HOLES IN I^{n}

NIKITA A. KARPENKO

Abstract. Let F be a field with char(F) ≠ 2. We prove Vishik’s conjecture stating that for any positive integer n and any anisotropic quadratic form ϕ over F lying in the n-th power I^{n} of the fundamental ideal I of the Witt ring of F, either dim(ϕ) ≥ 2^{n+1} or dim(ϕ) = 2^{n+1} − 2^{i} for some i ∈ {1, 2, …, n + 1}. This provides a complete solution of an old-standing problem in the algebraic theory of quadratic forms. The proof is based on computations in the Chow groups of direct products of projective quadrics (involving the Steenrod operations); the method developed can be also applied to other types of algebraic varieties.

Contents

1. Introduction 1 2. Notation and preliminary observations 2 3. Strategy of proof 7 4. Cycles on X^{2} 10 5. Cycles on X^{3} 18 6. Proof of Conjecture 1.1 22 7. Possible dimensions 27 References 29

1. Introduction

In this text F is a field with char(F) ≠ 2. Let I^{n} for some n ≥ 1 be the n-th power of the fundamental ideal I (of the classes of the even-dimensional quadratic forms) of the Witt ring W (F). An old-standing question in the algebraic theory of quadratic forms asks about the possible values of dimension of an anisotropic quadratic form ϕ over F such that [ϕ] ∈ I^{n}, where [ϕ] is the class of ϕ in W (F).

Examples with dim(ϕ) = 2^{n+1} − 2^{i} for each i ∈ {1, 2, …, n + 1} are easy to construct (see Remark 7.2). A classical theorem of J. Arason and A. Pfister [1, Hauptsatz] states that dim(ϕ) is never between 0 = 2^{n+1} − 2^{n+1} and 2^{n} = 2^{n+1} − 2^{n}. Also it is known that every value between 2^{n} and 2^{n} + 2^{n−1} = 2^{n+1} − 2^{n−1} is impossible (A. Pfister for n = 3, [18, Satz 14]; D. Hoffmann for n = 4, [6, main thm.]; A. Vishik for all n, [23], see also [22, thm. 1.5] ^{1}).

Date: December 2003.

Key words and phrases. Quadratic forms, Witt indices, Chow groups, Steenrod operations, correspondences. 2000 Mathematical Subject Classifications: 11E04; 14C25.

1Alternative proofs are given in [14, thm. 4.4] and in [12].

Finally, A. Vishik has shown that all even values ≥ 2^{n+1} are possible ([22, thm. 4.12], see also §7 here) and suggested the following

Conjecture 1.1 (Vishik, [22, conject. 4.11]). ^{2} If ϕ ∈ I^{n} and dim(ϕ) < 2^{n+1}, then dim(ϕ) = 2^{n+1} − 2^{i} for some i ∈ {1, 2, …, n + 1}.

In the present text we prove this conjecture (see §6), obtaining a complete answer to the question about possible dimensions of anisotropic quadratic forms whose classes lie in I^{n}. The proof closely follows the method of [12], but involves essentially more computations. As [12] as well, it makes use of an important property of the quadratic forms satisfying the hypotheses of Conjecture 1.1 established by A. Vishik in [22]. Here we give an extended version of this result (see Proposition 4.28) with an elementary, complete, and self-contained (in particular, independent of [22]) proof.

In the proof of Conjecture 1.1 we work with projective quadrics rather than with quadratic forms themselves. The method of proof is explained in §3; it certainly applies to other types of algebraic varieties (in place of quadrics).

Acknowledgements. This work was completed during the last week of September 2003 when I was visiting the University of Alberta; an important part of this work was written down during the first week of November 2003 when I was visiting the Universität Göttingen, where I gave a course of lectures on the subject. I am very grateful to both institutions for hospitality and stimulating conditions. Besides I would like to mention that numerous remarks of A. S. Merkurjev improved very much the final version of this text.

2. Notation and preliminary observations

Everywhere in the text, X is a smooth projective quadric over F of an even dimension D = 2d or of an odd dimension ^{3} D = 2d + 1 given by a non-degenerate quadratic form ϕ. We write X^{r} for the direct product X × ⋯ × X (over F) of r copies of X and we write C̄h(X^{r}) for the image of the restriction homomorphism Ch(X^{r}) → Ch(X̄^{r}) where X̄ = XF̄ with a fixed algebraic closure F̄ of F and Ch(.) stays for the modulo 2 total Chow group. We say that an element of Ch(X̄^{r}) is rational, if it lies in the subgroup C̄h(X^{r}) ⊂ Ch(X̄^{r}).

