SYMMETRIC CORRESPONDENCES ON QUADRICS

RAPHAËL FINO

Abstract. We prove a result comparing the rationality of some elementary algebraic cycles introduced by Alexander Vishik, defined on orthogonal grassmannians, with the rationality of some algebraic cycles defined on fiber products of the corresponding quadric. Keywords: Chow groups, quadratic forms, orthogonal flag varieties.

Contents

1. Introduction 1 2. Materials 2 3. Proof of the first part of Main Theorem 5 4. Elementary Discrete Invariant and first Witt index 10 5. Proof of the second part of Main Theorem 11 References 12

1. Introduction

Let X be a smooth projective quadric of dimension n over a field F associated with a non-degenerate F -quadratic form q and let K/F be a splitting field extension of q. Let us denote [n/2] as d. In his paper [4] dedicated to the Kaplansky’s conjecture on the u-invariant of a field, Alexander Vishik introduced the so-called Elementary Discrete Invariant. This invariant encodes information about rationality of algebraic cycles on the d + 1 orthogonal grassmannians associated with q, by means of certain elementary classes defined on these grassmannians over K: it is the collection of the codimensions of those elementary classes that are already defined at the level of the base field F (such an algebraic cycle is called rational).

In the current note, we relate the rationality of the highest elementary classes (highest in the sense that, for each grassmannian, the highest elementary class is the one with maximal codimension) with the rationality of certain algebraic cycles defined on some fiber products X × X × ⋯ × X.

More precisely, for any I ⊂ {0, …, d}, we write F(I) for the partial orthogonal flag variety associated with q. So, for any i ∈ {0, …, d}, the variety F(i) is the grassmannian of i-dimensional totally isotropic subspaces and we denote it by G_{i}. In particular G_{0} is the quadric X. For J ⊂ I, we write π with subindex I with J underlined inside it for the natural projection F(I) → F(J). In particular, for any i ∈ {0, …, d}, one can consider

Date: 6 November 2016.

2010 Mathematics Subject Classification. 14C25 ; 11E39.

\xymatrix{X & F(0, i) \ar[l]^{π_{(\underline{0},i)}} \ar[r]_{π_{(0,\underline{i})}} & G_{i},}

and we set

Z_{n−i}^{i} := π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(l_{0}) ∈ CH^{n−i}(G_{i}_{K}),

where l_{0} ∈ CH_{0}(X_{K}) is the class of a closed point x ∈ X_{K} of degree 1 and CH stands for the Chow ring with ℤ-coefficients. The cycle z_{n−i}^{i} := Z_{n−i}^{i} (mod 2) ∈ Ch^{n−i}(G_{i}_{K}) is the highest elementary class for the grassmannian G_{i} (with Ch the Chow ring with ℤ/2ℤ-coefficients). Note that A. Vishik proved in [4, Proposition 2.5] that the rationality of z_{n−i}^{i} implies the rationality of z_{n−j}^{j} for any j > i.

We now introduce the other type of algebraic cycles coming into play in the main statement of this note (Theorem 1.1). For any i ∈ {0, …, d}, let us denote by sym : CH^{∗}(X^{i+1}) → CH^{∗}(X^{i+1}) the homomorphism Σ_{s∈S_{i+1}}s_{∗}, where s : X^{i+1} → X^{i+1} is the isomorphism associated with a permutation s. We set

ρ_{i} := sym ((×_{j=0}^{i−1}h^{j}) × l_{0}) ∈ CH^{n+i(i−1)/2}(X_{K}^{i+1}),

where × is the external product and h^{j} is the j-th power of the hyperplane section class h ∈ CH^{1}(X) (always rational). Note that ρ_{1} is the Rost correspondence 1 × l_{0} + l_{0} × 1 of X in the sense of [1, §80] (we refer to [1, §62] for an introduction to correspondences). Note that the rationality of ρ_{i} implies the rationality of ρ_{j} for any j > i. Note also that ρ_{0} = l_{0} = Z_{n}^{0}.

The main result of this note is the following statement.

Theorem 1.1 (Main Theorem). Let i ∈ {0, …, d}. The cycle z_{n−i}^{i} is rational if and only if the cycle ρ_{i} (mod 2) is rational.

This result reduces certain questions about rationality of algebraic cycles on orthogonal grassmannians to the sole level of quadrics. For example, Theorem 1.1 allows one to reformulate Vishik Conjecture [3, Conjecture 3.11] and this reformulation may help prove the conjecture. Note that there are some quadrics for which none of the z_{n−i}^{i} is rational (hence none of the ρ_{i} is), this is for example the case for generic quadrics (see [3, Statement 3.6]).

The proofs in the present note mainly rely on computations of compositions of correspondences and the use of Chern classes of vector bundles over orthogonal grassmannians.

