AROUND RATIONALITY OF INTEGRAL CYCLES

RAPHAËL FINO

Abstract. In this article we prove a result comparing rationality of integral algebraic cycles over the function field of a quadric and over the base field. This is an integral version of the result known for ℤ/2ℤ-coefficients. Those results have already been proved by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

In many situations it can be important to know, if an element of the Chow group of some variety considered over an algebraic closure of its base field, is actually defined over the base field itself. In a previous paper ([4]), we already dealt with Chow groups modulo 2. In particular, we showed that for a Chow group modulo 2 element of codimension m, it is sufficient to check that it is defined over the function field of a sufficiently large quadric Q (of dimension > 2m), to get that it is defined over the base field (up to an element of exponent 2, see [4, Theorem 1.1]). Furthermore, we did this without the help of algebraic cobordism theory (it was the case in [7]) but with the only help of the Chow theory itself (including the Steenrod operations on Chow groups modulo 2). This allowed one to get a valid result in any characteristic different from 2 (however, the use of symmetric operations in algebraic cobordism theory by Alexander Vishik in [7] permitted him to obtain a statement without an exponent 2 element).

In this note, we prove an integral version of [4, Theorem 1.1] (generalizing in a way [8, Theorem 3.1] to any characteristic different from 2). We work again with the only help of Steenrod operations on Chow groups modulo 2, which is remarkable. This integral version (Theorem 3.1), whose the statement is very close to the ℤ/2ℤ-coefficients case, requires an additional condition on Q (aside from its size) saying that Q has a projective line defined over the generic point of Q.

I would like to gratefully thank Nikita Karpenko for sharing his great knowledge and his valuable advice. This work could not have been done without his help.

Contents

1. Preliminaries: decomposition of Chow groups 2 2. Preliminaries: Steenrod operations and correspondences 3 3. Main theorem 4 4. A stronger version of main theorem 10 References 13

Date: 11 March 2012.

1. Preliminaries: decomposition of Chow groups

In this paper, the word scheme means a separated scheme of finite type over a field and a variety is an integral scheme. Let F be a field of characteristic ≠ 2 and Y be an F -variety. We write CH(Y) for the integral Chow group of Y (see [3, Chapter X]) and we write Ch(Y) for CH(Y) modulo 2.

The main purpose of this section is to introduce the notion of coordinates for a cycle x ∈ CH(Q × Y), where Q is a smooth projective quadric over F and Y is a smooth F -variety. This notion will be useful during the proofs of Theorem 3.1 and Theorem 4.1.

Let Q be a smooth projective quadric over F of dimension n given by a quadratic form φ, and let us set i_{0}(Q) := i_{0}(φ), where i_{0}(φ) is the Witt index of φ.

For i = 0, ..., n, let us denote as h^{i} ∈ CH^{i}(Q) the ith power of the hyperplane section class (note that for any i, the cycle h^{i} is defined over the base field). For i < i_{0}(Q), let us denote as l_{i} ∈ CH_{i}(Q) the class of an i-dimensional totally isotropic subspace of ℙ(V), where V is the underlying vector space of φ. For i ≤ [n/2], we still write l_{i} ∈ CH_{i}(Q\overline{F}) for the class of an i-dimensional totally isotropic subspace of ℙ(V\overline{F}), where \overline{F} is an algebraic closure of F (if i < i_{0}(Q), the cycle l_{i} ∈ CH_{i}(Q\overline{F}) is the image of l_{i} ∈ CH_{i}(Q) under the change of field homomorphism CH(Q) → CH(Q\overline{F})). Let us notice that for i < [n/2], the cycle l_{i} (in CH_{i}(Q\overline{F}) or in CH_{i}(Q) if i < i_{0}(Q)) is canonical by [3, Proposition 68.2] (in case of even n, there are two classes of n/2-dimensional totally isotropic subspaces and we fix one of the two).

Let x be an element of CH^{r}(Q × Y). We write pr for the projection Q × Y → Y. For every i = 0, ..., i_{0}(Q) − 1, we have the following homomorphisms

\begin{matrix}CH^{r}(Q × Y) & ⟶ & CH^{r−i}(Y) \\ x & ⟼ & pr_{∗}(l_{i} · x) =: x^{i} \end{matrix} ,

and

\begin{matrix}CH^{r}(Q × Y) & ⟶ & CH^{r−n+i}(Y) \\ x & ⟼ & pr_{∗}(h^{i} · x) =: x_{i} \end{matrix} .

Definition 1.1. The cycle x^{i} ∈ CH^{r−i}(Y) is called the coordinate of x on h^{i} while x_{i} ∈ CH^{r−n+i}(Y) is called the coordinate of x on l_{i}.

Note that if r < [n/2], for any i = 0, ..., i_{0}(Q) − 1, one has x_{i} = 0 by dimensional reasons.

Remark 1.2. For any nonnegative integer k < i_{0}(Q), let us set x(k) := x − ∑_{i=0}^{k} h^{i} × x^{i} − ∑_{i=0}^{k} l_{i} × x_{i}. Note that for any i = 0, ..., k, the coordinate of x(k) on h^{i} (as well as its coordinate on l_{i}) is 0. The writing

x = x(k) + ∑_{i=0}^{k} h^{i} × x^{i} + ∑_{i=0}^{k} l_{i} × x_{i}

is called a decomposition of x.

