ON THE BASIC CORRESPONDENCE OF A SPLITTING VARIETY

MARKUS ROST

Contents

Preface 1 1. No Introduction 2 2. Definitions and a brief introduction 2 3. Expressing c(X) 6 4. A computation for the Q_{i} 12 5. Generalities for special correspondences 13 6. Existence of generic splitting varieties 16 7. Generalities for special correspondences II 19 8. Computing c(X) 27 9. Relations with characteristic numbers 34 References 42

Preface

This text should be considered as private notes, made public only to have a reference for discussions on the subject.

The technical ad hoc computations of Section 8 are obsolete; the computation of c(X) can be done in a much simpler way.

Date: September–November, 2006.

1. No Introduction

Looking for an introduction? See Remark 2.5.

2. Definitions and a brief introduction

Let p be a prime and let k be a field with char k ≠ p.

Moreover let n ≥ 2. We are going to discuss some part of the proof of the Bloch-Kato conjecture in weight n in [11].

For simplicity we will assume throughout char k = 0 and μ_{p} ⊂ k. Moreover for p = 2 we will often assume that −1 is a square.

Let B be a smooth variety over k and let X → B be a smooth morphism. We denote by X (X/B) the simplicial scheme

X (X/B) : X \overleftarrow{←} X ×_{B} X ← ; ← ; ← X ×_{B} X ×_{B} X ⋯

For the following we refer to [11, Proof of Lemma 6.5].

Lemma 2.1. Assume the Bloch-Kato conjecture holds in weights ≤ n − 1. Then for any smooth simplicial scheme X the sequence

0 → H^{n,n−1}(X , Z/p) → H_{et}^{n}(X , μ_{p}^{⊗(n−1)}) → H^{0}(X , \underline{H}_{et}^{n}(μ_{p}^{⊗(n−1)}))

is exact. Here the first map is given by changing from Nisnevich to etale topology and the second map is induced from the associated spectral sequence. □

Corollary 2.2. Assume the Bloch-Kato conjecture holds in weights ≤ n − 1. Let X → B be as above and surjective. Then there is a natural exact sequence

H^{n,n−1}(X (X/B), Z/p) ⊂ H_{et}^{n}(B, μ_{p}^{⊗(n−1)}) → ∐_{α} H_{et}^{n}(k(X_{α}), μ_{p}^{⊗(n−1)})

where the X_{α} are the components of X.

Proof. Since X → B has sections on an etale covering of B, it follows that

X (X/B) → X (B/B)

induces an isomorphism of the etale topological types. Hence X (X/B) and B have the same etale cohomology.

Moreover,

H^{0}(X , F) ⊂ H^{0}(X_{0}, F) = H^{0}(X, F) ⊂ ∐_{α} H^{0}(k(X_{α}), F )

as in [11, Proof of Lemma 6.5].

Now Lemma 2.1 yields the stated exact sequence. □

The Milnor operations

Q_{i}: H^{r,s}(−, Z/p) → H^{r+2p^{i}−1,s+p^{i}−1}(−, Z/p)

are defined by Q_{0} = Bockstein and Q_{i+1} = [P^{p^{i}}, Q_{i}] where

P^{h}: H^{r,s}(−, Z/p) → H^{r+2h(p−1),s+h(p−1)}(−, Z/p)

are the Steenrod operations in motivic cohomology (we assume that −1 is a square if p = 2). Let further

Q̃_{0} : H^{r,s}(−, Z/p) → H^{r+1,s}(−, Z)

be the integral valued Bockstein.

Let

δ ∈ H^{n,n−1}(X (X/B), Z/p)

be an element. As in [11, (5.2)], we consider the element

μ = μ(δ) = Q̂(δ) ∈ H^{2b+1,b}(X (X/B), Z)

where

Q̂ = Q̃_{0} ∘ Q_{1} ∘ ⋯ ∘ Q_{n−2}

and

b = \frac{p^{n−1} − 1}{p − 1}

Let

Δ_{X/B}: X (X/X) → X (X/B)

be the diagonal map. We consider the induced maps

Δ_{X/B}^{∗}: H^{r,s}(X (X/B)) → H^{r,s}(X (X/X)) = H^{r,s}(X)

and their kernels

H^{r,s}(X (X/B))^{[1]} = ker Δ_{X/B}^{∗} ⊂ H^{r,s}(X (X/B))

The natural filtration on the simplicial scheme X (X/B) yields then maps

H^{r,s}(X (X/B))^{[1]} ⟶^{proj} H^{1}(X/B, H^{r−1,s})

where H^{1}(X/B, F) is an abbreviation for the homology of the complex

F (X) ⟶^{π_{0}^{∗}−π_{1}^{∗}} F (X ×_{B} X) ⟶^{π_{0}^{∗}−π_{1}^{∗}+π_{2}^{∗}} F (X ×_{B} X ×_{B} X)

Note that H^{2b+1,b}(Y, Z) = 0 and H^{2b,b}(Y, Z) = CH^{b}(Y) (the classical Chow group) for any smooth variety Y. Therefore we get a map

H^{2b+1,b}(X (X/B), Z) ⟶^{proj} H^{1}(X/B, CH^{b})

We put

ρ = ρ(δ) = proj(μ(δ)) ∈ H^{1}(X/B, CH^{b})

Remark 2.3. We are particularly interested in the following situation. Let

u ∈ H_{et}^{n}(k, μ_{p}^{⊗(n−1)})

be a symbol and let X be a smooth splitting variety of u, i. e., u_{k(X)} = 0. By Corollary 2.2 with B = Spec k we can identify u with an element

δ ∈ H^{n,n−1}(X (X/k), Z/p)

as above and by the preceding discussion we get eventually an element

ρ ∈ H^{1}(X/k, CH^{b})

We call ρ the basic correspondence of the splitting variety X of the symbol u.

Example 2.4. Let n = 2 and take for X the Severi-Brauer variety associated with u. Then b = 1 and dim X = p − 1.

There is a standard class

ρ̃ ∈ CH^{1}(X^{2}) = Pic(X^{2})

It is represented by the line bundle

Hom_{A}(π_{0}^{∗}I, π_{1}^{∗}I)

on X. Here A is the central simple algebra of degree p associated with u and I is the tautological sheaf on X = SB(A) (the geometric points of X are the right ideals of A of rank p). It is easy to see that ρ̃ is in the kernel of

Pic(X^{2}) ⟶^{π_{0}^{∗}−π_{1}^{∗}+π_{2}^{∗}} Pic(X^{3})

One finds that ρ is represented by ρ̃ (up to sign).

Remark 2.5. Let us return to the general case with B = Spec k and X proper and irreducible. Consider a representative

ρ̃ ∈ CH^{b}(X^{2})

of ρ. Let

θ = ρ̃^{p−1} ∈ CH^{d}(X^{2})

with d = b(p − 1) = p^{n−1} − 1. Assume further that dim X = d and let

c(X) = c(X, u) = (π_{0})_{∗}(θ) ∈ CH^{0}(X) = Z

be the push forward of θ under the first projection (the other projection will also do the job).

I am pretty sure that (modulo p and up to sign) the number c(X) coincides with the number “c” appearing in [11, Lemma 5.15]. The nontriviality of “c” is crucial in [11]. Voevodsky uses [11, Theorem 3.8] in order to show that s_{d}(X)/p ≠ 0 mod p implies c ≠ 0 mod p. The proof of [11, Theorem 3.8] relies on [11, Lemma 2.2, Lemma 2.3]. However proofs of [11, Lemma 2.2, Lemma 2.3] are unfortunately not available.

The aim of this text is to find a smooth proper splitting variety X of u of dimension d such that

c(X) ≠ 0 mod p

This will be based on our construction of splitting varieties “with good characteristic number”.

I haven’t actually worked out a proof that c(X) agrees essentially with Voevodsky’s “c”, because that is not really necessary anymore.

Instead I plan to prove in a sequel^{1} that if c(X) ≠ 0 mod p, then X splits off a certain motive M (cf. Proposition 5.9, Section 7) which is very likely isomorphic to Voevodsky’s motive M_{p−1} constructed from the class μ in [11]. It will also turn out that c(X) mod p does not depend on the choice of ρ̃ provided that u is nonzero (cf. Corollary 5.8). In the forthcoming text we will use only classical methods for correspondences.

Given that motive M, one may use it to prove Bloch-Kato in a similar way as the motive of a Pfister quadric was used for the Milnor conjecture in [9]. One shows first that one has an “exact quadrangle” (with X = X (X/k))

M(X){pb} → M{b} ⟶^{ρ^{p−2}} M → M(X)

which means more precisely that there are exact triangles

M(X){d} → M → D → M(X){d}[1]
C → M → M(X) → C[1]

and an isomorphism

D{b} ≃ C

(The object D is very likely isomorphic to Voevodsky’s motive M_{p−2} in [11].) One may then prove [11, Lemma 6.13] in the same way as in [11], but now using the quadrangle in place of Voevodsky’s motives M_{i}. This reduces Bloch-Kato to the injectivity of

H^{2d+1,d+1}(M, Z_{(p)}) → H^{1,1}(k, Z_{(p)})

This follows from my results on the injectivity of the norm map on zero cycles with K_{1}-coefficients.

1This is meanwhile included in this text.

Remark 2.6. By the way, for n = 1 there is a well known analogous “quadrangle”: A (nonzero) element u ∈ H_{et}^{1}(k, Z/p) is a cyclic field extension of k of degree p. Let

G = Z/p = ⟨ t | t^{p} ⟩

be its Galois group. Then there is the exact sequence of G-modules

0 → Z → Z[G] → Z[G] → Z → 0

where the middle map is multiplication with 1 − t.

The G-module Z has infinite cohomological dimension, while the Gmodule Z[G] has cohomological dimension 0. This reflects the general situation: The motive M(X) has unbounded nontrivial motivic cohomology, while M is a direct summand of a finite-dimensional variety.

See also [11, Proof of Lemma 3.7] and [11, Example 5.6].

3. Expressing c(X)

In [6, Section 6] we considered a specific form (R, L, γ) of degree p which we will also use in this text. See also [7].

Given elements a_{1}, …, a_{n−1} ∈ k^{×}, let

v = (a_{1}) ⋯ (a_{n−1}) ∈ H_{et}^{n−1}(k, μ_{p}^{⊗(n−1)})

We have constructed in [6]:

(1) A smooth proper cellular variety R with dim R = p^{n−1} − p.

(2) Forms (R, L, γ), (R, L′_{i}, γ′_{i}), i = 1, …, n − 2, of degree p on line bundles L, L′_{i} over R.

(3) Let R_{0} ⊂ R be the singular locus of the forms γ, γ′_{i} and let B = R \ R_{0}. Moreover let (γ), (γ′_{i}) ∈ H_{et}^{1}(B, μ_{p}) be the classes of the corresponding forms.

Then for any x ∈ R_{0} one has

v_{κ(x)} = 0

in H_{et}^{n−1}(κ(x), μ_{p}^{⊗(n−1)}). Moreover, one has

v_{k(B)} = ((γ)(γ′_{1}) ⋯ (γ′_{n−2}))_{k(B)}

in H_{et}^{n−1}(k(B), μ_{p}^{⊗(n−1)}). Given such data, let

A = ⨁_{i=0}^{p−1} L^{⊗i}

be the (commutative) O_{R}-algebra of rank p associated to the form (R, L, γ), let

N_{A}: A → O_{R}

be its norm form, and let A(A) be the associated scheme (a bundle of affine spaces over R).