A basis of the group Ch(X̄) (over ℤ/2ℤ) consists of h^{i} and l_{i}, i ∈ {0, 1, …, d}, where h stays for the class of a hyperplane section of X̄ while l_{i} is the class of an i-dimensional linear subspace^{4} lying on X̄. Moreover, a basis of the group Ch(X̄^{r}) for every r ≥ 1 is given by the exterior products of the basis elements of Ch(X̄) (see, e.g., [11, §1] for an explanation why this is a basis). Speaking about a basis or a basis element (or a basis cycle: we will often apply the word “cycle” to an element of a Chow group) of Ch(X̄^{r}), we will always refer to the basis described above. By the decomposition of an element α ∈ Ch(X̄^{r}) we always mean its representation as a sum of basis cycles. We say that a basis cycle β is contained in the decomposition of α (or simply “is contained in α”), if β is a summand of the decomposition. More generally, for two cycles α′, α ∈ Ch(X̄^{r}), we say that α′ is contained in α, or that α′ is a subcycle of α (notation: α′ ⊂ α), if every basis element contained in α′ is also contained in α.

2A. Vishik announced (for the first time in June 2002) that he has a proof of Conjecture 1.1; his proof however is not available.

3Only the even-dimensional case is important for our main purpose; the odd-dimensional case is included for the sake of completeness.

4In the case of even D and i = d (and only in this case) the class l_{i} depends on the choice of the subspace:

more precisely, there are two different classes of d-dimensional subspaces on X̄ and no canonical choice of one of them is possible; we do not care about this however and we just choose one of them, call it l_{d} and “forget” about the other one which is equal to h^{d} − l_{d}.

A basis element of Ch(X̄^{r}) is called non-essential, if it is an exterior product of powers of h (including h^{0} = 1 = [X̄]); the other basis elements are called essential. An element of Ch(X̄^{r}) which is a sum of non-essential basis elements, is called non-essential as well. Note that all non-essential elements are rational simply because h is rational.^{5}

The multiplication table for the ring Ch(X̄) is determined by the rules h^{d+1} = 0, h·l_{i} = l_{i−1} (i ∈ {d, d−1, …, 0}; we adopt the agreement that l_{−1} = 0), and l_{d}^{2} = (D+1)·(d+1)·l_{0} (see [8, §2.1] and [9, §1.2.1]). The multiplication tables for the rings Ch(X̄^{r}) (for all r ≥ 2) follows by

(β_{1} × ⋯ × β_{r}) · (β′_{1} × ⋯ × β′_{r}) = β_{1}β′_{1} × ⋯ × β_{r}β′_{r} .

The cohomological action of the topological Steenrod algebra on Ch(X̄^{r}) (see [2] for the construction of the action of the topological Steenrod algebra on the Chow group of a smooth projective variety; originally Steenrod operations in algebraic geometry were introduced (in the wider context of motivic cohomology) by V. Voevodsky, [24]) is determined by the fact that the total Steenrod operation S: Ch(X̄^{r}) → Ch(X̄^{r}) is a (nonhomogeneous) ring homomorphism, commuting with the exterior products and satisfying the formulae (see [11, §2 and cor. 3.3])

S(h^{i}) = h^{i} · (1 + h)^{i}, S(l_{i}) = l_{i} · (1 + h)^{D−i+1}, i ∈ {0, 1, …, d}

(in order to apply these formulae, one needs a computation of the binomial coefficients modulo 2, done, e.g., in [13, lemma 1.1]).

The group C̄h(X) is easy to compute. First of all one has

Lemma 2.1. If the quadric X is anisotropic (that is, X(F) = ∅), then C̄h_{0}(X) ∌ l_{0}.

Proof. If l_{0} ∈ C̄h_{0}(X), then the variety X contains a closed point x of an odd degree [F (x) : F]. It follows that the quadratic form ϕ is isotropic over an odd degree extension of the base field (namely, over F (x)) and therefore, by the Springer-Satz (see [20, thm. 5.3]), is isotropic already over F. □

Corollary 2.2. If X is anisotropic, then the group C̄h(X) is generated by the nonessential basis elements.