Assume now that r < i_{0}(Q) and r ≤ k. Then, by [3, Theorem 66.2], one can write

x(k) = ∑_{i=0}^{r} h^{i} × w^{i}

with some w^{i} ∈ CH^{r−i}(Y). Since, for any i = 0, ..., r, the cycle w^{i} coincides with the coordinate x(k)^{i} of x(k) on h^{i}, we get that x(k) = 0.

Recall that one says that the quadric Q is completly split if i_{0}(Q) = [n/2] + 1.

Remark 1.3. Assume that Y = Q, r = n, and that k < [n/2] (what is the case if the quadric Q is not completly split). Let x be an element of CH^{n}(Q × Q). Since, for i = 0, ..., k, the group CH^{n−i}(Q) is free with basis {l_{i}} (because i < [n/2], see [3, §68]), one can uniquely write

x = x(k) + ∑_{i=0}^{k} b_{i} · h^{i} × l_{i} + ∑_{i=0}^{k} l_{i} × x_{i},

with some b_{i} ∈ ℤ.

Note that everything in Section 1 holds for Chow groups modulo 2 in place of the integral Chow groups.

2. Preliminaries: Steenrod operations and correspondences

In this section we continue to use notation introduced in the beginning of Section 1.

The Steenrod operations are the main tool of this note. We refer to [3, Chapter XI] for an introduction to the subject. We just recall here that for a smooth scheme X over a field F (of characteristic ≠ 2), there is a certain homomorphism S_{X} : CH(X) → CH(X) called the total Steenrod operation on X of cohomological type. Although S_{X} is constructed only for quasi-projective X in [3], Patrick Brosnan has extended the operation S_{X} to any scheme X in [2, §10]. In particular, the following proposition holds for any smooth scheme (not necessarily quasi-projective). This allows us to get rid of the assumption of quasiprojectivity for the main theorem of this note (Theorem 3.1) (Alexander Vishik needed that assumption in [8] because the algebraic cobordism theory is defined on the category of smooth quasi-projective schemes over F, see [6]).

In the following proposition, whose the statement and the proof are very close to [5, Lemma 3.1], we focus on how the Steenrod operations interact with the composition of correspondences (correspondences are defined in [3, §62]). This will be useful during the proof of Theorem 3.1.

Let X_{1}, X_{2}, X_{3} be smooth schemes over F (of characteristic ≠ 2), and assume that X_{2} is complete (so the push-forward associated with the projection X_{1}×X_{2}×X_{3} ⟶ X_{1}×X_{3} is well defined).

Proposition 2.1. For any correspondence α ∈ CH(X_{1} ×X_{2}) and for any correspondence β ∈ CH(X_{2} × X_{3}), one has

1) S_{X_{1}×X_{3}}(β ∘ α) = (S_{X_{2}×X_{3}}(β) · c(−T_{X_{2}})) ∘ S_{X_{1}×X_{2}}(α); 2) S_{X_{1}×X_{3}}(β ∘ α) = S_{X_{2}×X_{3}}(β) ∘ (S_{X_{1}×X_{2}}(α) · c(−T_{X_{2}})),

where T_{X_{2}} is the tangent bundle of X_{2} and c is the total Chern class.

Proof. For any i, j ∈ {1, 2, 3} such that i < j, let us write p_{ij} for the projection

X_{1} × X_{2} × X_{3} ⟶ X_{i} × X_{j}.

According to the composition rules of correspondences described in [3, §62], we have

β ∘ α = p_{13}_{∗}(p_{12}^{∗}(α) · p_{23}^{∗}(β)).

Therefore, by [3, Proposition 61.10] applied to p_{13}, we get

S_{X_{1}×X_{3}}(β ∘ α) = p_{13}_{∗}(S_{X_{1}×X_{2}×X_{3}}(p_{12}^{∗}(α) · p_{23}^{∗}(β)) · p_{12}^{∗}(pr_{2}^{∗}(c(−T_{X_{2}})))),

and since S conmmutes with the products and the pull-backs, we get

S_{X_{1}×X_{3}}(β ∘ α) = p_{13}_{∗}(p_{12}^{∗}(S_{X_{1}×X_{2}}(α)) · p_{23}^{∗}(S_{X_{2}×X_{3}}(β)) · ([X_{1}] × c(−T_{X_{2}}) × [X_{3}])),

this gives, on the one hand

S_{X_{1}×X_{3}}(β ∘ α) = p_{13}_{∗}(p_{12}^{∗}(S_{X_{1}×X_{2}}(α)) · p_{23}^{∗}(S_{X_{2}×X_{3}}(β) · c(−T_{X_{2}}))),

thus 1) is proved, and on the other hand, this gives

S_{X_{1}×X_{3}}(β ∘ α) = p_{13}_{∗}(p_{12}^{∗}(S_{X_{1}×X_{2}}(α) · c(−T_{X_{2}})) · p_{23}^{∗}(S_{X_{2}×X_{3}}(β))),

thus 2) is proved. □

3. Main theorem

In this section we continue to use notation introduced in the beginnings of Sections 1 and 2.