Now let a_{n} ∈ k^{×} and let

u = v ∪ (a_{n}) = (a_{1}) ⋯ (a_{n}) ∈ H_{et}^{n}(k, μ_{p}^{⊗n})

Furthermore let

X̂ ⊂ P(A ⊕ O_{R})

be the subscheme defined by the equation

N_{A}(x) = a_{n}t^{p} for [x, t] ∈ P(A ⊕ O_{R})

Let further A′ = A|B be the restriction to B. Note that A′ is a separable O_{B}-algebra of rank p. Let

U ⊂ A(A′)

be the subscheme defined by the equation

N_{A}(x) = a_{n} for x ∈ A(A′)

Then π: U → B is a torsor over the norm-1 torus of the algebra A′/B. Let L = k[z]/(z^{p} − a_{n}).

Lemma 3.1. Suppose that u ≠ 0. There exists a smooth completion X of U such that:

For x ∈ X \ U one has

v_{κ(x)} = 0

Moreover, for any closed point x ∈ (X \ U) × Spec L one has

[κ(x) : L] ∈ pZ

Proof. For X we may take any resolution of singularities of X̂. It suffices to check the claimed properties for X = X̂.

Let x ∈ X̂ \ U. If x lies over R_{0}, then v_{κ(x)} = 0 by (3). If x lies over B, then x = [y, 0] with [y] ∈ P(A′) and N_{A}(y) = 0. But then A_{π(x)} has zero divisors and is therefore split. Thus (γ(π(x))) = 0 and v_{κ(x)} = 0 by (3).

For the second claim it suffices to have

v_{L} ≠ 0 ∈ H_{et}^{n−1}(L, μ_{p}^{⊗(n−1)})

This follows from

N_{L/k}(v_{L} ∪ (z)) = v ∪ (a_{n}) = u ≠ 0

(Recall that p ≠ 2. If p = 2 this argument works also if −1 is a square.) □

Our aim is to express c(X) mod p in terms of the Chern classes of the line bundles L, L′_{i}.

Let ζ ∈ k be a primitive p-th root of 1. For

α ∈ H_{et}^{i}(−, μ_{p}^{⊗j}) = H_{et}^{i}(−, μ_{p}^{⊗j−1}) ⊗ μ_{p}

we write α/ζ for the element of H_{et}^{i}(−, μ_{p}^{⊗j−1}) with α = α/ζ ⊗ ζ. For the element

u/ζ = (a_{1}) ⋯ (a_{n})/ζ ∈ H_{et}^{n}(k, μ_{p}^{⊗(n−1)})

we have u/ζ = δ with

δ ∈ H^{n,n−1}(X (X/k), Z/p)

as above. (The notation δ is introduced for clarity.) Let also

ρ ∈ H^{1}(X/k, CH^{b})

be as above and let

ρ′ ∈ H^{1}(U/k, CH^{b})

be its image.

We will use the following morphisms:

b_{0} : Spec k → B

is some rational point. Moreover let

j: B × Spec L → U ⊂ A(A′)
j = z · 1_{A}

where z ∈ L, z^{p} = a_{n}. In particular we have the L-point

h = j ∘ (b_{0} × id_{L}): Spec L → U

of U.

The morphism

id_{U} × h: U × Spec L → U × U

and its induced map

(id_{U} × h)^{∗}: CH^{b}(U × U) → CH^{b}(U × Spec L)

yields a map

h̄: H^{1}(U/k, CH^{b}) → CH^{b}(U × Spec L)/ res_{L/k}(CH^{b}(U))

Let

ρ′′ = h̄(ρ′)

and let

ρ̃′′ ∈ CH^{b}(U × Spec L)

be a representative of ρ′′. Put

c(ρ̃′′) = deg(ρ̃^{′′p−1}) ∈ Z/p

Note here that the degree map over L

deg: CH_{0}(U × Spec L) → Z/p

is well defined by Lemma 3.1.

Note that c(X) is given by c(ρ̃′′). Hence our aim is to conclude c(ρ̃′′) ≠ 0. (Say, for some representative ρ̃′′ of ρ′′. One can however show that c(ρ̃′′) depends only on ρ′′).

Consider the morphism

f: U × Spec L → U × U × U
f(u, λ) = (u, j(b_{0}, λ), j(π(u), λ))

Let

ρ̃′ ∈ CH^{b}(U × U)

be a representative of ρ′.

Then

(π_{2} ∘ f)^{∗}(ρ̃′) ∈ CH^{b}(U × Spec L)

is obviously a representative of ρ′′.

Let us consider

(π_{0} ∘ f)^{∗}(ρ̃′) ∈ CH^{b}(U × Spec L)

The morphism

π_{0} ∘ f: U × Spec L → U × U
(u, λ) ↦ (j(b_{0}, λ), j(π(u), λ))

factors through U×Spec L → B×Spec L. Since B is an open subvariety of a cellular variety, the restriction map

res_{L/k}: CH^{∗}(B) → CH^{∗}(B × Spec L)

is surjective. Thus

(π_{0} ∘ f)^{∗}(ρ̃′) ∈ res_{L/k}(CH^{b}(U))

(The actual computation of c(X) later on will provide another argument avoiding the reference to the cellularity of R).

Finally let us consider

(π_{1} ∘ f)^{∗}(ρ̃′) ∈ CH^{b}(U × Spec L)

The morphism π_{1} ∘ f factors through

f̂: U × Spec L → U ×_{B} U ⊂ U × U
(u, λ) ↦ (u, j(π(u), λ))

On the other hand we have

(π_{0}^{∗} − π_{1}^{∗} + π_{2}^{∗})(ρ̃′) = 0

We conclude: Let

δ_{B} ∈ H^{n,n−1}(X (U/B), Z/p)

be the image of δ, let

ρ_{B} = proj(Q̂(δ_{B})) ∈ H^{1}(U/B, CH^{b})

and let

ρ̃_{B} ∈ CH^{b}(U ×_{B} U)

be a representative of ρ_{B}. Then

f̂^{∗}(ρ̃_{B}) ∈ CH^{b}(U × Spec L)

is a representative of ρ′′.

It follows that all we need to compute c(X) is the image δ_{B} of δ. We have in H_{et}^{n}(B, μ_{p}^{⊗(n−1)}):

(u/ζ)_{B} = (a_{1}) ⋯ (a_{n})/ζ = u_{1} ∪ (a_{n}) ± u_{2} ∪ (γ′_{1}) ⋯ (γ′_{n−2})

where

u_{1} = ((a_{1}) ⋯ (a_{n−1}) − (γ_{1})(γ′_{1}) ⋯ (γ′_{n−2}))/ζ ∈ H_{et}^{n−1}(B, μ_{p}^{⊗(n−2)})
u_{2} = (γ)/ζ ∪ (a_{n}) ∈ H_{et}^{2}(B, μ_{p})

Lemma 3.2. One has

π^{∗}(u_{2}) = 0

in H_{et}^{2}(U, μ_{p}).

Proof. Let q: B̃ = Spec A → B be the etale covering of degree p corresponding to (γ)/ζ ∈ H_{et}^{1}(B, Z/p). Then q^{∗}(γ) = 0 and

((γ) ∪ (a_{n}))_{U} = (γ)_{U} ∪ (N_{A}(x)) = q_{∗}(q^{∗}(γ)_{U} ∪ (x)) = 0
□

It follows that u_{2} vanishes over k(U). By Corollary 2.2, u_{2} lies in the subgroup

H^{2,1}(X (U/B), Z/p) ⊂ H_{et}^{2}(B, μ_{p})

If we map this to

H^{2,1}(X (U/U), Z/p) ⊂ H_{et}^{2}(U, μ_{p})

it follows by Lemma 3.2 that u_{2} = δ_{2} with

δ_{2} ∈ H^{2,1}(X (U/B), Z/p)^{[1]}

Let

ρ_{2} = proj(δ_{2}) ∈ H^{1}(U/B, H^{1,1}(−, Z/p))

The element u_{1} vanishes in the generic point of B. Hence u_{1} = δ_{1} with

δ_{1} ∈ H^{n−1,n−2}(X (B/B), Z/p) ⊂ H_{et}^{n−1}(B, μ_{p}^{⊗(n−2)})

Let

δ_{B} ∈ H^{n,n−1}(X (U/B), Z/p) ⊂ H_{et}^{n}(B, μ_{p}^{⊗(n−1)})

be the element corresponding to (u/ζ)_{B}.

Then

δ_{B} = (δ_{1})_{X (U/B)} ∪ (a_{n}) ± δ_{2} ∪ (γ′_{1}) ⋯ (γ′_{n−2})

in H^{n,n−1}(X (U/B), Z/p), with (a_{n}), (γ′_{i}) considered as elements of H^{1,1}(−, Z/p).

We have

Q̂(δ_{1}) ∈ H^{2b,b−1}(X (B/B), Z) = 0

and therefore

Q̂((δ_{1})_{X (U/B)} ∪ (a_{n})) = 0

Hence (up to sign)

ρ_{B} = proj(Q̂(δ_{B}))
= proj(Q̂(δ_{2} ∪ (γ′_{1}) ⋯ (γ′_{n−2})))
= Q̂(proj(δ_{2} ∪ (γ′_{1}) ⋯ (γ′_{n−2})))
= Q̂(ρ_{2} ∪ (γ′_{1}) ⋯ (γ′_{n−2}))

Let

ρ̃_{2} ∈ H^{1,1}(U ×_{B} U, Z/p)

be a representative of ρ_{2} and let

ω = f̂^{∗}(ρ̃_{2}) ∈ H^{1,1}(U × Spec L, Z/p)

It follows that

Q̂(ω ∪ (γ′_{1}) ⋯ (γ′_{n−2})) ∈ CH^{b}(U × Spec L)

is a representative of ρ′′. Hence all we have to show is that

deg ((Q̂(ω ∪ (γ′_{1}) ⋯ (γ′_{n−2})))^{p−1}) ≠ 0

Note that U × Spec L = T × Spec L, where T is the norm-1 torus of the algebra A′/B. One has

T = R_{A′/B}(G_{m})/G_{m}

and the norm map

N: R_{A′/B}(G_{m}) → G_{m}

yields a canonical class

α ∈ H^{1,1}(T, Z/p) = H_{et}^{1}(T, μ_{p})

Lemma 3.3. One has (up to sign)

ω = res_{L/k}(α) mod res_{L/k}(H^{1,1}(U, Z/p))

Proof. Well, what else can it be?

Seriously: This is a not so difficult exercise, but I haven’t found a nice way yet to write things down. Note that this is a very general statement for the torus T and T -torsor U associated to a smooth variety B and classes (A′/B) ∈ H_{et}^{1}(B, Z/p) and (a_{n}) ∈ H_{et}^{1}(k, μ_{p}). □

Corollary 3.4. In particular we have

c(X) = deg ((Q̂(α ∪ (γ′_{1}) ⋯ (γ′_{n−2})))^{p−1}) mod p

Here α ∈ H^{1,1}(T, Z/p), (γ′_{i}) ∈ H^{1,1}(B, Z/p), Q̂ operates as map

Q̂: H^{n−1,n−1}(T, Z/p) → H^{2b,b}(T, Z) = CH^{b}(T)

and

deg: CH_{0}(T) → Z/p

is the degree map (well defined because v ≠ 0). □

Hence eventually the proof of c(X) ≠ 0 resorts to an explicit computation for very concrete classes. The complexity of this computation is similar to the one of δ(L) ≠ 0 mod p in [6]. I have no doubts^{2} that indeed c(X) ≠ 0 and have checked this in some cases, for instance for n = 3.