In section 2, we introduce some materials which will be required to prove Theorem 1.1 in sections 3 and 5. In section 4, we apply Theorem 1.1 (actually, only the direct implication is needed) to relate the first Witt index of the quadric with the rationality of the cycle z_{n−i}^{i} (Proposition 4.1).

2. Materials

In this section, we use notations introduced in the introduction.

2.1. Rational cycles on powers of quadrics. We refer to [1, §68] for an introduction to cycles on powers of quadrics. For any i ∈ {1, …, d}, we set

Δ_{i} := sym ((×_{j=1}^{i−1}h^{j}) × 1 × l_{0}) + ∑_{k=i}^{d} sym ((×_{j=1}^{i−1}h^{j}) × h^{k} × l_{k}) ∈ CH^{n+i(i−1)/2}(X_{K}^{i+1}).

where l_{k} is the class in CH_{k}(X_{K}) of a k-dimensional totally isotropic subspace of ℙ ((V_{q})_{K}) (with V_{q} the F -vector space associated with q). Note that Δ_{1} is just the class of the diagonal in CH^{n}(X_{K}^{2}).

The following observation is crucial for our matter (it will be used in Corollary 3.15 to obtain the first part Theorem 1.1).

Lemma 2.1. For any i ∈ {1, …, d}, the cycle Δ_{i} ∈ CH^{n+i(i−1)/2}(X_{K}^{i+1}) is rational.

Proof. We use an induction on i. Since Δ_{1} is the class of the diagonal in CH^{n}(X_{K}^{2}), it is rational. Assume Δ_{i−1} rational and let σ ∈ S_{i+1} be a cyclic permutation (i ≥ 2). Then the cycle

(2.2) ∑_{j=0}^{i} σ_{∗}^{j}(Δ_{i−1} × h^{i−1}) = sym ((×_{j=1}^{i−1}h^{j}) × 1 × l_{0}) + ∑_{k=i−1}^{d} sym ((×_{j=1}^{i−1}h^{j}) × h^{k} × l_{k})

is also rational. Moreover, the cycle

(2.3)  sym ((×_{j=1}^{i−1}h^{j}) × h^{i−1} × l_{i−1})

is always rational. Indeed, since h^{i−1} appears exactly two times in every summand of (2.3), each distinct summand appears exactly two times, therefore it can be rewritten as

2 ∑_{s∈A_{i+1}} s_{∗}((×_{j=1}^{i−1}h^{j}) × h^{i−1} × l_{i−1}),

and one has 2l_{i−1} = h^{n−i+1}. Since Δ_{i} is the difference of (2.2) and (2.3), one get the conclusion. □

2.2. Cycles on orthogonal flag varieties. For 0 ≤ i ≤ d and n − i − d ≤ j ≤ n − i, we set

Z_{j}^{i} := π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(l_{n−i−j}) ∈ CH^{j}(G_{i}_{K})

and z_{j}^{i} := Z_{j}^{i} (mod 2) ∈ Ch^{j}(G_{i}_{K}). The cycles z_{j}^{i} are all the elementary classes defining the Elementary Discrete Invariant EDI(X).

For any nonnegative integer j and i > 0 such that j + i ≤ d, we set

W_{j}^{i} := π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(h^{j+i}) ∈ CH^{j}(G_{i}),

W_{j}^{0} := h^{j} and w_{j}^{i} := W_{j}^{i} (mod 2) ∈ Ch^{j}(G_{i}).

The proofs in the next sections will use the following lemma, which can easily be deduced from [4, Propositon 2.1 and Lemma 2.6] and its proofs. For 0 ≤ i ≤ d, let us denote by T_{i} the tautological vector bundle on G_{i}, i.e. T_{i} is given by the closed subvariety of the trivial bundle V 𝟙 = V_{q} ×G_{i} consisting of pairs (u, U) such that u ∈ U. For a vector bundle E over a scheme, we write c_{i}(E) for the i-th Chern class with value in CH.

Lemma 2.4 (Vishik). One has (i) π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(h^{i}) = W_{0}^{i} = [G(i)]; (ii)

π_{(\underline{i−1},i)}^{∗}(Z_{j}^{i−1}) = c_{1}(O(1)) · π_{(i−1,\underline{i})}^{∗}(Z_{j−1}^{i}) + π_{(i−1,\underline{i})}^{∗}(Z_{j}^{i}),

where O(1) is the standard sheaf on the projective bundle F(i − 1, i) = ℙ_{G_{i−1}}(T_{i}^{∨}), with T_{i}^{∨} the vector bundle dual to T_{i};

(iii) For 0 ≤ j < d − i + 1,

π_{(\underline{i−1},i)}^{∗}(W_{j}^{i−1}) = c_{1}(O(1)) · π_{(i−1,\underline{i})}^{∗}(W_{j−1}^{i}) + π_{(i−1,\underline{i})}^{∗}(W_{j}^{i});

(iv)

π_{(\underline{i−1},i)}^{∗}(W_{d−i+1}^{i−1}) = c_{1}(O(1)) · π_{(i−1,\underline{i})}^{∗}(W_{d−i}^{i}) + 2π_{(i−1,\underline{i})}^{∗}(Z_{d−i+1}^{i}).