Let F be a field of characteristic ≠ 2 and let Y be a smooth F -variety. We write \overline{Y} := Y\overline{F} where \overline{F} is an algebraic closure of F. Let X be a geometrically integral variety over F. An element \overline{y} of CH(\overline{Y}) (or of Ch(\overline{Y})) is F (X)-rational if its image \overline{y}_{\overline{F} (X)} under CH(\overline{Y}) → CH(Y_{\overline{F} (X)}) (resp. Ch(\overline{Y}) → Ch(Y_{\overline{F} (X)})) is in the image of CH(Y_{F (X)}) → CH(Y_{\overline{F} (X)}) (resp. Ch(Y_{F (X)}) → Ch(Y_{\overline{F} (X)})). An element \overline{y} of CH(\overline{Y}) (or of Ch(\overline{Y})) is called rational if it belongs to the subgroup \overline{CH}(Y) :=Im(CH(Y) → CH(\overline{Y})) (resp. \overline{Ch}(Y)). Note that since \overline{F} is algebraically closed, the homomorphism CH(\overline{Y}) → CH(Y_{\overline{F} (X)}) (as well as the homomorphism Ch(\overline{Y}) → Ch(Y_{\overline{F} (X)})) is injective by the specialization arguments.

Let Q be a smooth projective quadric over F of positive dimension n (in that case, Q is geometrically integral) given by a quadratic form φ. Since for isotropic Q, any F (Q)-rational element (in any codimension) is rational, we make the assumption that the quadric Q is anisotropic. In particular, Q is not completly split and one can consider the first Witt index i_{1}(φ) of φ, which we simply denote as i_{1}.

In a way, the following result is a generalization of [8, Theorem 3.1]. Indeed, the use of the Steenrod operations on the modulo 2 Chow groups allows one to obtain a valid result in any characteristic different from 2. Nevertheless, an exponent 2 element appears in our conclusion while it is not the case in [8, Theorem 3.1]. The proof is inspired by the proof of [8, Theorem 3.1].

Theorem 3.1. Assume that m < [n/2] and i_{1} > 1. Then any F (Q)-rational element of CH^{m}(\overline{Y}) is the sum of a rational element and an exponent 2 element.

Proof. The statement being trivial for negative m, we may assume that m ≥ 0 in the proof. Let \overline{y} be an F (Q)-rational element of CH^{m}(\overline{Y}). Since the quadric Q is isotropic over \overline{F}, the homomorphism CH(\overline{Y}) → CH(Y_{\overline{F} (Q)}) is surjective and is consequently an isomorphism. The element \overline{y} ∈ CH^{m}(\overline{Y}) being F (Q)-rational, there exists y ∈ CH^{m}(Y_{F (Q)}) mapped to \overline{y} under the homomorphism

CH^{m}(Y_{F (Q)}) → CH^{m}(Y_{\overline{F} (Q)}) ⟶^{∼} CH^{m}(\overline{Y}).

Let us fix an element x ∈ CH^{m}(Q × Y) mapped to y under the surjection (see [3, Corollary 57.11])

CH^{m}(Q × Y) ↠ CH^{m}(Y_{F (Q)}).

Since over \overline{F} the quadric Q becomes completly split and m < [n/2], by Remark 1.2 (applied with r = k = m), the image \overline{x} ∈ CH^{m}(\overline{Q} × \overline{Y}) of x decomposes as

(3.2)  \overline{x} = ∑_{i=0}^{m} h^{i} × x^{i}

where x^{i} ∈ CH^{m−i}(\overline{Y}) is the coordinate of \overline{x} on h^{i} (see Definition 1.1), and where x^{0} = \overline{y} by [7, Lemma 3.2].

Let π ∈ \overline{Ch}_{n+i_{1}−1}(Q^{2}) be the 1-primordial cycle (see [3, Definition 73.16] and paragraph right after [3, Theorem 73.26]). Since i_{1} > 1, by [3, Proposition 83.2], we get that the cycle (h^{0} × h^{i_{1}−1}) · π ∈ \overline{Ch}_{n}(Q^{2}) decomposes as

(3.3)  (h^{0} × h^{i_{1}−1}) · π = ∑_{p=0}^{r} ε_{p}(h^{2p} × l_{2p}) + ∑_{p=0}^{r} ε_{p}(l_{2p+i_{1}−1} × h^{2p+i_{1}−1}),

where ε_{p} ∈ \{0, 1\}, ε_{0} = 1, and r = [\frac{d−i_{1}+1}{2}] with d = [\frac{n}{2}]. Thus, one can choose a rational integral representative \overline{γ} ∈ \overline{CH}_{n}(Q^{2}) of (h^{0} × h^{i_{1}−1}) · π such that \overline{γ} decomposes as

(3.4)  \overline{γ} = ∑_{i=0}^{[ \frac{n}{2} ]} α_{i}(h^{i} × l_{i}) + ∑_{i=0}^{[ \frac{n}{2} ]} β_{i}(l_{i} × h^{i}) + δ(l_{[ \frac{n}{2} ]} × l_{[ \frac{n}{2} ]}),

with some integers α_{i}, β_{i} and δ, where α_{i} is even for all odd i and α_{0} is odd.

The element \overline{γ} being rational, there exists γ ∈ CH_{n}(Q^{2}) mapped to \overline{γ} under the restriction homomorphism CH_{n}(Q^{2}) → CH_{n}(\overline{Q}^{2}). The cycles γ and \overline{γ} are considered here as correspondences of degree 0.