Remark 3.5. For n = 3, the variety T is very close to SL_{1}(A) where A is a central simple algebra of degree p. In fact, there should be a morphism T → SL_{1}(A) of degree prime to p. I believe that my computations will show that for a certain standard element u ∈ CH^{p+1}(SL_{1}(A)) one has deg(u^{p−1}) ≠ 0 mod p. (The degree mod p is well defined, if A is non-division.)

4. A computation for the Q_{i}

For a start, let me formulate a general computation for Q̂ (with the index n shifted by 1). In the following X can be any smooth variety. Let

Q_{[n]} = Q_{n−1} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}

Note that for

α ∈ H^{1,1}(X, Z/p) = H_{et}^{1}(X, μ_{p})

2The computations for all n have been meanwhile included in this text. See Section 8.

the element

Q̃_{0}(α) ∈ H^{2,1}(X, Z) = Pic(X) = CH^{1}(X)

is just the first Chern class of the line bundle associated to α under

H_{et}^{1}(X, μ_{p}) → H_{et}^{1}(X, G_{m})

We will use standard properties of the Steenrod operations, see [8]. Let α ∈ H^{1,1}(X, Z/p) and β ∈ H^{2,1}(X, Z/p). One has Q_{i}^{2} = 0, the Q_{i} anti-commute and are derivations in the Z/2-graded sense (with α odd and β even). Moreover one has

P^{h}(α) = 0 (h ≥ 1)
P^{1}(β) = β^{p}
P^{h}(β) = 0 (h > 1)

and the total Steenrod operation

P^{•} = P^{0} + P^{1} + P^{2} + ⋯

is multiplicative.

Lemma 4.1. Let α_{i} ∈ H^{1,1}(X, Z/p) (i = 1, …, n) and let β_{i} = Q_{0}(α_{i}). Then

Q_{[n]}(α_{1} ⋯ α_{n}) = ∑_{σ∈S_{n}} sgn(σ) ∏_{i=1}^{n}(β_{σ(i)})^{p^{i−1}} mod p

in CH^{b}(X) ⊗ Z/p with

b = \frac{p^{n} − 1}{p − 1}

Proof. This is an exercise using the standard properties of the Steenrod operations mentioned above. □

5. Generalities for special correspondences

Let

b = \frac{p^{n} − 1}{p − 1}

Let X be a smooth proper irreducible variety of dimension

d = b(p − 1) = p^{n} − 1

Let

ρ ∈ CH^{b}(X^{2})

and let

c(ρ) = (π_{0})_{∗}(ρ^{p−1}) ∈ CH^{0}(X) = Z

Recall the complex

CH^{b}(X) ⟶^{π_{0}^{∗}−π_{1}^{∗}} CH^{b}(X^{2}) ⟶^{π_{0}^{∗}−π_{1}^{∗}+π_{2}^{∗}} CH^{b}(X^{3})

whose homology we have denoted by H^{1}(X/B, CH^{b}).

Definition 5.1. ρ is called a special correspondence if

(π_{0}^{∗} − π_{1}^{∗} + π_{2}^{∗})(ρ) = 0 (1) c(ρ) ≠ 0 mod p (2)

As we have seen before, special correspondences appear naturally for splitting varieties of symbols. So what can we do with such correspondences? In this section we derive some first properties of special correspondences.

Let h: Spec L → X be an L-point of X where L/k is a field extension. We write X_{L} = X × Spec L. Let

H = (id_{X} × h)^{∗}(ρ) ∈ CH^{b}(X_{L})

Then

c(ρ) = deg_{X_{L}/L}(H^{p−1})

Lemma 5.2. One has

ρ_{L} = H × X − X × H

in CH^{b}(X_{L}^{2})

Proof. Just pull back the cocycle condition (1) under

id_{X} × id_{X} × h: X × X × Spec L → X × X × X
□

Let

ρ^{j} ∈ CH^{jb}(X^{2})

be the j-th power of ρ in the Chow ring of X. Considering ρ^{j} as an element of End(X) = CH^{∗}(X^{2}) in the category of Chow correspondences, we get a map

(ρ^{j})_{∗}: CH^{r}(X) → CH^{r−(p−1−j)b}(X)
(ρ^{j})_{∗}(α) = (π_{0})_{∗}(π_{0}^{∗}(α)ρ^{j})

Definition 5.3. Let 0 ≤ h ≤ p − 1. For α ∈ CH^{bh}(X) we define its ρ-degree d_{ρ}(α) ∈ Z by

d_{ρ}(α) · [X] = (ρ^{p−1−h})_{∗}(α)

in CH^{0}(X) = [X]Z.

ON THE BASIC CORRESPONDENCE OF A SPLITTING VARIETY 15

For h = 0 we simply have

d_{ρ}([X]) = c(ρ)

by definition. Lemma 5.4. For α ∈ CH^{bh}(X) one has

d_{ρ}(α) = deg_{X_{L}/L}(α_{L}H^{p−1−h})

Proof. One has

(ρ^{p−1−h})_{∗}(α)_{L} = (ρ_{L}^{p−1−h})_{∗}(α_{L})
= (π_{0})_{∗}((α_{L} × X)(H × X − X × H)^{p−1−h})
= deg(α_{L}H^{p−1−h})H^{0}
□

Lemma 5.5. For α ∈ CH^{bh}(X) and p − 1 − h ≤ j ≤ p − 1 one has

(ρ^{j})_{∗}(α)_{L} ∈ Z_{(p)}^{×} · d_{ρ}(α)H^{j+h−(p−1)}

Proof. One has

(ρ^{j})_{∗}(α)_{L} = (ρ_{L}^{j})_{∗}(α_{L})

= (π_{0})_{∗}((α_{L} × X)(H × X − X × H)^{j})

= \binom{j}{p − 1 − h} deg(α_{L}H^{p−1−h})H^{j+h−(p−1)}

□

For α ∈ CH^{bh}(X) with 1 ≤ h ≤ p − 1 the element (ρ^{p−h})_{∗}(α) is in CH^{b}(X). Thus

Ω(α) = α((ρ^{p−h})_{∗}(α))^{p−1−h}

is a zero cycle. Lemma 5.6.

deg(Ω(α)) ∈ Z_{(p)}^{×} · d_{ρ}(α)^{p−h}

Proof. Over L we have by Lemma 5.5

Ω(α)_{L} = α_{L}(ud_{ρ}(α)H)^{p−1−h}

for some u ∈ Z_{(p)}^{×}. Taking degrees yields the claim (see Lemma 5.4). □

Let

I(X) = deg(CH_{0}(X)) ⊂ Z

Corollary 5.7. Suppose I(X) ⊂ pZ. Then for any α ∈ CH^{bh}(X) with 1 ≤ h ≤ p − 1 one has

d_{ρ}(α) ∈ pZ

Proof. By assumption p divides deg(Ω(α)). By Lemma 5.6 it follows that p divides some power of d_{ρ}(α). Since p is prime, the claim follows.

□

Corollary 5.8. Suppose I(X) ⊂ pZ. Let α ∈ CH^{b}(X) and let

ρ′ = ρ + (π_{0}^{∗} − π_{1}^{∗})(α)

Then

c(ρ′) = c(ρ) mod p

Proof. Just compute over L and use Lemma 5.4 and Corollary 5.7. □

Proposition 5.9. Let ρ be a special correspondence. Then there exists a projector π ∈ End(X) ⊗ Z_{(p)} such that

(1) π_{L} = c(ρ)^{−1}ρ_{L}^{p−1} mod p CH^{d}(X_{L}^{2}) ⊗ Z_{(p)}

(2) For M = (X, π) one has

M ⊗ X = ⨁_{i=0}^{p−1} Z_{(p)}{bi} ⊗ X

in the category of Chow motives over Z_{(p)}.

Proof. [See Section 7 for another way.]

Only a brief sketch for p = 2: After replacing ρ by ρc(ρ)^{−1} we may assume c(ρ) = 1. Let

π′ = ρ − X × (π_{0})_{∗}(ρ^{2})

Over L this is a projector, because

π′_{L} = H × X + X × H

Since L can be any residue class field of X, it follows by nilpotence (see [5, Proposition 1]) that there is a projector π with π′_{L} = π_{L}.

For general p one proceeds similarly, starting from ρ^{p−1} ∈ CH^{d}(X^{2}).

□

6. Existence of generic splitting varieties

See also [11, Theorem 7.3].

Let

u ∈ H_{et}^{n}(k, μ_{p}^{⊗(n−1)})

be a symbol and let Z be a not necessarily connected scheme over k such that u_{κ(z)} = 0 for all z ∈ Z. We call such a scheme a splitting scheme of u. A smooth splitting scheme is nothing else than a disjoint union of smooth splitting varieties. We will assume smoothness in the following.

As discussed in Section 2, there is a natural element

ρ = ρ_{Z} ∈ H^{1}(Z/k, CH^{b})

with

b = \frac{p^{n−1} − 1}{p − 1}

This element is functorial in Z. Namely, if f: Z′ → Z is a morphism, then

ρ_{Z′} = f^{∗}(ρ_{Z})

We are particularly interested in the following situation: Let X be a smooth irreducible proper splitting variety of u of dimension d = b(p − 1) = p^{n−1} − 1 and let Y be any smooth irreducible splitting variety of u. Let

Z = X ∪ Y (disjoint union)

We have

CH^{b}(Z^{2}) = CH^{b}(X^{2}) ⊕ CH^{b}(X × Y) ⊕ CH^{b}(Y × X) ⊕ CH^{b}(Y^{2})

Let E = k(Y) be the function field of Y. The projection maps

CH^{b}(Z^{2}) → CH^{b}(X × Y) → CH^{b}(X_{E})

(with X_{E} = X × Spec E) induce a map

Ψ_{X,E}: H^{1}(Z/k, CH^{b}) → CH^{b}(X_{E})/ CH^{b}(X)_{E}

(since b > 0 one has CH^{b}(Spec E) = 0). Let

ρ_{X,E} = Ψ_{X,E}(ρ_{Z}) ∈ CH^{b}(X_{E})/ CH^{b}(X)_{E}

Note that ρ_{X,E} is functorial in X and E.

Choose a representative

ρ̃_{X,E} ∈ CH^{b}(X_{E})

of ρ_{X,E}, let

θ_{X,E} = (ρ̃_{X,E})^{p−1} ∈ CH^{d}(X_{E})

and let

c(X, E) = deg_{X_{E}/E}(θ_{X,E}) mod p ∈ Z/p

Lemma 6.1. Assume I(X) ⊂ pZ. Then

c(X, E) = c(X) mod p

In particular, c(X, E) does not depend on E or Y at all.