Proof. If the decomposition of an element α ∈ C̄h(X) contains an essential basis element l_{i} for some i ≠ D/2, then l_{i} ∈ C̄h(X) because l_{i} is the i-dimensional homogeneous component of α (and C̄h(X) is a graded subring of Ch(X̄)). If the decomposition of an element α ∈ C̄h(X) contains the essential basis element l_{i} for i = D/2, then the (D/2)- dimensional homogeneous component of α is either l_{D/2} or l_{D/2} + h^{D/2} and we still have l_{i} ∈ C̄h(X). It follows that l_{0} = l_{i} · h^{i} ∈ C̄h(X), a contradiction with Lemma 2.1. □

5There are at least two direct ways to show that h is rational: (1) h is the pull-back of the hyperplane class H with respect to the embedding of X̄ into the projective space, and H is rational; (2) h is the first Chern class of [OX̄(1)] ∈ K_{0}(X̄), and [OX̄(1)] = res([O_{X}(1)]) is rational.

Now assume for a moment that the quadric X is isotropic but not completely split (that is, 𝔦_{0}(X) ≤ d), write a for the Witt index 𝔦_{0}(X) of X (defined as the Witt index 𝔦_{0}(ϕ) of ϕ, see [20, def. 5.10 of ch. 1]), and let X_{0} be the projective quadric given by the anisotropic part ϕ_{0} of ϕ (one has dim(X_{0}) = dim(X) − 2a; the case dim(X_{0}) = 0 is possible). We consider a group homomorphism pr: Ch(X̄) → Ch(X̄_{0}) determined on the basis by the formulae h^{i} ↦ h^{i−a} and l_{i} ↦ l_{i−a} (here we adopt the agreement h^{i} = 0 and l_{i} = 0 for all negative i). Also we consider a backward group homomorphism in : Ch(X̄_{0}) → Ch(X̄) determined by the formulae h^{i} ↦ h^{i+a} and l_{i} ↦ l_{i+a} for i ∈ {0, 1, …, d − a}.

Let r be a positive integer. For every length r sequence i_{1}, …, i_{r} of integers satisfying i_{j} ∈ [0, a] ∪ [D − a + 1, D], we define a group homomorphism

pr_{i_{1}...i_{r}}: Ch(X̄^{r}) → Ch(X̄_{0}^{s})

with s = #{i_{j}| i_{j} = a}, called projection. Let {j_{1} < ⋯ < j_{s}} be the set of indices such that i_{j_{s}} = a. We put J_{l} = {j| i_{j} < a} and J_{h} = {j| i_{j} > a}. Then we define pr_{i_{1}...i_{r}}(α_{1} × ⋯ × α_{r}) for a basis element α_{1} × ⋯ × α_{r} as pr(α_{j_{1}}) × ⋯ × pr(α_{j_{s}}) as far as α_{j} = l_{i_{j}} for any j ∈ J_{l} and α_{j} = h^{D−i_{j}} for any j ∈ J_{h}; we set pr_{i_{1}...i_{r}}(α_{1} × ⋯ × α_{r}) = 0 otherwise.

Also we define a backward group homomorphism in_{i_{1}...i_{r}} : Ch(X̄_{0}^{s}) → Ch(X̄^{r}), called inclusion, by

in_{i_{1}...i_{r}}(β_{1} × ⋯ × β_{s}) = α_{1} × ⋯ × α_{r}

for a basis element β_{1} × ⋯ × β_{s}, where α_{j} = l_{i_{j}} for j ∈ J_{l}, α_{j} = h^{D−i_{j}} for j ∈ J_{h}, and α_{j_{k}} = in(β_{k}) for k = 1, 2, …, s.

Proposition 2.3 (cf. [12, lemma 2.2]). In the notation introduced right above, the direct sum

⨁_{(i_{1},...,i_{r})} pr_{i_{1}...i_{r}}: Ch(X̄^{r}) → ⨁_{(i_{1},...,i_{r})} Ch(X̄_{0}^{s(i_{1},...,i_{r})})

of all projections is an isomorphism with the inverse given by the sum of all inclusions. Under these isomorphisms, rational cycles correspond to rational cycles.