Lemma 3.5. For any i = 0, ..., m, one can choose a rational integral representative s^{i} ∈ CH^{m+i}(\overline{Q} × \overline{Y}) of S^{i}((\overline{x} mod 2) ∘ (\overline{γ} mod 2)) such that

1) for any 0 ≤ j ≤ m, 2s^{i,j} is rational , where s^{i,j} ∈ CH^{m+i−j}(\overline{Y}) is the coordinate of s^{i} on h^{j};

2) for any odd 0 ≤ j ≤ m, s^{i,j} is rational.

Proof. First of all, since m < [n/2], for any j = 0, ..., m, one has h^{n−j} = 2l_{j} (see [3, §68]). Therefore, for any rational cycle s ∈ CH(\overline{Q} × \overline{Y}), the element 2pr_{∗}(l_{j} · s) (where pr is the projection Q × Y → Y) is rational and 1) is proved.

Assume now that j is odd. By Proposition 2.1 1), for any i = 0, ..., m, one has

(3.6) S^{i}((\overline{x} mod 2) ∘ (\overline{γ} mod 2)) = ∑_{k=0}^{m} ∑_{t=0}^{m}(S^{t}(\overline{x} mod 2) · c_{i−k−t}(−T_{Q})) ∘ S^{k}(\overline{γ} mod 2).

For every k = 0, ..., m, let ã^{k} ∈ CH^{n+k}(\overline{Q} × \overline{Q}) be a rational integral representative of S^{k}(\overline{γ} mod 2) ∈ Ch^{n+k}(\overline{Q} × \overline{Q}). We write ã^{k,j} ∈ CH^{n+k−j}(\overline{Q}) for the coordinate of ã^{k} on h^{j}. For every k = 0, ..., m and every t = 0, ..., m, we choose a rational integral representative d_{k,t} ∈ CH^{m+i−k}(\overline{Q} × \overline{Y}) of S^{t}(\overline{x} mod 2) · c_{i−k−t}(−T_{Q}) ∈ Ch^{m+i−k}(\overline{Q} × \overline{Y}). Thus, by the equation (3.6), the cycle

s^{i} := ∑_{k=0}^{m} ∑_{t=0}^{m} d_{k,t} ∘ ã^{k} ∈ CH^{m+i}(\overline{Q} × \overline{Y})

is a rational integral representative of S^{i}((\overline{x} mod 2) ∘ (\overline{γ} mod 2)).

Moreover, for any 0 ≤ k ≤ m, one has by (3.3)

S^{k}(\overline{γ} mod 2) = ∑_{p=0}^{r} ε_{p}S^{k}(h^{2p} × l_{2p}) + ∑_{p=0}^{r} ε_{p}S^{k}(l_{i_{1}−1+2p} × h^{i_{1}−1+2p}).

Therefore, for any 0 ≤ k ≤ m, denoting as a^{k,j} ∈ Ch^{n+k−j}(\overline{Q}) the coordinate of S^{k}(\overline{γ} mod 2) on h^{j}, we have

a^{k,j} = ∑_{(p,t)∈E_{k,j}} ε_{p}\binom{2p}{t}S^{k−t}(l_{2p})

where E_{k,j} = {(p, t) ∈ ⟦0, r⟧ × ⟦0, k⟧ |2p + t = j}.

Furthermore, since j is odd, for any (p, t) ∈ E_{k,j}, the binomial coefficient \binom{2p}{t} is even. Therefore, for any 0 ≤ k ≤ m, we have a^{k,j} = 0 and, consequently, the cycle ã^{k,j} ∈ CH^{n+k−j}(\overline{Q}) is divisible by 2. Since j − k < [n/2], the group CH^{n+k−j}(\overline{Q}) is generated by l_{j−k} and 2l_{j−k} = h^{n+k−j} (see [3, §68]). Hence, for any 0 ≤ k ≤ m, the cycle ã^{k,j} is rational.

According to the composition rules of correspondences described in [3, §62], we have the identity

h^{j} × s^{i,j} = ∑_{k=0}^{m} ∑_{t=0}^{m} d_{k,t} ∘ (h^{j} × ã^{k,j}) = ∑_{k=0}^{m} ∑_{t=0}^{m} h^{j} × pr_{∗}(ã^{k,j} · d_{k,t}).

Therefore, since for any 0 ≤ k ≤ m and for any 0 ≤ t ≤ m, the cycles ã^{k,j} and d_{k,t} are rational, we get that s^{i,j} is rational and 2) is proved. □

Furthermore, we fix a smooth subquadric P of Q of dimension m; we write in for the imbedding

(P ↪ Q) × id_{Y} : P × Y ↪ Q × Y.

Then, considering x as a correspondence, we set

z := in^{∗}(x ∘ γ) ∈ CH^{m}(P × Y).

According to the composition rules of correspondences described in [3, §62] and in view of decompositions (3.2) and (3.4), we get that the image \overline{z} ∈ CH^{m}(\overline{P} × \overline{Y}) of z can be written as

\overline{z} = ∑_{i=0}^{m} α_{i} · h^{i} × x^{i}

(we recall that the integer α_{i} is even for all odd i and that α_{0} is odd). For every i = 0, ..., m, we set z^{i} := α_{i} · x^{i} ∈ CH^{m−i}(\overline{Y}).