Proof. Let L = k(X) and K = k(X × Y). By functoriality we have

ρ_{X,E} → ρ_{X,K} ← ρ_{X,L}

under

\frac{CH^{b}(X_{E})}{CH^{b}(X)_{E}} → \frac{CH^{b}(X_{K})}{CH^{b}(X)_{K}} ← \frac{CH^{b}(X_{L})}{CH^{b}(X)_{L}}

It follows that for any representatives

ρ̃_{X,E} ∈ CH^{b}(X_{E}), ρ̃_{X,L} ∈ CH^{b}(X_{L})

there exists α ∈ CH^{b}(X) with

(ρ̃_{X,E})_{K} = (ρ̃_{X,L})_{K} + α_{K}

in CH^{b}(X_{K}).

We next look at ρ_{X,L}. Since for

Z = X ∪ X (disjoint union)

the simplicial schemes X (Z/k) and X (X/k) have the same cohomology, the element ρ_{X,L} equals the image of

ρ = ρ_{X} ∈ H^{1}(X/k, CH^{b})

in

\frac{CH^{b}(X_{L})}{CH^{b}(X)_{L}}

Hence

c(X) = deg_{X_{L}/L}((ρ̃_{X,L})^{p−1})

By Corollary 5.8 we have

c(X) = deg_{X_{L}/L}((ρ̃_{X,L} + α_{L})^{p−1})

Hence

c(X) = deg_{X_{K}/K}(((ρ̃_{X,E})_{K})^{p−1}) = deg_{X_{E}/E}((ρ̃_{X,E})^{p−1})
□

Corollary 6.2. Assume that I(X) ⊂ pZ and that c(X) ≠ 0 (mod p). Then X is a p-generic splitting variety of u.

Proof. Let E = k(Y) be any splitting field of u. Then

θ_{X,E} ∈ CH^{d}(X_{E})

is a zero cycle of degree prime to p. □

The core of the argument is entirely due to Voevodsky. On [11, page 37] he defines X = X (Y/k), where Y is the union of all splitting varieties up to isomorphism and draws the conclusion in [11, Theorem 7.3].

It is amusing to consider the case n = 2. In this case ρ can be defined easily with the classical relation between the kernel of H^{2}(k, G_{m}) → H^{2}(k(X), G_{m}) and Pic(X). Taking for X the Severi-Brauer variety of dimension p − 1, one may then check by inspection (see Example 2.4) that c(X) ≠ 0 (mod p). Corollary 6.2 shows then the p-genericity of the Severi-Brauer variety, without using much of the theory of central simple algebras.

7. Generalities for special correspondences II

7.1. Preliminary Definitions. Let X be a smooth variety over a field k.

For n ≥ 0 and 0 ≤ i ≤ n let

π_{i}: X^{n+1} → X^{n}
π_{i}(x_{0}, …, x_{n}) = (x_{0}, …, x_{i−1}, x_{i+1} …, x_{n})

Then we have complexes

(7.1.1) ⋯ ⟶ CH^{r}(X^{n}) ⟶^{δ_{n}} CH^{r}(X^{n+1}) ⟶^{δ_{n+1}} CH^{r}(X^{n+2}) ⟶ ⋯

with

δ_{n} = ∑_{i=0}^{n}(−1)^{i}π_{i}^{∗}

Here CH^{r}(Z) is the classical Chow group of r-codimensional cycles. We put

Z^{n}(X/k, CH^{b}) = ker δ_{n+1}
H^{n}(X/k, CH^{b}) = ker δ_{n+1}/ im δ_{n}

We will use the following notations: Let I be an index set. For i ∈ I let X_{i} = X be a copy of X and let X^{I} = ∏_{i} X_{i}. We will sometimes understand

X^{n+1} = X_{0} × ⋯ × X_{n}

where the X_{i} are copies of X.

For a sequence i_{1}, …, i_{n} in I let

π̂_{i_{1},...,i_{n}}: X^{I} → X^{n}
π̂_{i_{1},...,i_{n}}((x_{i})_{i∈I}) = (x_{i_{1}}, …, x_{i_{n}})

For u ∈ CH^{∗}(X^{n}) we write

u_{[i_{1},...,i_{n}]} = π̂_{i_{1},...,i_{n}}^{∗}(u) ∈ CH^{∗}(X^{I})

7.2. Symmetry of special correspondences. We assume that X is proper of dimension d = b(p − 1).

Let ρ ∈ CH^{b}(X^{2}) be a special correspondence. Let τ be the exchange involution.

Lemma 7.1. τ^{∗}(ρ) = −ρ

Proof. We have

ρ_{[1,2]} − ρ_{[0,2]} + ρ_{[0,1]} = 0

in CH^{b}(X^{3}) by the cocycle condition. Pulling back with the diagonals X^{2} → X^{3}, (x, y) ↦ (x, x, y), (x, y, x) yields

ρ_{[0,1]} − ρ_{[0,1]} + ρ_{[0,0]} = 0, ρ_{[1,0]} − ρ_{[0,0]} + ρ_{[0,1]} = 0

which gives

ρ_{[0,0]} = 0, ρ_{[1,0]} + ρ_{[0,1]} = 0
□

7.3. Construction of the motive for p = 2. I first used nilpotence to get the projector, see the proof of Proposition 5.9. Meanwhile I noticed that there is a much more direct argument.

For an illustration, we first consider the case p = 2. Then b = d. We assume c(ρ) = 1. Let

β = −(π_{0})_{∗}(ρ^{2}) ∈ CH_{0}(X)

Then

β_{L} = −(π_{0})_{∗}((H_{0} − H_{1})^{2}) = −(π_{0})_{∗}(−2H_{0}H_{1}) = 2H

and therefore

deg(β) = 2

Lemma 7.2.

ρ ∘ ρ = −ρ + β × X
ρ ∘ ρ ∘ ρ = ρ

Proof. One has by Lemma 7.1:

(π_{1})_{∗}(ρ) = (π_{0} ∘ τ)_{∗}(ρ) = −(π_{0})_{∗}(ρ) = −c(ρ)[X] = −[X]

One has in CH^{d}(X_{0} × X_{2}) with

π_{1} : X_{0} × X_{1} × X_{2} → X_{0} × X_{2}
(x_{0}, x_{1}, x_{2}) → (x_{0}, x_{2})

and X_{i} = X:

(ρ ∘ ρ)_{[0,2]} = (π_{1})_{∗}(ρ_{[0,1]}ρ_{[1,2]})
= (π_{1})_{∗}(ρ_{[0,1]}(ρ_{[0,2]} − ρ_{[0,1]}))
= (π_{1})_{∗}(ρ_{[0,1]})ρ_{[0,2]} − (π_{1})_{∗}((ρ_{[0,1]})^{2})
= −ρ_{[0,2]} + β_{[0]}

This proves the first claim. For the second claim note that

(β × X) ∘ ρ = β × X

for any β. Thus

ρ ∘ ρ ∘ ρ = −ρ ∘ ρ + (β × X) ∘ ρ = ρ − β × X + β × X = ρ
□

Corollary 7.3. ρ ∘ ρ is a projector. □

Note that

(ρ ∘ ρ)_{L} = H × X + X × H

so ρ ∘ ρ is the desired projector.

7.4. Correspondences generated by ρ. We work over Z_{(p)}.

In this section we show that the correspondences in CH^{∗}(X^{n}) generated by a special correspondence ρ ∈ CH^{b}(X^{2}) via products, (simplicial) pull backs and push forwards form for every n a free Z_{(p)}-module. It is denoted by R_{n}.

We consider similarly the correspondences in CH^{∗}(X_{L}^{n}) generated by H ∈ CH^{b}(X_{L}) (where ρ_{L} = H × X − X × H). The resulting ring is denoted by S_{n}.

It turns out that the natural map R_{n} → S_{n} is injective. This way, identities for correspondences for X generated by ρ can be easily verified by passing to L.

The rings R_{n} → S_{n} are independent of the weight of the symbol. In fact, one has

R_{n} = CH^{∗}(X^{n}), S_{n} = CH^{∗}((P^{p−1})^{n})

where X is a nonsplit Severi-Brauer variety of dimension p − 1.

The following discussion is simple in nature, but a bit tedious and tiring. Voevodsky’s framework of motives over simplicial schemes indicates that there should be simpler organization and explanation.

For i = 0, …, p − 1 let

β′_{i} = (π_{0})_{∗}(ρ^{p−1+i}) ∈ CH^{bi}(X)

Then

(β′_{i})_{L} = (π_{0})_{∗}((H_{0} − H_{1})^{p−1+i}) = c(ρ)\binom{p − 1 + i}{i}(−1)^{i}H^{i}

Note that for 1 ≤ i ≤ p − 1 one has

\binom{p − 1 + i}{i} ∈ pZ_{(p)}^{×}

We put for 1 ≤ i ≤ p − 1

β_{i} = c(ρ)^{−1}(−1)^{i} \frac{p}{\binom{p−1+i}{i}}β′_{i} ∈ CH^{bi}(X) ⊗ Z_{(p)}

so that

(β_{i})_{L} = pH^{i}

We put

β_{0} = c(ρ)^{−1}β′_{0} = [X]

Let

ε = c(ρ)^{−1}β_{p−1} ∈ CH^{d}(X) ⊗ Z_{(p)}

The zero cycle ε has degree p.

For i = 0, …, p − 1 let

α_{i} = (ρ^{i})_{∗}(ε) ∈ CH^{bi}(X)

Note that

(α_{i})_{L} = pH^{i}

In particular α_{0} = p[X].

Lemma 7.4. In the Chow ring CH^{∗}(X) one has

α_{i}α_{j} = pα_{i+j}

Proof. Note that ε_{[1]}ε_{[2]}ρ_{[1,2]} = 0 by dimension reasons. Hence

(α_{i}α_{j})_{[0]} = (π_{1,2})_{∗}(ε_{[1]}ε_{[2]}ρ_{[0,1]}^{i}ρ_{[0,2]}^{j})
= (π_{1,2})_{∗}(ε_{[1]}ε_{[2]}ρ_{[0,1]}^{i}ρ_{[0,1]}^{j})
= α_{0}α_{i+j}
□

Lemma 7.5. For i = 0, …, p−1, the element α_{i} is in the Z_{(p)}-module generated by the elements

β_{j}β_{i−j}, 0 ≤ j ≤ i

Proof. For a certain unit u ∈ Z_{(p)}^{×} one has

u(α_{i})_{[0]} = (π_{1,2})_{∗}(ρ_{[0,1]}^{i}ρ_{[1,2]}^{2p−2})

= (π_{1,2})_{∗}(ρ_{[0,1]}^{i}(ρ_{[0,1]} + ρ_{[0,2]})^{2p−2})

= ∑_{j+k=2p−2} \binom{2p − 2}{j}(π_{1})_{∗}(ρ_{[0,1]}^{i+j})(π_{2})_{∗}(ρ_{[0,2]}^{k})

□

Corollary 7.6. For i = 1, …, p − 1 one has α_{i} = β_{i}.

Proof. Note that

(α_{i})_{L} = pH^{i} = (β_{i})_{L}

Since H is not a torsion element, it suffices to show that α_{i} = uβ_{i} for some u ∈ Z_{(p)}.