The following statement is a direct consequence of the previous lemma.

Lemma 2.5. For any 1 ≤ i ≤ d, i ≤ k ≤ d and 0 ≤ m ≤ k, one has

π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(W_{k−i}^{i} · Z_{n−i−m}^{i}) = ∑_{j=max(i−m,0)}^{min(k−m,i)} W_{k−m−j}^{i−1} · π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−2i+j}^{i}).

Proof. This is obtained by applying repeatedly Lemma 2.4(ii) after Lemma 2.4(iii), and using the Projection Formula combined with dimensional considerations. □

Finally, the following description of the Chern classes modulo 2 of the tautological vector bundles on orthogonal grassmannians will be needed.

Lemma 2.6. For any 1 ≤ i ≤ d and 0 ≤ j ≤ i, one has

c_{j}(T_{i−1}) (mod 2) = π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(z_{n−2i+j}^{i})

in Ch^{j}(G_{i−1}_{K}).

Proof. We use an induction on j. Since Z_{n−2i}^{i} = π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(l_{i}), the cycle π_{(i−1,\underline{i})}^{∗}(Z_{n−2i}^{i}) is the class in CH (F(i − 1, i)_{K}) of the closure of {(y, < y, x >)|x ∈ (y^{⊥}\y) ∩ L_{i}} in F(i − 1, i)_{K}, with L_{i} a fixed i-dimensional totally isotropic subspace of ℙ ((V_{q})_{K}). Consequently, one has π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−2i}^{i}) = 1. Let 1 ≤ j ≤ i. By [4, Proposition 2.1], for any l > d − i + 1, the element c_{l}(V 𝟙/T_{i−1}) is divisible by 2 in CH^{l}(G_{i−1}_{K}). Therefore, by Whitney Sum Formula ([1, Proposition 54.7]), the Chern class c_{j}(T_{i−1}) is congruent modulo 2 to

c_{j}(V 𝟙/T_{i−1}) + ∑_{k=max(1,j−d+i−1)}^{j−1} c_{j−k}(V 𝟙/T_{i−1}) · c_{k}(T_{i−1}).

By the induction hypothesis and [4, Proposition 2.1], the latter cycle is congruent modulo 2 to

(2.7)  c_{j}(V 𝟙/T_{i−1}) + ∑_{k=max(1,j−d+i−1)}^{j−1} W_{j−k}^{i−1} · π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−2i+k}^{i}).

Moreover, by Lemma 2.5, one has

(2.8) π_{(\underline{i−1},i)}_{∗} ∘π_{(i−1,\underline{i})}^{∗}(W_{d−i}^{i} ·Z_{n−i−d+j}^{i}) = ∑_{k=max(j−d+i,0)}^{j} W_{j−k}^{i−1} ·π_{(\underline{i−1},i)}_{∗} ∘π_{(i−1,\underline{i})}^{∗}(Z_{n−2i+k}^{i}).

If j − d + i − 1 < 0, combining (2.7) with (2.8), one get that c_{j}(T_{i−1}) is congruent modulo 2 to

π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−2i+j}^{i}) + π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(W_{d−i}^{i} · Z_{n−i−d+j}^{i}).

Futhermore, by applying Lemma 2.4(iv) after Lemma 2.4(ii) and using the Projection Formula combined with dimensional considerations, one obtains

(2.9)  π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(w_{d−i}^{i} · z_{n−i−d+j}^{i}) = w_{d−i+1}^{i−1} · π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(z_{n−i−d+j−1}^{i}).

Since j − d + i − 1 < 0, the cycle π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−i−d+j−1}^{i}) is trivial by dimensional reasons.

If j − d + i − 1 = 0, combining (2.7) with (2.8), one get that c_{d−i+1}(T_{i−1}) is congruent modulo 2 to

π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−i+d+1}^{i}) + W_{d−i+1}^{i−1} + π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(W_{d−i}^{i} · Z_{n−2i+1}^{i})

and the two last summands of the previous cycle are equal modulo 2 by identity (2.9) (valid for any 1 ≤ j ≤ d) and the base of the induction.

Otherwise – if j − d + i − 1 > 0 – one proceeds similarly combining (2.7) with (2.8) and using identity (2.9).

□

3. Proof of the first part of Main Theorem

We use notations and materials introduced in sections 1 and 2. For any i ∈ {1, …, d}, we denote by θ_{i} the class of the subvariety

{(y, x_{1}, …, x_{i+1}) | x_{1}, … x_{i+1} ∈ y} ⊂ G_{i} × X^{i+1}

in CH(G_{i} × X^{i+1}). Viewing the cycle θ_{i} as a correspondence G_{i} ⇝ X^{i+1}, we set

α_{i} := (θ_{i})_{∗}(Z_{n−i}^{i}) + ρ_{i} ∈ CH(X_{K}^{i+1}),

and we view α_{i} as a correspondence X_{K} ⇝ X_{K}^{i}.