Note that since the Steenrod operations of cohomological type commute with in^{∗} (see [3, Theorem 61.9]), for every i = 0, ..., m, the cycle in^{∗}(s^{i}) ∈ CH^{m+i}(\overline{P} × \overline{Y}) (with s^{i} as in Lemma 3.5) is a rational integral representative of S^{i}(\overline{z} mod 2) ∈ Ch^{m+i}(\overline{P} × \overline{Y}).

Lemma 3.7. For any [(m + 1)/2] ≤ m′ ≤ m, the cycle

∑_{i=0}^{m′} \binom{m′ + i + 1}{i}s^{m′−i,m′−i} ∈ CH^{m}(\overline{Y})

is the sum of a rational element \overline{δ_{m′}} and an exponent 2 element.

Proof. For any [(m+1)/2] ≤ m′ ≤ m, we can fix a smooth subquadric P′ of P of dimension m′; we write in_{m′} for the imbedding

(P′ ↪ P) × id_{Y} : P′ × Y ↪ P × Y.

By [4, Lemma 1.2], one has

S^{m′}pr_{m′}_{∗}in_{m′}^{∗}(z mod 2) = ∑_{i=0}^{m′} pr_{m′}_{∗}(c_{i}(−T_{P′}) · in_{m′}^{∗}S^{m′−i}(z mod 2)) in Ch^{m}(Y)

(where T_{P′} is the tangent bundle of P′, c_{i} are the Chern classes, and pr_{m′} is the projection P′ × Y → Y).

If m′ ≥ [(m + 1)/2] + 1, since pr_{m′}_{∗}in_{m′}^{∗}(z mod 2) ∈ Ch^{m−m′}(Y) and m − m′ < m′, we have S^{m′}pr_{m′}_{∗}in_{m′}^{∗}(z mod 2) = 0. Therefore, we get

∑_{i=0}^{m′} pr_{m′}_{∗}(c_{i}(−T_{P′}) · in_{m′}^{∗}S^{m′−i}(z mod 2)) = 0 in Ch^{m}(Y).

Furthermore, by [3, Lemma 78.1], for any i = 0, ..., m′, one has c_{i}(−T_{P′}) ≡ \binom{m′+i+1}{i}h^{i} (mod 2). We deduce that the cycle

∑_{i=0}^{m′} \binom{m′ + i + 1}{i}pr_{m′}_{∗}(h^{i} · in_{m′}^{∗}in^{∗}(s^{m′−i}))

is twice a rational element \overline{δ_{m′}} ∈ CH^{m}(\overline{Y}). Since, by the projection formula ([3, Proposition 56.9]), for any i = 0, ..., m′, one has pr_{m′}_{∗}(h^{i}·in_{m′}^{∗}in^{∗}(s^{m′−i})) = pr_{∗}(h^{n−m′+i}·s^{m′−i}) = 2s^{m′−i,m′−i}, we are done with the case m′ ≥ [(m + 1)/2] + 1.

If m′ = [(m + 1)/2] and m is odd, we still have m − m′ < m′ and we can do the same reasoning as in the first case. If m′ = [(m + 1)/2] and m is even, we have m − m′ = m′ = m/2, and in this case, we have

S^{m/2}pr_{m/2}_{∗}in_{m/2}^{∗}(z mod 2) = (pr_{m/2}_{∗}in_{m/2}^{∗}(z mod 2))^{2}.

Therefore, by the same reasoning as in the first case, there exists δ_{m/2} ∈ CH^{m}(Y) such that

2 ∑_{i=0}^{m/2} \binom{\frac{m}{2} + i + 1}{i}s^{\frac{m}{2} −i, \frac{m}{2} −i} = 2\overline{δ_{m/2}} + (pr_{m/2}_{∗}in_{m/2}^{∗}(\overline{z}))^{2}.

Moreover, we have

(pr_{m/2}_{∗}in_{m/2}^{∗}(\overline{z}))^{2} = (2z^{\frac{m}{2}} )^{2} = 2 · (2z^{\frac{m}{2}} ^{2}),

and since for any i = 0, ..., m, the cycle 2z^{i} = pr_{m}_{∗}(h^{m−i} · \overline{z}) is rational, the cycle

2z^{\frac{m}{2}} ^{2} = pr_{m}_{∗}(\overline{z}^{2}) − 4 ∑_{0≤i≤m_{i≠ \frac{m}{2}}} z^{i} · z^{m−i}

is rational also and we are done with the proof of Lemma 3.7. □

Lemma 3.8. For any j = 0, ..., m, one can choose an integral representative v^{j} ∈ CH^{m}(\overline{Y}) of S^{j}(z^{j} mod 2) such that

1) the cycle 2v^{j} is rational;

2) the cycle v^{j} is rational for odd j;

3) for any k = 0, ..., m, one has s^{k,k} = ∑_{j=0}^{k} a_{j}^{k}v^{j}, where a_{j}^{k} is the binomial coefficient \binom{j}{k−j}.