We proceed by induction on i. Suppose α_{j} = β_{j} for 1 ≤ j < i. By Lemma 7.5, α_{i} is a linear combination of β_{0}β_{i} = β_{i} and two-fold products β_{j}β_{i−j} with 1 ≤ j < i. By induction we have α_{j} = β_{j}, and by Lemma 7.4, we have β_{j}β_{i−j} = pα_{i}. Hence α_{i} is a linear combination of β_{i} and pα_{i}. □

For n ≥ 1 let

S_{n} = Z_{(p)}[H_{1}, …, H_{n}] ⊂ CH^{∗}(X_{L}^{n}) ⊗ Z_{(p)}

be the subring generated by the H_{i} = H_{[i]}. The ring S_{n} has the presentation

Z_{(p)}[H_{1}, …, H_{n}]/(H_{1}^{p}, …, H_{n}^{p})

In particular, it is free of rank p^{n}.

We define

R_{n} ⊂ CH^{∗}(X^{n}) ⊗ Z_{(p)}

as the submodule generated by the monomials

(β_{i})_{[1]}ρ_{[1,2]}^{h_{2}} ⋯ ρ_{[1,n]}^{h_{n}}

with 0 ≤ i ≤ p − 1, 0 ≤ h_{k} ≤ p − 1, k = 2, …, n.

The image of R_{n} in CH^{∗}(X_{L}^{n}) ⊗ Z_{(p)} is the subring

Z_{(p)}[H_{2} − H_{1}, …, H_{n} − H_{1}] + pS_{n} ⊂ S_{n}

(note that (H_{i} − H_{1})^{p} = 0 mod p). It is easy to see that the specified monomials form a Z_{(p)}-basis of R_{n}. Furthermore, R_{n} → S_{n} is injective.

Lemma 7.7. R_{n} contains the elements (β_{i})_{[k]} for k = 1, …, n.

Proof. We have

(β′_{i})_{[2]} − (β′_{i})_{[1]} = (π_{0})_{∗}(ρ_{[0,2]}^{p−1+i} − ρ_{[0,1]}^{p−1+i})

= (π_{0})_{∗}((ρ_{[0,1]} + ρ_{[1,2]})^{p−1+i} − ρ_{[0,1]}^{p−1+i})

= ∑_{k=1}^{p−1+i} \binom{p − 1 + i}{k}(π_{0})_{∗}(ρ_{[0,1]}^{p−1+i−k})ρ_{[1,2]}^{k}

= ∑_{k=1}^{i} \binom{p − 1 + i}{k}(β′_{i−k})_{[1]}ρ_{[1,2]}^{k}

□

Lemma 7.8.

pc(ρ)ρ = (π_{0}^{∗} − π_{1}^{∗})(β′_{1})

Proof. Just specialize the previous computations to the case i = 1. □

Corollary 7.9. For any [ρ] ∈ H^{1}(X/k, CH^{b}) one has

pc(ρ)[ρ] = 0
□

Lemma 7.10. R_{2} contains the element ρ_{[1,2]}^{p}.

Proof. This is the most tricky part.

Note that ρ^{2p−1} = 0 by dimension reasons. We have

c(ρ)ρ_{[1,2]}^{p} = (π_{0})_{∗}(ρ_{[0,2]}^{p−1}ρ_{[1,2]}^{p})

= (π_{0})_{∗}(ρ_{[0,2]}^{p−1}(ρ_{[0,2]} − ρ_{[0,1]})^{p})

= (−1)^{p}(π_{0})_{∗}(ρ_{[0,2]}^{p−1}ρ_{[0,1]}^{p}) + pΩ

= (−1)^{p}(π_{0})_{∗}((ρ_{[0,1]} + ρ_{[1,2]})^{p−1}ρ_{[0,1]}^{p}) + pΩ

= ∑_{i=0}^{p−2} u_{i}(β′_{i+1})_{[1]}ρ_{[1,2]}^{p−1−i} + pΩ

Here u_{i} ∈ Z and Ω is a linear combination of the elements

γ_{i} = (π_{0})_{∗}(ρ_{[0,2]}^{p−1+i}ρ_{[0,1]}^{p−i})

with i = 1, …, p − 1.

Hence it suffices to show

pγ_{i} ∈ R_{2}

The element ε is a zero cycle of degree p. Since we work over Z_{(p)}, we may assume that k has no extensions of degree prime to p. Hence we may assume that there exists a field L with [L : k] = p and a point Spec L → X. (The following use of the field L of degree p may and should be replaced by a more formal discussion using the cycle ε directly.)

Then

p(γ_{i})_{[1,2]} = N_{L/k}(((γ_{i})_{[1,2]})_{L})
= N_{L/k}((π_{0})_{∗}((H_{0} − H_{2})^{p−1+i}(H_{0} − H_{1})^{p−i}))

Since i ≥ 1 and H^{p} = 0, the element p(γ_{i})_{[1,2]} is a linear combination of the elements

pN_{L/k}(H_{1}^{k}H_{2}^{p−k})

with k = 1, …, p − 1. We have

((β_{k})_{[1]}ρ_{[1,2]}^{p−k})_{L} = pH_{1}^{k}(H_{1} − H_{2})^{p−k}

Since H^{p} = 0, these elements generate the same module as the elements

pH_{1}^{k}H_{2}^{p−k}

Therefore the element p(γ_{i})_{[1,2]} is a linear combination of the elements

p(β_{k})_{[1]}ρ_{[1,2]}^{p−k}

But those elements are obviously in R_{2}. □

We summarize:

Corollary 7.11. R_{n} is the ring generated over Z_{(p)} by (β_{i})_{[k]}, (α_{i})_{[k]}, ρ_{[k,l]} with 1 ≤ k, l ≤ n and 0 ≤ i ≤ p − 1. □

Corollary 7.12. The family of subrings R_{n} ⊂ CH^{∗}(X^{n}) is closed under external products CH^{∗}(X^{n}) × CH^{∗}(X^{m}) → CH^{∗}(X^{n+m}). It is also closed under pull backs and push forwards along any simplicial morphism X^{n} → X^{m}. □

Let

Π_{ρ} = c(ρ)^{−1} ∑_{i=0}^{p−1} H_{0}^{i}H_{1}^{p−1−i} ∈ S_{2} ⊂ CH^{∗}(X_{L}^{2})

One has Π_{ρ} ∈ CH^{d}(X_{L}^{2}) = End(X_{L}). Since c(ρ) = deg(H^{p−1}), it is easy to see that Π_{ρ} is an idempotent when considered as an element of End(X_{L}).

Note that

(ρ_{[0,1]}^{p−1})_{L} = Π_{ρ} mod pS_{2}

Hence Π_{ρ} ∈ R_{2} ⊂ CH^{∗}(X^{2}). In fact, one has Π_{ρ} ∈ CH^{d}(X^{2}) = End(X).

It is also an idempotent of End(X), since R_{2} → S_{2} is injective and the R_{n} are closed under composition operations.

Definition 7.13. The element Π_{ρ} ∈ R_{n} is called the projector of ρ.

The Chow motive

M = M_{ρ} = (X, Π_{ρ})

is called the motive of ρ.

Proposition 7.14. One has

X ⊗ M = ⨁_{i=0}^{p−1} X ⊗ Z_{(p)}{bi}

in the category of Chow motives over Z_{(p)}.

Proof. Let f: Y → X be a morphism. For 0 ≤ i ≤ p − 1 let

φ_{i}: CH^{h}(Y) → CH^{h+bi}(Y × X)
φ_{i}(α) = f^{∗}(α)(f × id_{X})^{∗}(ρ^{i})

Φ = ∑_{i=0}^{p−1} φ_{i}: ⨁_{i=0}^{p−1} CH^{h−bi}(Y) → CH^{h}(Y × X)

and

ψ_{i}: CH^{h+bi}(Y × X) → CH^{h}(Y)
ψ_{i}(β) = (π_{X})_{∗}(β(f × id_{X})^{∗}(ρ^{p−1−i}))

Ψ = ∑_{i=0}^{p−1} ψ_{i}: CH^{h}(Y × X) → ⨁_{i=0}^{p−1} CH^{h−bi}(Y)

Then for α ∈ CH^{h}(Y) one has

(ψ_{j} ∘ φ_{i})(α) = (π_{X})_{∗}(α_{Y} ρ_{Y,X}^{p−1−j+i})
= αf^{∗}((π_{1})_{∗}(ρ_{[0,1]}^{p−1−j+i}))

Hence Ψ ∘ Φ is a triangular matrix with invertible elements on the diagonal, and therefore an isomorphism.

Note that (id_{Y} × Π_{ρ}) ∘ Φ = Φ. It remains to show that

Φ: ⨁_{i=0}^{p−1} CH^{h−bi}(Y) → CH^{h}(Y × M_{ρ})

is surjective. The projection Y × M_{ρ} → Y induces a spectral sequence (see [4])

E_{2}^{r,s} = A^{r}(Y, A^{s}(M_{ρ})) ⟹ A^{r+s}(Y × M_{ρ})

Since CH^{s}((M_{ρ})_{L}) = H^{i}Z_{(p)} for s = bi and CH^{s}((M_{ρ})_{L}) = 0 otherwise, the claim follows. □

8. Computing c(X)

In the following we refer to the notations in [6]. All Chow groups are understood with coefficients Z/p.

8.1. Recalling the R_{i}. See [6, Section 6, p. 15–16]. We switch from the index n in [6] to m, to avoid conflicts below.

Given forms (S, H_{i}, α_{i}), i = 1, …, m, we defined forms

(R_{i}/R_{i+1}, J_{i}, γ_{i}), (R_{i}/R_{i+1}, J′_{i}, γ′_{i})

We write R = R_{1} and assume S = Spec k. Then we have a sequence of bundles

R = R_{1} → R_{2} → ⋯ → R_{m} = S = Spec k

with relative dimensions dim(R_{i}/R_{i+1}) = p^{i+1} − p^{i}. Thus

dim R = p^{m} − p = p(p^{m−1} − 1)

Remark 8.1. The extension R(\sqrt[p]{γ_{1}})/R is a model for the extension Y^{p}/C^{p}Y for a norm variety Y of the symbol {α_{1}, …, α_{m}}, where C^{p}Y = (Y^{p})/(Z/p) is the p-th cyclic symmetric product of Y.

We denote:

s_{i} = c_{1}(J_{i}) ∈ CH^{1}(R_{i}) ⊂ CH^{1}(R)
t_{i} = c_{1}(J′_{i}) ∈ CH^{1}(R_{i}) ⊂ CH^{1}(R)

We have by definition

R_{i}/R_{i+1} = C_{p} = C_{p}(α_{1}, …, α_{i}, γ_{i+1})

8.2. Recalling the C_{r}. See [6, Section 5, p. 10]. We fix the index i above and put n = i + 1.

Let us recall some parts of the definition of C_{p}. We have:

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

With the notations from [6] we have a sequence of bundles

R_{i} = C_{p} = S_{p} → S_{p−1} → ⋯ → S_{0} = S′ = R_{i+1}

with relative dimensions dim S_{r}/S_{r−1} = p^{n−1} − p^{n−2}.

Remark 8.2. S_{r}/S_{r−1} parameterizes one of the semi-elementary moves in subsymbols. One uses p moves of the given type, which are in total parameterized by S_{p}/S_{0} = R_{i}/R_{i+1}. In particular,

dim(R_{i}/R_{i+1}) = p(p^{n−1} − p^{n−2}) = p(p^{i} − p^{i−1})

as mentioned before (recall n = i + 1).