The following computations are the key point in the proof of the first part of Theorem 1.1 (see Corollary 3.15).

Proposition 3.1. For any i ∈ {1, …, d}, one has

(α_{i} (mod 2))_{∗} (h^{k}) = \{ \begin{matrix}sym ((×_{j=1}^{i−1}h^{j}) × 1) & if k = 0; \\ 0 & if 1 ≤ k ≤ i − 1; \\ sym ((×_{j=1}^{i−1}h^{j}) × h^{k}) & if i ≤ k ≤ d. \end{matrix}

Proof. The following lemma provides an appropriate formula for ((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗}. We write p with underlined target for projections.

Lemma 3.2. For any x ∈ CH(X_{K}), one has

((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗} (x) = p_{G_{i}×\underline{X^{i}}} _{∗} (π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(x) · Z_{n−i}^{i} · η_{i}) ,

where η_{i} is the class in CH(G_{i} × X^{i}) of the subvariety {(y, x_{1}, …, x_{i}) | x_{1}, …, x_{i} ∈ y} ⊂ G_{i} × X^{i}.

Proof. By definition, one has

(θ_{i})_{∗}(Z_{n−i}^{i}) = p_{G_{i}×\underline{X^{i+1}}} _{∗} ((Z_{n−i}^{i} × [X^{i+1}]) · θ_{i})

so

((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗} (x) = p_{X×\underline{X^{i}}} _{∗} ((x × [X^{i}]) · p_{G_{i}×\underline{X^{i+1}}} _{∗} ((Z_{n−i}^{i} × [X^{i+1}]) · θ_{i})) .

Therefore, using the Projection Formula (see [1, Proposition 56.9]) and the following fiber product diagram with respect to projections

\xymatrix{G_{i} × X × X^{i} \ar[r] \ar[d] & X × X^{i} \ar[d] \\ G_{i} × X^{i} \ar[r] & X^{i}} ,

one get that

((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗} (x) = p_{G_{i}×\underline{X^{i}}} _{∗} ∘ p_{\underline{G_{i}}×X×\underline{X^{i}}} _{∗} ((Z_{n−i}^{i} × x × [X^{i}]) · θ_{i}) .

Moreover, the cycle θ_{i} can be rewritten as

θ_{i} = p_{\underline{G_{i}×X}×X^{i}}^{∗} ([F(i, 0)]) · p_{\underline{G_{i}}×X×\underline{X^{i}}}^{∗}(η_{i}),

where [F(i, 0)] is the class in CH(G_{i} × X) of the subvariety F(i, 0) ⊂ G(i) × X. Hence, using the Projection Formula twice, one get that

((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗} (x) = p_{G_{i}×\underline{X^{i}}} _{∗} ((p_{\underline{G_{i}}×X}_{∗} ((Z_{n−i}^{i} × x) · [F(i, 0)]) × [X^{i}]) · η_{i})
= p_{G_{i}×\underline{X^{i}}} _{∗} (p_{\underline{G_{i}}×X}_{∗} (p_{G_{i}×\underline{X}}^{∗}(x) · [F(i, 0)]) · Z_{n−i}^{i} · η_{i}) .

Furthermore, by denoting the closed embedding F(i, 0) ↪ G_{i} × X as in, one has in_{∗}(1) = [F(i, 0)]. It follows again from the Projection Formula that p_{G_{i}×\underline{X}}^{∗}(x)·[F(i, 0)] = in_{∗} ∘ in^{∗} (p_{G_{i}×\underline{X}}^{∗}(x)). Consequently, one has

p_{\underline{G_{i}}×X}_{∗} (p_{G_{i}×\underline{X}}^{∗}(x) · [F(i, 0)]) = π_{(0,\underline{i})}_{∗} ∘ π_{(\underline{0},i)}^{∗}(x)

and the lemma is proven. □

For any x ∈ CH^{k}(X_{K}) with k ≤ i−1, the cycle π_{(0,\underline{i})}_{∗} ∘π_{(\underline{0},i)}^{∗}(x) is trivial by dimensional reasons. Thus, by Lemma 3.2, the cycle ((θ_{i})_{∗}(Z_{n−i}^{i}))_{∗} (x) is also trivial. Therefore, since (ρ_{i})_{∗}(h^{k}) = sym ((×_{j=1}^{i−1}h^{j}) × 1) for k = 0 and is trivial for 0 < k ≤ d, one get the conclusion of Proposition 3.1 for the cases k ≤ i − 1.