Proof. We induct on j. For j = 0, one has 2z^{0} = pr_{m}_{∗}(h^{m} · \overline{z}), so we choose v^{0} := z^{0}. For j = 1, one has

S^{1}((\overline{x} mod 2)∘(\overline{γ} mod 2)) = ∑_{i=0}^{m} h^{i}×S^{1}(z^{i} mod 2)+∑_{i=0}^{m} i·h^{i+1}×(z^{i} mod 2) ∈ Ch^{m+1}(\overline{Q}×\overline{Y}).

In the latter expression, the coordinate on h^{1}, whose s^{1,1} is an integral representative, is S^{1}(z^{1} mod 2). Since, by Lemma 3.5, the cycle s^{1,1} is rational, we choose v^{1} := s^{1,1}. Assume that the representatives v^{0}, v^{1}, ..., v^{j−1} are already built.

One has

S^{j}((\overline{x} mod 2) ∘ (\overline{γ} mod 2)) = ∑_{k=0}^{j} ∑_{i=0}^{m} S^{k}(h^{i}) × S^{j−k}(z^{i} mod 2) ∈ Ch^{m+j}(\overline{Q} × \overline{Y}).

In the latter expression, the coordinate on h^{j}, whose s^{j,j} is an integral representative, is

a_{j}^{j} · S^{j}(z^{j} mod 2) + a_{j−1}^{j} · S^{j−1}(z^{j−1} mod 2) + ⋯ + a_{0}^{j} · S^{0}(z^{0} mod 2),

where a_{i}^{j} = \binom{i}{j−i} for any 0 ≤ l ≤ j. Therefore, the cycle

v^{j} := s^{j,j} − (a_{j−1}^{j} · v^{j−1} + ⋯ + a_{0}^{j} · v^{0})

is an integral representative of S^{j}(z^{j} mod 2). Moreover, the element

2s^{j,j} = 2(v^{j} + a_{j−1}^{j} · v^{j−1} + ⋯ + a_{0}^{j} · v^{0})

is rational by Lemma 3.5. By the induction hypothesis, we get that the cycle 2v^{j} is rational. Furthermore, if j is odd, then the cycle s^{j,j} is rational by Lemma 3.5, and for any even 0 ≤ l ≤ j, the binomial coefficient a_{l}^{j} is even. Therefore, by the induction hypothesis, we get that the cycle v^{j} is rational. We are done with the proof of Lemma 3.8. □

Finally, the following lemma will lead to the conclusion. Denote as η(X) the power series ∑_{i≥0} η_{i} · X^{i} in variable X, where η_{l} = (−1)^{l}\binom{2l+1}{l}.

Lemma 3.9. For any polynomial f ∈ ℤ[X] of degree ≤ [m/2], the linear combination

∑_{j=0}^{m} g_{m−j} · v^{j}

is the sum of a rational element and an exponent 2 element, where g(X) = ∑_{l} g_{l} · X^{l} is the power series f(X) · η(X).

Proof. Let f = ∑ f_{k} · X^{k} ∈ ℤ[X] be some polynomial of degree ≤ [m/2]. Consider the element

ε := ∑_{m′=[ \frac{m+1}{2} ]}^{m} f_{m−m′} · δ_{m′} ∈ CH^{m}(Y),

with δ_{m′} as in Lemma 3.7. Then, we have

2\overline{ε} = 2 ∑_{m′=[ \frac{m+1}{2} ]}^{m} f_{m−m′} ∑_{i=0}^{m′} \binom{m′ + i + 1}{i}s^{m′−i,m′−i}.

Furthermore, by Lemma 3.8 3), for any k = 0, ..., m, one has s^{k,k} = ∑_{j=0}^{k} a_{j}^{k}v^{j}. Hence, we get the identity

2\overline{ε} = 2 ∑_{m′=[ \frac{m+1}{2} ]}^{m} f_{m−m′} ∑_{j=0}^{m′} ( ∑_{l=0}^{m′−j} \binom{m′ + l + 1}{l}\binom{j}{m′ − l − j}) v^{j},

and the latter identity can be rewritten as

2\overline{ε} = 2 ∑_{i=0}^{[ \frac{m}{2} ]} ∑_{j=0}^{m} f_{i} · c_{i,j} · v^{j},

where c_{i,j} := ∑_{l=0}^{m−i−j} \binom{m−i+l+1}{l}\binom{j}{m−i−j−l}. If m−i−j < 0, then we have c_{i,j} = η_{m−i−j} = 0. Otherwise – if m − i − j ≥ 0 – we set k := m − i − j, and we have

c_{i,j} ≡ ∑_{l=0}^{k} \binom{−k − j − 2}{l}\binom{j}{k − l} (mod 2),

which is congruent modulo 2 to \binom{−k−2}{k} by the Chu-Vandermonde Identity (see [1, Corollary 2.2.3]). Therefore, since \binom{−k−2}{k} ≡ \binom{2k+1}{k} (mod 2), we get that, for any i = 0, ..., [m/2] and for any j = 0, ..., m,

c_{i,j} ≡ η_{m−i−j} (mod 2).

