R_{K/k}(T)

with (deg g, p) = 1.

We would like to conclude that (deg f, p) = 1. To compute deg f we may do base change k → K, so that K = k × ⋯ × k. Then the diagram looks as

X_{ω} ⟵^{g} W ⟶^{f} \overline{T} := ∏_{i=1}^{p} N_{A}^{−1}(ω_{i})

with ω = (ω_{1}, …, ω_{p}), and ω_{i} ∈ k are p independent generic elements. (deg f, p) = 1 follow from Proposition 35 (4) and the “higher degree formula”.

Let us take it for granted, so that deg f is prime to p. Extension from the generic point of 𝔸(K) to 𝔸(K) provides a diagram

𝔸(K)  𝔸(K)  𝔸(K)
N_{A}M_{1}...M_{p−1}↑  ↑  ↑^{R_{K/k}(N_{A})}
𝔸(L) ⟵^{ḡ} \overline{W} ⟶^{f̄} R_{K/k}(𝔸(A))

with (deg ḡ, p) = (deg f̄, p) = 1.

Assume now that k is p-special.

Let x ∈ Z_{K} be a point of degree p and let δ ∈ κ(x)^{∗}. Our aim is to show that

N_{K/k}([δ]) ∈ A_{0}(Z, K_{1})

is represented by a sum of elements concentrated in points of Z of degree p (over k). If the symbol is split over K, this is easy to check (assuming CH_{0}(Z_{K}) = ℤ, when K is a splitting field). So we may assume {α_{1}, …, α_{n}}_{K} ≠ 0.

Let ω = N_{κ(x)/K}(δ) ∈ K^{∗}. By multiplying ω with some p-th power in (K^{∗})^{p}, we may arrange that ω lies in any given open dense subset of 𝔸(K).

As we have seen, there is a K-rational point P ∈ R such that

(R, J, γ)|P

represents the Kummer extension κ(x)/K. Hence ω is a value of N_{A} over K. Hence ω is in the image of R_{K/k}(𝔸(A)) under R_{K/k}(N_{A}). Since (deg f̄, p) = 1, ω is also in the image of 𝔸(L) under N_{A}M_{1} … M_{p−1}.

The p projections 𝔸(L) → R (via ℙ(A), ℙ(B_{i})) give us p points in R, whence p points z_{i} ∈ Z of degree p. Furthermore we have elements δ_{i} ∈ κ(z_{i}) such that

δ = N_{κ(z_{0})/k}(δ_{0})N_{K⊗κ(z_{1})/K}(1 + z_{1}\sqrt[p]{ϵ}) ⋯ N_{K⊗κ(z_{p−1})/K}(1 + z_{p−1}\sqrt[p]{ϵ})

in K^{∗}.

By multiplicativity we see that

[δ] = [δ_{0}]_{K} + [1 + z_{1}\sqrt[p]{ϵ}] + ⋯ + [1 + z_{p−1}\sqrt[p]{ϵ}].

in A_{0}(Z_{K}, K_{1}). But then

N_{K/k}([δ]) = p[δ_{0}] + [N_{K⊗κ(z_{1})/κ(z_{1})}(1 + z_{1}\sqrt[p]{ϵ})] + …

is concentrated in the points z_{i}. ???

References

[1] E. Bierstone and P. D. Milman, Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math. 128 (1997), no. 2, 207–302.

[2] E. E. Floyd, Actions of (Z_{p})^{k} without stationary points, Topology 10 (1971), 327–336.

[3] 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.

[4] V. Voevodsky, The Milnor conjecture, preprint, Max-Planck-Institute for Mathematics, Bonn, 1996, http://www.math.uiuc.edu/K-theory/0170, 1996.