In [6, Proof of Theorem 5.2, p. 13] we defined certain elements

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

subject to the relations

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

Here L = J_{i+1} is the line bundle for γ_{i+1}. Hence

x_{0} = s_{i+1}^{p^{i−1}}

We have x_{r}, z_{r} ∈ CH^{p^{i−1}}(S_{r}). Moreover

(8.2.1)  (S_{r} → S_{r−1})_{∗}(z_{r}^{i}) = \begin{cases}0 & for i = 0, …, p − 2 \\ 1 & for i = p − 1 \end{cases}

We have to sharpen some computations for the rings R_{r}(a, b) of [6, Section 5].

Lemma 8.3. One has

x_{r}^{p^{2}−1} = x_{r}^{p−1}

in R_{r}(0, 1).

Proof. By [6, Corollary 5.5] one has

z_{r+1}^{p^{2}} = z_{r+1}^{p}

in R_{r+1}(0, 1). Since z_{r+1}^{p} = z_{r+1}x_{r}^{p−1} we have

z_{r+1}x_{r}^{p^{2}−1} = z_{r+1}x_{r}^{p−1}

Since R_{r+1}(0, 1) is the free R_{r}(0, 1)-module with basis z_{r+1}^{i}, 0 ≤ i ≤ p − 1, the claim follows. □

Remark 8.4. This proof of Lemma 8.3 is a bit strange since one uses the ring R_{r+1}(a, b) to get an identity for R_{r}(a, b).

Alternatively, one may argue within R_{r}(a, b) as follows. By [6, Proposition 5.4], the ring R_{r}(a, b) is isomorphic to a product of rings of the form

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

An element u ∈ K_{k} can be written as u = a + v with a ∈ F_{p} and v^{p} = 0. Then

u^{p^{2}−1} = (a + v)^{p(p−1)}u^{p−1} = a^{p−1}u^{p−1}

If u is a unit, then a ≠ 0 and a^{p−1} = 1. Thus, if u is a unit, or if u = 0, we have

u^{p^{2}−1} = u^{p−1}

One may now inspect the isomorphisms

R_{r}(a, b) ≃ ∏_{i∈F_{p}} R_{r−1}(b, ib + a)

(for b ≠ 0) resp.

R_{r}(a, 0) ≃ F_{p}[z_{1}]/(z_{1}^{p}) ⊗ R_{r−1}(0, 1)

from the proof of [6, Proposition 5.4] to see that x_{r} will in each component either go to 0 or to unit · x_{r−1}. This way the claim follows by induction on r.

Remark 8.5. Meanwhile I found a much much simpler way to prove [6, Corollary 5.5] and Lemma 8.3.

Namely one has

(8.2.2)  x_{r}^{p}x_{r−1} − x_{r}x_{r−1}^{p} = 0
(8.2.3)  ∑_{i=0}^{p}(x_{r}^{p−1})^{i}(x_{r−1}^{p−1})^{p−i} = 1

in R_{r}(0, 1).

Proof of (8.2.2). Let

f(x, y) = x^{p} − xy^{p−1}, g(x, y) = yf(x, y) = x^{p}y − xy^{p}

Then

f(x_{r}, x_{r−1}) = (z_{r} + x_{r−2})^{p} − (z_{r} + x_{r−2})x_{r−1}^{p−1}
= (z_{r}x_{r−1}^{p−1} + x_{r−2}^{p}) − (z_{r} + x_{r−2})x_{r−1}^{p−1}
= x_{r−2}^{p} − x_{r−2}x_{r−1}^{p−1} = f(x_{r−2}, x_{r−1})

Hence

g(x_{r}, x_{r−1}) = g(x_{r−2}, x_{r−1}) = −g(x_{r−1}, x_{r−2})

and

±g(x_{r}, x_{r−1}) = g(x_{0}, x_{−1}) = g(1, 0) = 0
□

Proof of (8.2.3). Let

h(x, y) = ∑_{i=0}^{p}(x^{p−1})^{i}(y^{p−1})^{p−i}

Then

h(x, y) = f(x, y)^{p−1} + y^{(p−1)p}

Hence, by the above,

h(x_{r}, x_{r−1}) = h(x_{r−2}, x_{r−1}) = h(x_{r−1}, x_{r−2}) = ⋯ = h(1, 0) = 1

□

One finds now

x_{r}^{p−1} = x_{r}^{p−1}h(x_{r}, x_{r−1})
= x_{r}^{p^{2}−1} + x_{r}^{p−1}f(x_{r−1}, x_{r})^{p−1}
= x_{r}^{p^{2}−1} + g(x_{r−1}, x_{r})^{p−1} = x_{r}^{p^{2}−1}

Moreover one has

u^{p^{2}} = u^{p}

for all u ∈ R_{r}(0, 1), because this holds for all the ring generators x_{s} of R_{r}(0, 1) over F_{p}.

The ring R′ is the homogeneous variant R_{p}(0, x_{0}) of R_{p}(0, 1), see the lines before [6, Corollary 5.7].

Corollary 8.6. One has

x_{r}^{p^{2}−1} = x_{r}^{p−1}x_{0}^{p^{2}−p}

in R′. □

Lemma 8.7. Let π_{i}: R_{i} → R_{i+1} be the projection. Then for h, k ≥ 0 one has

(π_{i})_{∗}(x_{p−1}^{(p−1)k}x_{p}^{(p−1)h}) = \begin{cases}0 & if h = 0 or k + h < p \\ x_{0}^{(p−1)(k+h−p)} & else \end{cases}

Proof. If h = 0 or k + h < p, the claim follows by dimension reasons. Otherwise we may by Corollary 8.6 reduce to the case h ≤ p. As in the proof of [6, Lemma 5.3], one sees then that

x_{p}^{(p−1)h}

has the same leading term as

x_{p}^{(p−1)}x_{p−1}^{(p−1)(h−1)}

We may therefore assume h = 1.

Again by Corollary 8.6 reduce to the case k ≤ p. And again as in the proof of [6, Lemma 5.3], one sees then that

x_{p}^{(p−1)}x_{p−1}^{(p−1)k}

has the same leading term as

x_{p}^{(p−1)}x_{p−1}^{(p−1)}x_{p−2}^{(p−1)(k−1)}

Iterating this, we end up with

x_{p}^{(p−1)}x_{p−1}^{(p−1)} ⋯ x_{1}^{(p−1)}x_{0}^{(p−1)(k+1−p)}

This proves the claim. □

8.3. The final computation. By Corollary 3.4 we want to compute the number

c = deg ((Q̂((γ′_{1}) ⋯ (γ′_{m−1}) ∪ α))^{p−1}) mod p

Here T → B ⊂ R is the norm-1 torus defined by the form (R_{1}, J_{1}, γ_{1}) and α ∈ H^{1,1}(T, Z/p) is its standard class.

The torus T → B is an open subscheme of the projective bundle P(A) → R. Let

w ∈ CH^{1}(P(A))

be the first Chern class of the tautological line bundle. Then

CH^{∗}(P(A)) = CH^{∗}(R)[w]/(w^{p} − ws_{1}^{p−1})

One finds

Q_{0}(α) = w|T

in CH^{1}(T).

By Lemma 4.1 we have (mod p)

c = ( ∑_{σ∈S_{m}} sgn(σ) ∏_{i=1}^{m}(β_{σ(i)})^{p^{i−1}})^{p−1}

in CH_{0}(P(A)) = Z with β_{1} = t_{1}, …, β_{m−1} = t_{m−1} and β_{m} = w. From

w^{p} = ws_{1}^{p−1}

we get

c = w^{p−1}( ∑_{σ∈S_{m}} φ_{σ})^{p−1}

where

φ_{σ} = sgn(σ)t_{m−1}^{p^{σ(1)−1}}t_{m−2}^{p^{σ(2)−1}} ⋯ t_{1}^{p^{σ(m−1)−1}}s_{1}^{p^{σ(m)−1}−1}

Hence

c = ( ∑_{σ∈S_{m}} φ_{σ})^{p−1}

in CH_{0}(R) = Z.

We have s_{1} = x_{p}, t_{1} = x_{p−1} (with i = 1 above). By Lemma 8.7 we have

(R_{1} → R_{2})_{∗}(x_{p−1}^{(p−1)k}x_{p}^{(p−1)h}) = \begin{cases}0 & if h = 0 or k + h < p \\ s_{2}^{(p−1)(k+h−p)} & else \end{cases}

Note that if we replace formally in the sum

∑_{σ∈S_{m}} φ_{σ}

the variables s_{1}, t_{1} with s_{2}, we get 0. Thus the same statement holds also for

( ∑_{σ∈S_{m}} φ_{σ})^{p−1}

For

ω = (σ_{1}, …, σ_{p−1}) ∈ (S_{m})^{p−1}

let

Φ_{ω} = φ_{σ_{1}} ⋯ φ_{σ_{p−1}}

If σ_{1}(m) = ⋯ = σ_{p−1}(m) = 1, then (use (8.2.1) for r = p and i = 0)

(R_{1} → R_{2})_{∗}(Φ_{ω}) = 0

Otherwise

(R_{1} → R_{2})_{∗}(Φ_{ω})

is formally obtained from Φ_{ω} by replacing the variables s_{1}, t_{1} with s_{2} and then dividing by s_{2}^{p(p−1)}. Hence we have (up to sign)

c = ( ∑_{σ∈S_{m}, σ(m)=1} sgn(σ)t_{m−1}^{p^{σ(1)−1}} ⋯ t_{2}^{p^{σ(m−2)−1}}s_{2}^{p^{σ(m−1)−1}−1})^{p−1}

in CH_{0}(R_{2}) = Z. Or:

c = ( ∑_{σ∈S_{m−1}} sgn(σ)t_{m−1}^{p^{1+σ(1)−1}} ⋯ t_{2}^{p^{1+σ(m−2)−1}}s_{2}^{p^{1+σ(m−1)−1}−1})^{p−1}

= ( ∑_{σ∈S_{m−1}} sgn(σ)(t_{m−1}^{p})^{p^{σ(1)−1}} ⋯ (t_{2}^{p})^{p^{σ(m−2)−1}}(s_{2}^{p})^{p^{σ(m−1)−1}−1})^{p−1}

We have s_{2}^{p} = x_{p}, t_{2}^{p} = x_{p−1} (with i = 2 above). By Lemma 8.7 we have

(R_{2} → R_{3})_{∗}(x_{p−1}^{(p−1)k}x_{p}^{(p−1)h}) = \begin{cases}0 & if h = 0 or k + h < p \\ (s_{3}^{p})^{(p−1)(k+h−p)} & else \end{cases}

Proceeding further this way, we end up with

c = (∑_{σ∈S_{1}} sgn(σ))^{p−1} = 1

8.4. Further computations. The remarks of this section have been found some weeks after typing the Subsection 8.3. In the end they will simplify the presentation of the computation of c(X) considerably.

We keep the notations from Subsection 8.3.

Let

H = Q̂((γ′_{1}) ⋯ (γ′_{m−1}) ∪ α) ∈ CH^{b}(T)

Let

\overline{s}_{i} = s_{i}^{p^{i−1}} ∈ CH^{p^{i−1}}(R_{i}) ⊂ CH^{p^{i−1}}(R)
\overline{t}_{i} = t_{i}^{p^{i−1}} ∈ CH^{p^{i−1}}(R_{i}) ⊂ CH^{p^{i−1}}(R)

There is the following remarkable decomposition of H:

Lemma 8.8. One has (up to sign)

H = w ∏_{i=1}^{m−1}(\overline{t}_{i}^{p} − \overline{t}_{i}\overline{s}_{i}^{p−1})

Proof. Once this statement is formulated, it can be easily proved by the arguments used before in Subsection 8.3. In fact, the proof is much less tedious. We omit the details for now. □

Corollary 8.9. c(X) = 1 mod p.