Thus, since by Lemma 3.8 1), for any j = 0, ...m, the cycle 2v^{j} is rational, we get that there exists an element δ ∈ CH^{m}(Y) such that

2\overline{δ} = 2 ∑_{i=0}^{[ \frac{m}{2} ]} ∑_{j=0}^{m} f_{i} · η_{m−i−j} · v^{j} = 2 ∑_{j=0}^{m} g_{m−j} · v^{j},

where g(X) = ∑_{l} g_{l} · X^{l} is the power series f(X) · η(X). Hence, there exists an exponent 2 element λ ∈ CH^{m}(\overline{Y}) such that

∑_{j=0}^{m} g_{m−j} · v^{j} = \overline{δ} + λ,

and we are done. □

We finish now the proof of Theorem 3.1. Since for any j = 0, ..., m, the cycle 2v^{j} is rational, v^{j} is rational for all odd j, and v^{0} is an odd multiple of \overline{y}, it follows from Lemma 3.9 that it is sufficient now to find a polynomial f ∈ ℤ[X] of degree ≤ [m/2] such that the power series g(X) := f(X) · η(X) has an odd coefficient at X^{m} and even coefficients at smaller monomials of the same parity. By [8, Lemma 3.13], such a polynomial exists. □

4. A stronger version of main theorem

In this section we continue to use notation introduced in the beginning of Section 3. The following result is stronger than Theorem 3.1 although its statement is less eloquent.

Let K/F be an extension and X be an F -variety. In the following proof, an element x ∈ CH^{∗}(X_{K}) is called rational if it is in the image of the restriction homomorphism CH^{∗}(X) → CH^{∗}(X_{K}).

In the same way as before, the following theorem is a generalization of [8, Proposition 3.7] (although, putting aside characteristic, Theorem 4.1 is still weaker than the original version in the sense that an exponent 2 element appears in the conclusion).

Theorem 4.1. Assume that m < [n/2] and i_{1} > 1, and let E/F be an extension such that i_{0}(Q_{E}) > m. Then, for any y ∈ CH^{m}(Y_{F (Q)}) there exists δ ∈ CH^{m}(Y) and an exponent 2 element λ ∈ CH^{m}(Y_{E(Q)}) such that y_{E(Q)} = δ_{E(Q)} + λ.

Proof. We proceed the same way as in the proof of Theorem 3.1.

Let us fix an element x ∈ CH^{m}(Q × Y) mapped to y under the surjection

CH^{m}(Q × Y) ↠ CH^{m}(Y_{F (Q)}).

Since i_{0}(Q_{E}) > m, by Remark 1.2 (applied with r = k = m), the image x_{E(Q)} ∈ CH^{m}(Q_{E(Q)} × Y_{E(Q)}) of x decomposes as

x_{E(Q)} = ∑_{i=0}^{m} h^{i} × x^{i}

where x^{i} ∈ CH^{m−i}(Y_{E(Q)}) is the coordinate of x_{E(Q)} on h^{i} (see Definition 1.1).

The image of x under the composition

CH^{m}(Q × Y) → CH^{m}(Q_{E} × Y_{E}) → CH^{m}(Y_{E(Q)})

is x^{0}. Therefore, by the commutativity of the diagram

\xymatrix{CH^{m}(Q_{E} × Y_{E}) \ar[r] & CH^{m}(Y_{E(Q)}) \\ CH^{m}(Q × Y) \ar[u] \ar[r] & CH^{m}(Y_{F (Q)}) \ar[u]}

we get that x^{0} = y_{E(Q)} and we want to prove that there exists δ ∈ CH^{m}(Y) and an exponent 2 element λ ∈ CH^{m}(Y_{E(Q)}) such that x^{0} = δ_{E(Q)} + λ.

Let π ∈ Ch_{n+i_{1}−1}(Q^{2}) be an element mapped to the 1-primordial cycle under the restriction homomorphism Ch^{∗}(Q) → Ch^{∗}(\overline{Q}). By [3, Proposition 83.2], there is no cycle of type h^{j} × l_{j} with odd j appearing in the decomposition of (h^{0} × h^{i_{1}−1}) · π_{\overline{E}(Q)} ∈ Ch_{n}(Q_{\overline{E}(Q)}^{2}) (and the cycle h^{0} × l_{0} appears).

Moreover, since the coefficients near the cycles contained in the decomposition of (h^{0} × h^{i_{1}−1}) · π_{E(Q)} ∈ Ch_{n}(Q_{E(Q)}^{2}) given by Remark 1.3 (with k = m) do not change when going over \overline{E}(Q), the cycle (h^{0} × h^{i_{1}−1}) · π_{E(Q)} can be uniquely written as a linear combination of cycles of type h^{j} × l_{j} with even j ∈ ⟦0, m⟧ (and the coefficient near h^{0} × l_{0} is 1) , of cycles of type l_{j} × h^{j} (where j ∈ ⟦0, m⟧), and of a cycle ρ ∈ Ch^{n}(Q_{E(Q)}^{2}) whose coordinate on h^{j} (as well as coordinate on l_{j}) is 0 for j ∈ ⟦0, m⟧.

Thus, fixing a rational integral representative γ_{E(Q)} ∈ CH_{n}(Q_{E(Q)}^{2}) of (h^{0}×h^{i_{1}−1})·π_{E(Q)}, we get that the integral coefficient α_{j} near the cycle h^{j} ×l_{j} contained in the decomposition of γ_{E(Q)} (given by Remark 1.3 , with k = m), is even for all odd j, and that α_{0} is odd.