Note that for i ≤ k ≤ d, Lemma 3.2 provides the following identity

(3.3)  (α_{i} (mod 2))_{∗} (h^{k}) = p_{G_{i}×\underline{X^{i}}} _{∗} (w_{k−i}^{i} · z_{n−i}^{i} · η_{i}) in Ch(X_{K}^{i}),

where we abuse notation and write η_{i} for η_{i} (mod 2).

We prove the cases i ≤ k ≤ d of Proposition 3.1 by backward induction on i.

For any 1 ≤ i ≤ d, by the very definitions, one has

(3.4)  η_{i} = ∏_{j=1}^{i} (Id_{G_{i}} × p_{X_{j}^{i}})^{∗} ([F(i, 0)]) in CH(G_{i} × X^{i}),

with p_{X_{j}^{i}} the projection from X^{i} to the j-th coordinate, and

(3.5)  [F(i, 0)] = ∑_{m=0}^{d} Z_{n−i−m}^{i} × h^{m} + ∑_{m=i}^{d} W_{m−i}^{i} × l_{m} in CH(G_{i}_{K} × X_{K}).

The base of the backward induction i = d (so i = k = d) is obtained by combining the identities (3.3), (3.4) and (3.5) for i = d (recall also that w_{0}^{i} = 1 for any 0 ≤ i ≤ d by Lemma 2.4(i)) with the fact that, for any integers 0 ≤ a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{d} ≤ d, one has

deg (∏_{j=0}^{d} z_{n−d−a_{j}}^{d}) = \{ \begin{matrix}1 & if \{a_{0}, a_{1}, …, a_{d}\} = \{0, 1, …, d\} ; \\ 0 & otherwise, \end{matrix}

where deg : Ch(G_{d}_{K}) → Ch(Spec(K)) = ℤ/2ℤ is the homomorphism associated with the push-forward of the structure morphism, see [1, Lemma 87.6].

The backward induction step will follow from Lemma 3.10, which make use of the following statement.

Lemma 3.6. For any 2 ≤ i ≤ d, one has

z_{n−i}^{i}·η_{i} = (π_{(i−1,\underline{i})} × Id_{X^{i}})_{∗} ((π_{(i−1,\underline{i})} × p_{X_{i}^{i}})^{∗} ([F(i, 0)]) · (π_{(\underline{i−1},i)} × p_{Xi ; \overline{i}})^{∗} (z_{n−i+1}^{i−1} · η_{i−1})) .

Proof. It follows from the identity (3.5) and Lemma 2.4(ii), (iii) and (iv) that

(π_{(\underline{i−1},i)} × Id_{X})^{∗} ([F(i − 1, 0)]) = (c_{1}(O(1)) × 1) · (π_{(i−1,\underline{i})} × Id_{X})^{∗} ([F(i, 0)])

+ (π_{(i−1,\underline{i})} × Id_{X})^{∗} ((1 × h) · ([F(i, 0)] − Z_{n−i−d}^{i} × h^{d}))

+ 2π_{(i−1,\underline{i})}^{∗}(Z_{d−i+1}^{i}) × l_{d}

in CH(F(i − 1, i)_{K} × X_{K}). Considered modulo 2, this gives (3.7)

(π_{(\underline{i−1},i)} × Id_{X})^{∗} ([F(i − 1, 0)]) = (c_{1}(O(1)) × 1 + 1 × h) · (π_{(i−1,\underline{i})} × Id_{X})^{∗} ([F(i, 0)])

in Ch(F(i − 1, i)_{K} × X_{K}).

Moreover, by (3.4), one has (3.8)

(π_{(\underline{i−1},i)} × p_{Xi ; \overline{i}})^{∗} (z_{n−i+1}^{i−1} · η_{i−1}) = π_{(\underline{i−1},i)}^{∗}(z_{n−i+1}^{i−1}) · ∏_{j=1}^{i−1} (π_{(\underline{i−1},i)} × p_{X_{j}^{i}})^{∗} ([F(i − 1, 0)])

with p_{Xi ; \overline{i}} the projection from X^{i} to the i − 1 first coordinates. By (3.7), the right member of the previous equation can be rewritten as

π_{(\underline{i−1},i)}^{∗}(z_{n−i+1}^{i−1}) · ∏_{j=1}^{i−1} (c_{1}(O(1)) × [X^{i}] + 1 × p_{X_{j}^{i}}^{∗}(h)) · ∏_{j=1}^{i−1} (π_{(i−1,\underline{i})} × p_{X_{j}^{i}})^{∗} ([F(i, 0)]) .