Proof. This is now easy: Note that

\overline{t}_{i}^{p} − \overline{t}_{i}\overline{s}_{i}^{p−1} = f(\overline{t}_{i}, \overline{s}_{i})

where f is as in the proof of (8.2.2). The proof of (8.2.2) yields

(\overline{t}_{i}^{p} − \overline{t}_{i}\overline{s}_{i}^{p−1})^{p−1} + (\overline{s}_{i})^{(p−1)p} = h(\overline{t}_{i}, \overline{s}_{i}) = \overline{s}_{i+1}^{p−1}

From this one gets

(R_{i} → R_{i+1})_{∗}((\overline{t}_{i}^{p} − \overline{t}_{i}\overline{s}_{i}^{p−1})^{p−1}) = ±1

Plugging this into the formula of Lemma 8.8 yields the claim. □

Lemma 8.8 was found by a mere computation. I don’t have yet a geometric explanation of it.

Here is a related remark, which also deserves some explanation:

Lemma 8.10. One has

(S_{r+1} → S_{r})_{∗}(x_{r+1}^{2p−1}) = f(x_{r−1}, x_{r})

9. Relations with characteristic numbers

The material of this section is not needed for the proof of the Bloch-Kato conjecture.

Lemma 9.2 of this section was inspired by [3, Lemma 4.4].

Let

S_{•} = S_{0} + S_{1} + ⋯: CH_{∗}(X) ⊗ Z/p → CH_{∗}(X) ⊗ Z/p
S_{h}: CH_{j}(X) ⊗ Z/p → CH_{j−h(p−1)}(X) ⊗ Z/p

and

P^{•} = P^{0} + P^{1} + ⋯: CH^{∗}(X) ⊗ Z/p → CH^{∗}(X) ⊗ Z/p
P^{h}: CH^{j}(X) ⊗ Z/p → CH^{j+h(p−1)}(X) ⊗ Z/p

be the Steenrod operations of Brosnan [1]. The two variants are related by

S_{•}(α) = P^{•}(α) · b_{•}(T X)

where b_{•}(−) is the characteristic class with

b_{•}(V) = ∑_{i≥0} b_{i(p−1)}(V)
b_{•}(V ⊕ W) = b_{•}(V)b_{•}(W)
b_{•}(L) = (1 + c_{1}(L)^{p−1})^{−1}

for vector bundles V, W and line bundle L. (For p = 2 one gets the Segre classes b_{•}(V) = c_{•}(−V).)

Let

b = \frac{p^{n} − 1}{p − 1}

Let X be a smooth proper irreducible variety of dimension

d = b(p − 1) = p^{n} − 1

Theorem 9.1. Suppose p = 2 or n = 1.

Suppose I(X) ⊂ pZ. Suppose further that there exists a special correspondence ρ ∈ CH^{b}(X^{2}). Then

\frac{b_{d}(X)}{p} ≠ 0 mod p

The same statement holds without the restriction p = 2 or n = 1, see below.

Proof. For Y/k proper with

I(Y) = deg_{Y/k}(CH_{0}(Y)) ⊂ pZ

let

DEG_{Y} : CH_{0}(Y)/p → pZ/p^{2}Z
DEG_{Y} (u mod p) = deg_{Y/k}(u) mod p^{2}

for u ∈ CH_{0}(Y).

We consider the following commutative diagram

CH_{d}(X^{2}) ⟶^{S_{d}} CH_{0}(X^{2})/p ⟶^{DEG_{X^{2}}} pZ/p^{2}Z
↓^{(π_{0})_{∗}}  ↓^{(π_{0})_{∗}}  ∥
CH_{d}(X) ⟶^{S_{d}} CH_{0}(X)/p ⟶^{DEG_{X}} pZ/p^{2}Z

One has

S_{d}([X]) = b_{d}(T X)

and

(π_{0})_{∗}((ρ)^{p−1}) = c(ρ)[X]

One has

S_{•}(ρ^{p−1}) = P^{•}(ρ^{p−1})b_{•}(T X^{2}) = P^{•}(ρ)^{p−1}b_{•}(T X^{2})

For n = 1 we have b = 1. In this case one has

P^{•}(ρ) = ρ + ρ^{p}

Hence

S_{•}(ρ^{p−1}) = (ρ^{p−1} + (p − 1)ρ^{2(p−1)})(1 + b_{d}(T X_{0}) + b_{d}(T X_{1}))

Therefore

c(ρ)b(X) = (1 + (−1)^{p−1})c(ρ)b(X)

+ (−1)^{p−1}(p − 1)\binom{2(p − 1)}{p − 1}c(ρ)^{2} mod pI(X)

This proves the claim.

For p = 2 we have b = d. It suffices to show that if we represent S_{d}(ρ) by an integral zero cycle, then the degree of this zero cycle is not divisible by 4.

We have

(9.0.1)  S_{d}(ρ) = ρb_{d}(T X^{2}) + P^{1}(ρ)b_{d−1}(T X^{2}) + ⋯ + P^{d}(ρ)

For the last term we have P^{d}(ρ) = ρ^{2} mod 2 and one has

deg(ρ^{2}) = −2c(ρ)^{2}

It suffices to show that all the other terms on the right hand side of (9.0.1) have integral representatives with degrees divisible by 4.

Let X_{0} = X_{1} = X and write X^{2} = X_{0} × X_{1}. Let F be the function field of X.

For the first term on the right hand side of (9.0.1) we have over F by dimension reasons

ρ_{F} b_{d}(T X^{2}) = (H × X − X × H) ∑_{j+r=d} b_{j}(T X_{0})b_{r}(T X_{1})
= (H × b_{d}(T X_{1})) − (b_{d}(T X_{0}) × H)

This zero cycle has obviously degree 0.

As for the remaining terms: Let 1 ≤ i ≤ d − 1. Note that for H ∈ CH^{d}(X_{F}) ⊗ Z/2 we have P^{i}(H) = 0 by dimension reasons. Since ρ_{F} = H × X − X × H we get

(9.0.2)  P^{i}(ρ)_{F} = 0

in CH^{d+i}(X_{F}^{2} ) ⊗ Z/2.

Let β ∈ CH^{d+i}(X^{2}) be an integral representative of P^{i}(ρ). We have to show that deg(βb_{d−i}(T X^{2})) ∈ Z is 4-divisible. By (9.0.2) there exists γ ∈ CH^{d+i}(X_{F}^{2} ) such that β_{F} = 2γ. It remains to show that deg(γb_{d−i}(T X^{2})) ∈ Z is 2-divisible. Now

b_{d−i}(T X^{2}) = ∑_{j+r=d−i} b_{j}(T X_{0})b_{r}(T X_{1})

It remains to show that all the numbers deg(γb_{j}(T X_{0})b_{r}(T X_{1})) ∈ Z are 2-divisible. We have j + r = d − i > 0. We assume r > 0 (the case j > 0 is similar). Then

deg(γb_{j}(T X_{0})b_{r}(T X_{1})) = deg_{X_{1}_{F} /F} ((π_{0})_{∗}(γb_{j}(T X_{0}))b_{r}(T X_{1}))

This number is 2-divisible because of the following Lemma 9.2. □

Lemma 9.2. Let p = 2. Suppose that there exists a special correspondence ρ ∈ CH^{b}(X^{2}). Let r > 0. Let α ∈ CH^{r}(X) and β ∈ CH^{d−r}(X_{F}). Then

deg_{X_{F} /F} (α_{F} β) ∈ I(X) ⊗ Z_{(2)}

Proof. Let φ ∈ CH^{d−r}(X^{2}) be a preimage of β under the natural map

CH^{d−r}(X^{2}) → CH^{d−r}(X × Spec F)

and consider the zero cycle

ω = ρ(α × X)φ ∈ CH^{2d}(X^{2})

Its degree is in I(X^{2}) = I(X). We compute its degree over F. One has

ω_{F} = (H × X)(α_{F} × X)φ_{F} − (X × H)(α_{F} × X)φ_{F}

The first term vanishes because Hα_{F} = 0 by dimension reasons. As for the second term note that

(π_{0})_{∗}((α_{F} × X)φ_{F} ) = deg_{X_{F} /F} (α_{F} β)[X]

Hence

deg(ω) = −c(ρ) deg_{X_{F} /F} (α_{F} β)
□

Here is the variant of Lemma 9.2 for any p.

Let

J(X) = { n ∈ Z | n^{p−1} ∈ I(X) }

If I(X) ⊂ pZ, then J(X) ⊂ pZ.

Lemma 9.3. Let r > 0. Suppose that there exists a special correspondence ρ ∈ CH^{b}(X^{2}). Let α ∈ CH^{r}(X) and β ∈ CH^{d−r}(X_{F}). Then

deg_{X_{F} /F} (α_{F} β) ∈ J(X) ⊗ Z_{(p)}

Proof. Let φ ∈ CH^{d−r}(X^{2}) be a preimage of β under the natural map

CH^{d−r}(X^{2}) → CH^{d−r}(X × Spec F)

Then

(π_{0})_{∗}(α_{[0]}φ_{[0,1]}) = deg_{X_{F} /F} (α_{F} β)[X_{1}]

Since

(π_{1})_{∗}([X_{1}]ρ_{[1,2]}^{p−1}) = c(ρ)[X_{2}]

we get

c(ρ) deg(α_{F} β)[X_{2}] = (π_{0,1})_{∗}(α_{[0]}φ_{[0,1]}ρ_{[1,2]}^{p−1})
= (π_{0,1})_{∗}(α_{[0]}φ_{[0,1]}(ρ_{[0,2]} − ρ_{[0,1]})^{p−1})
= ∑_{i=0}^{p−1} \binom{p − 1}{i}(−1)^{i}(π_{0})_{∗}(ρ_{[0,2]}^{p−1−i}(π_{1})_{∗}(α_{[0]}φ_{[0,1]}ρ_{[0,1]}^{i}))

= ∑_{i=1}^{p−1} \binom{p − 1}{i}(−1)^{i}d_{ρ}((π_{1})_{∗}(α_{[0]}φ_{[0,1]}ρ_{[0,1]}^{i}))[X_{2}]

Here the term for i = 0 vanishes. Namely, since r > 0 one has (π_{1})_{∗}(φ_{[0,1]}) = 0 by dimension reasons and therefore (π_{1})_{∗}(α_{[0]}φ_{[0,1]}) = α_{[0]}(π_{1})_{∗}(φ_{[0,1]}) = 0. The claim follows from Corollary 5.7. □

In the following we are going to prove Theorem 9.1 without the restriction p = 2 or n = 1. We assume that we are given a special correspondence ρ ∈ CH^{b}(X^{2}). We also assume I(X) ⊂ pZ. Moreover, as before, we use the notations ρ_{L} = H × X − X × H with L = F = k(X).