Let γ ∈ CH_{n}(Q^{2}) mapped to γ_{E(Q)} under the restriction homomorphism CH_{n}(Q^{2}) → CH_{n}(Q_{E(Q)}^{2}). We have the following lemma, whose the statement and the proof are very close to Lemma 3.5.

Lemma 4.2. For any i = 0, ..., m, one can choose a rational integral representative s^{i} ∈ CH^{m+i}(Q_{E(Q)} × Y_{E(Q)}) of S^{i}((x_{E(Q)} mod 2) ∘ (γ_{E(Q)} mod 2)) such that

1) for any 0 ≤ j ≤ m, 2s^{i,j} is rational , where s^{i,j} is the coordinate of s^{i} on h^{j};

2) for any odd 0 ≤ j ≤ m, s^{i,j} is rational.

Proof. We use same notation as those introduced during the proof of Lemma 3.5. One can prove 1) exactly as the same way as Lemma 3.5 1). We need the following proposition to prove 2).

Proposition 4.3. Let X be a smooth F -variety and let ρ be an element of Ch(Q × X) such that for any j = 0, ..., r, its coordinate ρ^{j} on h^{j} is 0. Then, for any integer k and for any j = 0, ..., r, the coordinate of S^{k}(ρ) on h^{j} is 0.

Proof. We induct on k. For k = 0, one has S^{0} = Id. Assume that the statement is true till the rank k and let j ∈ ⟦0, r⟧. By [3, Corollary 61.15] (Cartan Formula), one has

S^{k+1}(l_{j} · ρ) = l_{j} · S^{k+1}(ρ) + ∑_{i=1}^{k+1} S^{i}(l_{j}) · S^{k+1−i}(ρ).

Since for any i = 1, ..., k + 1, the cycle S^{i}(l_{j}) is a multiple of l_{j−i} (see [3, Corollary 78.5]), by the induction hypothesis, we get

pr_{∗}(l_{j} · S^{k+1}(ρ)) = pr_{∗}(S^{k+1}(l_{j} · ρ)).

Furthermore, by [3, Proposition 61.10], one has

S^{k+1} ∘ pr_{∗}(l_{j} · ρ) = ∑_{i=0}^{k+1} pr_{∗}(c_{k+1−i}(−T_{Q}) · S^{i}(l_{j} · ρ)),

and since pr_{∗}(l_{j} · ρ) = 0, we deduce that

pr_{∗}(l_{j} · S^{k+1}(ρ)) = ∑_{i=0}^{k} a_{i},

where a_{i} = pr_{∗}(c_{k+1−i}(−T_{Q})·S^{i}(l_{j} ·ρ)). We are going to prove that for any i = 0, ..., k, one has a_{i} = 0. Let i be an integer in ⟦0, k⟧. Since by [3, Lemma 78.1], the cycle c_{k+1−i}(−T_{Q}) is a multiple of h^{k+1−i}, it suffices to show that pr_{∗}(h^{k+1−i} · S^{i}(l_{j} · ρ)) = 0.

By the Cartan Formula and [3, Corollary 78.5], the cycle pr_{∗}(h^{k+1−i} · S^{i}(l_{j} · ρ)) is a linear combination of cycle of type pr_{∗}(h^{k+1−i} · l_{j−t} · S^{i−t}(x)), where t ∈ ⟦0, i⟧. Since by [3, Proposition 68.1], for any t = 0, ..., i, one has h^{k+1−i} · l_{j−t} = l_{j−t−(k+1−i)}, we are done by the induction hypothesis. □

We finish now the proof of Lemma 4.2. Assume that j is odd. Since by Proposition 4.3, for any k = 0, ..., m, the coordinate of S^{k}(ρ) on h^{j} is 0, the only fact that we have to explain here to prove 2) (i.e what is new compared to the proof of Lemma 3.5) is why the corresponding cycle ã^{k,j} ∈ CH^{n+k−j}(Q_{E(Q)}) is rational.

For the same reasons as in the proof of Lemma 3.5, the cycle ã^{k,j} ∈ CH^{n+k−j}(Q_{E(Q)}) is divisible by 2. Moreover, since one has j − k ≤ m < i_{0}(Q_{E}), the cycle l_{j−k} is defined over E and it is consequently defined over E(Q). Furthermore, since j − k ≤ m < [n/2], the group CH^{n+k−j}(Q_{E(Q)}) is free with basis \{l_{j−k}\} (as well as the group CH^{n+k−j}(Q\overline{E(Q)})) and therefore the restriction homomorphism

CH^{n+k−j}(Q_{E(Q)}) ⟶ CH^{n+k−j}(Q\overline{E(Q)})

is injective (it is even an isomorphism). Since 2l_{j−k} = h^{n+k−j}, we deduce that any cycle of CH^{n+k−j}(Q_{E(Q)}) divisible by 2 is rational. Thus, for any 0 ≤ k ≤ m, the cycle ã^{k,j} is rational and we finish as in the proof of Lemma 3.5. □

Now, one can finish the proof of Theorem 4.1 exactly the same way as the proof of Theorem 3.1 replacing \overline{F} by E(Q). □

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[3] Elman, R., Karpenko, N., and Merkurjev, A. The algebraic and geometric theory of quadratic forms, vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI, 2008.

[4] Fino, R. Around rationality of cycles. Linear Algebraic Groups and Related Structures (preprint serveur), 450 (2011, Nov 16), 11 pages.

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