Hence, by multiplying the equation (3.8) by (π_{(i−1,\underline{i})} × p_{X_{i}^{i}})^{∗} ([F(i, 0)]) and using Lemma 2.4(ii), one get

(3.9) (π_{(i−1,\underline{i})} × p_{X_{i}^{i}})^{∗} ([F(i, 0)]) · (π_{(\underline{i−1},i)} × p_{Xi ; \overline{i}})^{∗} (z_{n−i+1}^{i−1} · η_{i−1}) =

(c_{1}(O(1)) × [X^{i}]) · ∏_{j=1}^{i−1} (c_{1}(O(1)) × [X^{i}] + 1 × p_{X_{j}^{i}}^{∗}(h)) · (π_{(i−1,\underline{i})} × Id_{X^{i}})^{∗} (z_{n−i}^{i} · η_{i}).

Furthermore, it follows from the Projective Bundle Theorem (see [1, Theorem 53.10]) applied to π_{(i−1,\underline{i})} : F(i − 1, i) → G_{i} that π_{(i−1,\underline{i})}_{∗}(c_{1}(O(1))^{i}) = 1. Since dim F(i − 1, i) − dim G_{i} = i, one obtains the conclusion of the lemma by composing the equation (3.9) by (π_{(i−1,\underline{i})} × Id_{X^{i}})_{∗} and using the Projection Formula.

□

Lemma 3.10. For any 2 ≤ i ≤ d and i ≤ k ≤ d, the cycle in identity (3.3) can be rewritten as

∑_{m=0}^{k} ∑_{j=max(i−m,0)}^{min(k−m,i)} p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (w_{k−m−j}^{i−1} · σ_{i−1}^{j} · z_{n−i+1}^{i−1} · η_{i−1}) × h^{m},

with σ_{i−1}^{j} = π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(Z_{n−2i+j}^{i}) (mod 2) ∈ Ch^{j}(G_{i−1}_{K}).

Proof. It follows from Lemma 3.6 and the Projection Formula that the cycle in identity (3.3) can be rewritten as

p_{G_{i−1}×\underline{X^{i}}} _{∗} ((π_{(\underline{i−1},i)} × Id_{X^{i}})_{∗} ∘ (π_{(i−1,\underline{i})} × p_{X_{i}^{i}})^{∗} (w_{k−i}^{i} · [F(i, 0)]) · z_{n−i+1}^{i−1} · (η_{i−1} × 1)) .

Moreover, by decomposition (3.5) and dimensional considerations, the latter cycle is equal to

∑_{m=0}^{k} p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (π_{(\underline{i−1},i)}_{∗} ∘ π_{(i−1,\underline{i})}^{∗}(w_{k−i}^{i} · z_{n−i−m}^{i}) · z_{n−i+1}^{i−1} · η_{i−1}) × h^{m}.

Lemma 2.5 completes the proof. □

Let 2 ≤ i ≤ d. On the one hand, by identity (3.3) and backward induction hypothesis, for any i ≤ k ≤ d, one has

(3.11)  p_{G_{i}×\underline{X^{i}}} _{∗} (w_{k−i}^{i} · z_{n−i}^{i} · η_{i}) = sym ((×_{j=1}^{i−1}h^{j}) × h^{k}) .

Therefore, in particular, the coordinate of (3.11) on top right h^{i−1}, i.e.,

p_{\underline{X^{i−1}}×X} _{∗} ((p_{G_{i}×\underline{X^{i}}} _{∗} (w_{k−i}^{i} · z_{n−i}^{i} · η_{i})) · [X^{i−1}] × l_{i−1})

is equal to

(3.12)  sym ((×_{j=1}^{i−2}h^{j}) × h^{k}) .

On the other hand, by Lemma 3.10, this coordinate is also equal to

(3.13)  ∑_{j=1}^{min(k−i+1,i)} p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (w_{k−i+1−j}^{i−1} · σ_{i−1}^{j} · z_{n−i+1}^{i−1} · η_{i−1}) .

Since W_{k−i+1−j}^{i−1} = c_{k−i+1−j}(V 𝟙/T_{i−1}) (see [4, Proposition 2.1]) and σ_{i−1}^{j} =c_{j}(T_{i−1}) (mod 2) (see Lemma 2.6), by Whitney Sum Formula, one has

∑_{j=0}^{min(k−i+1,i)} w_{k−i+1−j}^{i−1} · σ_{i−1}^{j} = ∑_{j=0}^{min(k−i+1,i)} c_{k−i+1−j}(V 𝟙/T_{i−1}) · c_{j}(T_{i−1}) = c_{k−i+1}(V 𝟙) = 0.

Consequently, in view of (3.12) and (3.13), one get

(3.14)  p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (w_{k−i+1}^{i−1} · z_{n−i+1}^{i−1} · η_{i−1}) = sym ((×_{j=1}^{i−2}h^{j}) × h^{k}) .