38 MARKUS ROST

Definition 9.4. For i ≥ 0 define

e_{i}(ρ) ∈ Z/p

as follows. Define h_{i} ∈ Q by

i(p − 1) + b(h_{i} + 1) = b(p − 1)

If h_{i} is a nonnegative integer, we set

e_{i}(ρ) = deg(P^{i}(H)H^{h_{i}}) ∈ Z/p

Otherwise we set e_{i}(ρ) = 0.

For i = 0 we get h_{0} = p − 2 and

e_{0}(ρ) = c(ρ)

Note that if i > 0, then h_{i} < p − 2. Lemma 9.5. For i > 0 one has

e_{i}(ρ) = 0

Proof. Let

α = (π_{0})_{∗}(P^{i}(ρ)ρ^{h_{i}+1}) ∈ CH^{b}(X) ⊗ Z/p

Then

α_{L} = (π_{0})_{∗}((P^{i}(H_{0}) − P^{i}(H_{1}))(H_{0} − H_{1})^{h_{i}+1})

Since h_{i} + 1 < p − 1 one has

α_{L} = ue_{i}(ρ)H

for some unit u ∈ Z_{(p)}^{×}. The claim follows from Corollary 5.7. □ We consider the elements (θ_{i})_{[0,1]} = c(ρ)^{−2}(π_{2,3})_{∗}(P^{i}(ρ_{[2,3]})ρ_{[0,2]}^{p−1}ρ_{[1,3]}^{p−1}) ∈ CH^{b+i(p−1)}(X^{2}) ⊗ Z/p

and

ω_{i} = c(ρ)^{−1}(π_{1})_{∗}(P^{i}(ρ)ρ^{p−1}) ∈ CH^{b+i(p−1)}(X) ⊗ Z/p

Lemma 9.6.

P^{i}(ρ) = θ_{i} − (π_{0}^{∗} − π_{1}^{∗})(ω_{i})

Proof.

(θ_{i})_{[0,1]} = c(ρ)^{−2}(π_{2,3})_{∗}((P^{i}(ρ_{[0,1]}) − P^{i}(ρ_{[0,2]}) + P^{i}(ρ_{[1,3]}))ρ_{[0,2]}^{p−1}ρ_{[1,3]}^{p−1})
= P^{i}(ρ_{[0,1]}) − (ω_{i})_{[0]} + (ω_{i})_{[1]}
□

Lemma 9.7. For i > 0 one has

(θ_{i})_{L} = 0

in CH^{b+i(p−1)}(X_{L}^{2}) ⊗ Z/p.

Proof.

(θ_{i})_{L} = c(ρ)^{−2}(π_{2,3})_{∗}((P^{i}(H_{2}) − P^{i}(H_{3}))(H_{0} − H_{2})^{p−1}(H_{1} − H_{3})^{p−1})

The claim follows from Lemma 9.5. □

Lemma 9.8. Let j, k > 0 with j + k + rb = 2d for some r ≤ p − 1. Let α ∈ CH^{j}(X), β ∈ CH^{k}(X). Then

deg(π_{0}^{∗}(α)π_{1}^{∗}(β)ρ^{r}) ∈ J(X)^{2} ⊗ Z_{(p)}

Proof. Computing this degree over L, one gets essentially d_{ρ}(α)d_{ρ}(β). The claim follows from Corollary 5.7. □

Theorem 9.9. Suppose I(X) ⊂ pZ. Suppose further that there exists a special correspondence ρ ∈ CH^{b}(X^{2}). Then

\frac{b_{d}(X)}{p} ≠ 0 mod p

Proof. Let A be the 2d-dimensional component (so A is a zero cycle) of

S_{•}(ρ^{p−1}) = P^{•}(ρ)^{p−1}b_{•}(T X^{2})
= (ρ + P^{1}(ρ) + ⋯ + P^{b−1}(ρ) + ρ^{p})^{p−1}b_{•}(T X_{0})b_{•}(T X_{1})

The zero cycle A is a sum of terms

U = P^{i_{1}}(ρ) ⋯ P^{i_{p−1}}(ρ)b_{j}(T X_{0})b_{k}(T X_{1})

of total degree 2d. We consider various cases.

Case 1. i_{1} = ⋯ = i_{p−1} = 0. In this case

U = ρ^{p−1}(b_{d}(T X_{0}) + b_{d}(T X_{1}))

The degree of this integral zero cycle is (1 + (−1)^{p−1})c(ρ)b_{d}(X).

Case 2. After a permutation of the i_{j} one has i_{1} = ⋯ = i_{p−2} = 0, i_{p−1} = b. (So this case appears exactly p − 1 times). In this case

U = ρ^{2(p−1)}

The degree of this integral zero cycle is upc(ρ)^{2} for some p-unit u.

It suffices to show that for all other cases the zero cycle U has an integral representative with degree divisible by p^{2}.

After plugging the formula of Lemma 9.6 into U we get zero cycles of the form

V = θ_{i_{1}} ⋯ θ_{i_{r}}π_{0}^{∗}(α)π_{1}^{∗}(β)

with α ∈ CH^{j}(X), β ∈ CH^{k}(X). The α, β are products of the b_{j}(T X) and of integral representatives of the ω_{i}.

We choose integral representatives θ̃_{i_{j}} of θ_{i_{j}} and put

Ṽ = θ̃_{i_{1}} ⋯ θ̃_{i_{r}}π_{0}^{∗}(α)π_{1}^{∗}(β)

which is an integral zero cycle. We have to show that deg(Ṽ) is p^{2}- divisible in all the cases not coming from Case 1 or Case 2.

Case 3. r ≥ 2 and i_{1}, i_{2} > 0. In this case one computes the degree over L. The p^{2}-divisibility follows from Lemma 9.7.

Case 4. j = k = 0. This case appears only when r = p − 1 (because all the ω_{i} are positive dimensional). But since we have excluded Case 1 and Case 2, we will be in Case 3.

Case 5. i_{1} = ⋯ = i_{r} = 0. Then r < p − 1, because otherwise we would come from Case 1.

Suppose j = 0. Then

(π_{0})_{∗}(V) = (π_{0})_{∗}(ρ^{r})β = 0

since (π_{0})_{∗}(ρ^{r}) = 0.

Similarly for k = 0.

Suppose j, k > 0. Then

V = ρ^{r}π_{0}^{∗}(α)π_{1}^{∗}(β)

The p^{2}-divisibility follows Lemma 9.8.

There remains the following case:

Case 6. k > 0 or j > 0, r ≥ 1, i_{1} ≥ 1. We assume k > 0. Let

W = (π_{0})_{∗}(θ̃_{i_{1}} ⋯ θ̃_{i_{r}}π_{0}^{∗}(α))

Then

W_{L} = 0 mod p

by Lemma 9.7.

The p^{2}-divisibility follows Lemma 9.3.

□

Here is a very general fact about smooth proper varieties X of dimension d = b(p − 1).

Lemma 9.10. Suppose that d ≠ p^{m} − 1 for all m. Then

b_{d}(X) ∈ pI(X) + p^{2}Z_{(p)}

Proof. First recall that

b_{d}(X) ∈ pZ

for all d > 0 and X. By the multiplicativity property of the series b_{i}, this implies

b_{d}(Y × Z) ∈ p^{2}Z

if dim Y, dim Z > 0.

Let Ω^{∗} be the complex cobordism ring. Suppose that d ≠ p^{m} − 1 for all m. Let x_{0} ∈ Ω^{d} with s_{d}(x_{0}) ≠ 0 mod p. Consider the characteristic class

f(V) = b_{d}(V) − b_{d}(x_{0})s_{d}(x_{0})^{−1}s_{d}(V)

over Z_{(p)}. Then f(x_{0}) = 0. Moreover f(yz) ∈ p^{2}Z_{(p)} for positive dimensional y, z ∈ Ω^{∗}. By the known structure of Ω^{∗}, this implies

f(x) ∈ p^{2}Z_{(p)}

for x ∈ Ω^{d}. By the Hattori-Stong theorem and by Riemann-Roch it follows that

f(X) ∈ p^{2}Z_{(p)}

for any X over any field. The claim follows from

b_{d}(X) = f(X) + \frac{b_{d}(x_{0})}{p}s_{d}(x_{0})^{−1}ps_{d}(X)

□

Example 9.11. Let p = 2 and d = 2. Then

b_{2} = c_{1}^{2} − c_{2} = (c_{1}^{2} + c_{2}) − 2c_{2}

and therefore

b_{2}(X) = 4 Todd(X) − 2c_{2}(X)

Corollary 9.12. Suppose I(X) ⊂ pZ. Suppose further that there exists a special correspondence ρ ∈ CH^{b}(X^{2}). Then

d = p^{m} − 1

for some m.

For the case of quadrics see [2, Theorem 6.1], [3, Theorem 5.1].

Proof. This is clear from Theorem 9.9 and Lemma 9.10. □

Here is the variant of Lemma 9.10 for d = p^{m} − 1.

Lemma 9.13. Suppose that d = p^{m} − 1 for some m. Then

b_{d}(X) = ±s_{d}(X) mod (pI(X) + p^{2}Z_{(p)})

Proof. Let x_{0} ∈ Ω^{d} with s_{d}(x_{0}) = p. Consider the characteristic class

f(V) = b_{d}(V) − \frac{b_{d}(x_{0})}{p}s_{d}(V)

As in the proof of Lemma 9.10 one gets

f(X) ∈ p^{2}Z

We have

b_{d}(X) = f(X) + \frac{b_{d}(x_{0})}{p}s_{d}(X)

The number

\frac{b_{d}(x_{0})}{p} mod p

is the coefficient of P^{b} when expressing q_{m} in the P^{i} (by some well known formula). This coefficient is equal to ±1 . □

References

[1] P. Brosnan, Steenrod operations in Chow theory, Trans. Amer. Math. Soc. 355 (2003), no. 5, 1869–1903 (electronic).

[2] O. Izhboldin and A. Vishik, Quadratic forms with absolutely maximal splitting, Quadratic forms and their applications (Dublin, 1999), Contemp. Math., vol. 272, Amer. Math. Soc., Providence, RI, 2000, pp. 103–125.

[3] N. Karpenko and A. Merkurjev, Rost projectors and Steenrod operations, Doc. Math. 7 (2002), 481–493 (electronic).

[4] M. Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393 (electronic).

[5] , The motive of a Pfister form, Preprint, 1998, ⟨www.math.uni-biele feld.de/˜ rost/motive.html⟩.

[6] , Chain lemma for splitting fields of symbols, Preprint, 1998, ⟨www. math.uni-bielefeld.de/˜ rost/chain-lemma.html⟩.

[7] A. Suslin and S. Joukhovitski, Norm varieties, J. Pure Appl. Algebra 206 (2006), no. 1-2, 245–276.

[8] V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. (2003), no. 98, 1–57.

[9] , Motivic cohomology with Z/2-coefficients, Publ. Math. Inst. Hautes Études Sci. (2003), no. 98, 59–104.

[10] , Motives over simplicial schemes, Preprint, 2003, K-theory Preprint Archives, No. 638, ⟨www.math.uiuc.edu/K-theory/0638/⟩.

[11] , Motivic cohomology with Z/l-coefficients, Preprint, 2003, K-theory Preprint Archives, No. 639, ⟨www.math.uiuc.edu/K-theory/0639/⟩.

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

E-mail address: rost at math.uni-bielefeld.de
URL: http://www.math.uni-bielefeld.de/~rost