Note that, by identities (3.3) and (3.14), it only remains to prove that

p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (z_{n−i+1}^{i−1} · η_{i−1}) = sym ((×_{j=1}^{i−1}h^{j}))

to complete the backward induction step. On the one hand, by backward induction hypothesis, the coordinate of (3.11) on top right h^{k} is

sym ((×_{j=1}^{i−1}h^{j}))

and on the other hand, by Lemma 3.10, it is also equal to

p_{G_{i−1}×\underline{X^{i−1}}} _{∗} (z_{n−i+1}^{i−1} · η_{i−1}) .

Proposition 3.1 is proven.

□

We are now able to prove the first part of the main result of this note (Theorem 1.1), which we restate below.

Corollary 3.15. Let i ∈ {0, …, d}. If z_{n−i}^{i} is rational then ρ_{i} (mod 2) is also rational.

Proof. Since the conclusion is obvious for i = 0, we assume that i ≥ 1. In view of the ring structure of CH(X_{K}^{i+1}) (see [1, §68]), and knowing that the cycle α_{i} is symmetric, one deduces from Proposition 3.1 that

α_{i} (mod 2) = Δ_{i} (mod 2) + β

with β a sum of nonessential elements (a nonessential element is an external product of powers of the hyperplane class, it is always rational). Since α_{i} = (θ_{i})_{∗}(Z_{n−i}^{i}) + ρ_{i}, the corollary is proved. □

The following statement is a consequence of Proposition 3.1 and its proof.

Proposition 3.16. Let 1 ≤ i ≤ d − 1, i + 1 ≤ k ≤ d and 1 ≤ m ≤ i. For any integers 0 ≤ a_{1} ≤ a_{2} ≤ ⋯ ≤ a_{i} ≤ d, the integer

deg ((Z_{n−i}^{i} · ∏_{l=1}^{i} Z_{n−i−a_{l}}^{i}) · (∑_{j=0}^{i−m} W_{k−m−j}^{i} · c_{j}(T_{i})))

is congruent to 1 (mod 2) if {a_{1}, …, a_{i}} = {k} ∪ ({1, 2, …, i}\{m}) and to 0 (mod 2) otherwise.

Proof. Proposition 3.1 and Lemma 3.10 provide two descriptions of the coordinate of the cycle (3.3) on top right h^{m} (note that one can decompose the cycle η_{i} ∈ CH(G_{i}_{K} × X_{K}^{i}) appearing in Lemma 3.10 as a sum of external products by combinng equations (3.4) and (3.5)). The conclusion is obtained by comparing these two descriptions and applying the Whitney Sum Formula to that given by Lemma 3.10. □

Remark 3.17. We also retrieve [4, Statement 2.15] by proceeding the same way as in the previous proof but considering the coordinate on top right h^{k} instead of top right h^{m}. Proposition 3.16 is completed for i = d by [1, Lemma 87.6].

4. Elementary Discrete Invariant and first Witt index

In this section, we continue to use notations and materials introduced in the previous sections. Corollary 3.15 implies the following condition on the first Witt index i_{1} of the quadric X.

Proposition 4.1. Assume X anisotropic and let i ∈ {1, …, d}. If z_{n−i}^{i} is rational then i_{1} ≤ i.

Proof. In view of Corollary 3.15, the statement follows from the next lemma.

Lemma 4.2. Assume X anisotropic and let i ∈ {1, …, d}. If ρ_{i} (mod 2) is rational then i_{1} ≤ i.

Proof. Suppose that ρ_{i} (mod 2) is rational and that i_{1} > i. We claim that this implies that ρ_{i−1} (mod 2) is also rational.

Indeed, let us denote by π ∈ Ch^{n−i_{1}+1}(X_{K}^{2}) the 1-primordial cycle (see [1, Definition 73.16] and paragraph right after [1, Theorem 73.26], it is a rational cycle). Even if it means adding a rational cycle to π, one can assume that π decomposes as

π = 1 × l_{i_{1}−1} + l_{i_{1}−1} × 1 + ∑_{j=i_{1}}^{d−i_{1}+1} a_{j} (h^{j} × l_{j+i_{1}−1} + l_{j+i_{1}−1} × h^{j})

for some a_{j} ∈ ℤ/2ℤ (the fact that one can choose to make the previous sum start from j = i_{1} is due to [1, Proposition 73.27]). Since i_{1} > i, the computation of the composition of rational correspondences (ρ_{i} (mod 2))_{K} ∘ ((1 × h^{i_{1}−i}) · π) gives the identity

(ρ_{i} (mod 2))_{K} ∘ ((1 × h^{i_{1}−i}) · π) = 1 × (ρ_{i−1} (mod 2))_{K} .

Therefore, pulling back the latter algebraic cycle with respect to the diagonal morphism

\begin{matrix}X^{i} & ⟶ & X^{i+1} \\ (x_{1}, x_{2}, x_{3}, …, x_{i}) & ⟼ & (x_{1}, x_{1}, x_{2}, x_{3} …, x_{i}) \end{matrix}

(for example), one get that ρ_{i−1} (mod 2) is also a rational cycle.

It follows from the claim that l_{0} is rational, i.e., X is isotropic. □

The proposition is proved. □

Remark 4.3. Combining Proposition 4.1 with [4, Propositon 2.5], one get that, representing the Elementary Discrete Invariant EDI(X) of an anisotropic quadric X as a (d + 1) × (d + 1) coordinate square (see [4, Definition 2.3 and the paragraph right after]), there is no marked integral node below the i_{1}-th diagonal of this square. That is to say, the invariant EDI(X) looks as

\begin{aligned}d & × & & & & × \\ i_{1} & × \\ & ∘ \\ & & ∘ \\ & ∘ \\ 0 & ∘ & ∘ & ∘ & × & × \end{aligned} ,

where ∘ means that the node is unmarked. We recall that each row is associated with an orthogonal grassmannian, starting with the quadric at the bottom row, and that each integral node corresponds to an elementary classes as defined by A. Vishik in [4], with codimension decreasing from left to right (the cycles z_{n−i}^{i} ∈ Ch^{n−i}(G(i)_{K}) correspond to the first column on the left).

5. Proof of the second part of Main Theorem

We use notations and materials introduced in previous sections. In the following statement, the fact that if ρ_{1} is rational then Z_{n−1}^{1} is also rational has already been shown by A. Vishik in the proof of [2, Theorem 4.4]. The second part of Main Theorem (Theorem 1.1) is actually valid at the level of integral Chow groups.

Proposition 5.1. Let i ∈ {0, …, d}. If ρ_{i} is rational then Z_{n−i}^{i} is also rational.

Proof. Since the conclusion is obvious for i = 0, we assume that i ≥ 1. Consider the rational cycle

θ′_{i} := p_{\underline{X}×X×\underline{F(0,i)}}^{∗} ∘ (Id_{X} × π_{(\underline{0},i)})^{∗} (Δ_{1}) · p_{X×\underline{X}×\underline{F(0,i)}}^{∗} ∘ (Id_{X} × π_{(0,\underline{i})})^{∗} ([F(0, i)]_{K})

in CH^{2n−i}(X_{K}^{2} × F(0, i)_{K}), with Δ_{1} the class of the diagonal in CH^{n}(X_{K}^{2}) (see §2.1). We view it as a correspondence θ′_{i} : X_{K}^{2} ⇝ F(0, i)_{K}. Using decomposition (3.5), one get that for any (α, β) ∈ {1, h, …, h^{i−1}, l_{0}}^{2}, with α ≠ β, one has

θ′_{i}_{∗}(α × β) = \{ \begin{matrix}π_{(0,\underline{i})}^{∗}(Z_{n−i}^{i}) & if α × β = 1 × l_{0} ; \\ 0 & otherwise. \end{matrix}

Consequently, since ρ_{i} is assumed to be rational, one obtains that the cycle

(Id_{X^{i−1}} × θ′_{i})_{∗} (ρ_{i}) = sym ((×_{j=1}^{i−1}h^{j})) × π_{(0,\underline{i})}^{∗}(Z_{n−i}^{i})

is rational. By multiplying the previous cycle by [X^{i−1}] × π_{(\underline{0},i)}^{∗}(h^{i}) and then composing by (Id_{X^{i−1}} × π_{(0,\underline{i})})_{∗}, one get, using the Projection Formula and Lemma 2.4(i), that the cycle

sym ((×_{j=1}^{i−1}h^{j})) × Z_{n−i}^{i}

is rational. Therefore, it suffices to prove that, for any 1 ≤ k ≤ i − 1, the rationality of sym ((×_{j=1}^{i−k}h^{j})) × Z_{n−i}^{i} implies the rationality of sym ((×_{j=1}^{i−k−1}h^{j})) × Z_{n−i}^{i} to conclude. This follows from the next formula (which uses decomposition (3.5)) p_{\underline{X^{i−k−1}}×X×\underline{G_{i}}}_{∗} ([X^{i−k−1}] × (h^{k} × [G_{i}] · [F(0, i)]_{K}) · sym ((×_{j=1}^{i−k}h^{j})) × Z_{n−i}^{i}) =

sym ((×_{j=1}^{i−k−1}h^{j})) × Z_{n−i}^{i} .
□

References

[1] 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.

[2] Vishik, V. Symmetric operations (in russian). In Trudy Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., pp. 92-105. English transl: Proc of the Steklov Institute of Math. 246, 79–92 (2004).

[3] Vishik, V. Generic points of quadrics and Chow groups. Manuscripta Math 122, 3 (2007), 365–374.

[4] Vishik, V. Fields of u-invariant 2^{r} + 1. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. II, vol. 270 of Progr. Math. Birkhauser Boston Inc., Boston, MA, 2009, pp. 661-685.

Instituto de Matemáticas, Ciudad Universitaria, UNAM, DF 04510, México E-mail address: fino at im.unam.mx