Proof. By the Rost motivic decomposition theorem for isotropic quadrics (original proof is in [19], generalizations are obtained in [10] and [4]), there is a motivic decomposition (in the category of the integral Chow motives)

(∗)  X ≃ ℤ ⊕ ℤ(1) ⊕ ⋯ ⊕ ℤ(a − 1) ⊕ X_{0}(a) ⊕ ℤ(D − a + 1) ⊕ ⋯ ⊕ ℤ(D)

(where ℤ is the motive of Spec F, while M(i) is the i-th Tate twist of a motive M). Rasing to the r-th power, we get a motivic decomposition of the variety X^{r}; each summand of this decomposition is a twist of the motive of X_{0}^{s} with s varying between 0 and r. If we numerate the summands of the decomposition (∗) by their twists, then the summands of the decomposition of X^{r} are numerated by the sequences

i_{1}, …, i_{r} with i_{j} ∈ [0, a] ∪ [D − a + 1, D] .

Moreover, the (i_{1}, …, i_{r})-th summand is X_{0}^{s}(i_{1} + ⋯ + i_{r}), where s = #{i_{j}| i_{j} = a}.

In order to finish the proof of Proposition 2.3, it suffices to show that the projection morphism to the (i_{1}, …, i_{r})-th summand considered on the Chow group and over F̄ coincides with pr_{i_{1}...i_{r}} while the inclusion morphism of the (i_{1}, …, i_{r})-th summand considered on the Chow group and over F̄ coincides with in_{i_{1}...i_{r}}. Clearly it suffices to check this for r = 1 only. For i ≠ D/2, this is particularly easy to do because of the relation dim_{ℤ/2ℤ}(Ch_{i}(X̄)) ≤ 1. Indeed, Ch_{i}(ℤ(k)) = 0 for k ≠ i. Therefore for any i with a ≤ i ≤ D − a, i ≠ D/2, the projection and the inclusion between Ch_{i}(X̄) and Ch_{i−a}(X̄_{0}) are isomorphisms and, as a consequence, they interchange the only non-zero elements of these two groups (which are l_{i} and l_{i−a} if i < D/2, or h^{D−i} and h^{D−i−a} if i > D/2). For i < a, the projection and the inclusion are isomorphisms between Ch_{i}(X̄) and ℤ/2ℤ = Ch_{i}(ℤ(i)) and the only non-zero element of the first Chow group is l_{i}. Finally, for i > D − a, the projection and the inclusion are isomorphisms between Ch_{i}(X) and ℤ/2ℤ and the only non-zero element of the Chow group is h^{D−i}.

For i = D/2 (here we are in the case of even D, of course), the basis of the group Ch_{i}(X̄) is given by the elements h^{d} and l_{d}, while the basis of the group Ch_{i−a}(X̄_{0}) is given by the elements h^{d−a} and l_{d−a}. The subgroups C̄h_{d}(X) ⊂ Ch_{d}(X̄) and C̄h_{d−a}(X_{0}) ⊂ Ch_{d−a}(X̄_{0}), however, are 1-dimensional, generated by h^{d} and h^{d−a} (because l_{d−a} ∉ C̄h(X_{0}) by Corollary 2.2). Since these subgroups are interchanged by the projection and the inclusion, h^{d} corresponds to h^{d−a}. Now there are only two possibilities for the element of Ch_{d}(X̄) corresponding to l_{d−a} ∈ Ch_{d−a}(X̄_{0}): either this is l_{d} or this is l_{d} + h^{d}. Which one of these two possibilities takes place depends on the construction of the motivic decomposition (∗); but a given motivic decomposition can be always corrected in such a way that the first possibility takes place (one can simply use an automorphism of the variety X_{0} interchanging l_{d−a} with l_{d−a} + h^{d−a}). □

The “most important” summand in the motivic decomposition of X^{r} is, of course, X_{0}^{r}. We introduce a special notation for the projection and the inclusion related to this summand: pr^{r} = pr_{a...a} and in^{r} = in_{a...a}.

Corollary 2.4. The (mutually semi-inverse) homomorphisms

pr^{r}: Ch(X̄^{r}) → Ch(X̄_{0}^{r}) and in^{r}: Ch(X̄_{0}^{r}) → Ch(X̄^{r})

(for any r ≥ 1) map rational cycles to rational cycles; moreover, the induced homomorphism pr^{r}: C̄h(X^{r}) → C̄h(X_{0}^{r}) is surjective. □

Now we get an extended version of Corollary 2.2 which reads as follows:

Corollary 2.5. For an arbitrary quadric X (isotropic or not) and any integer i one has: l_{i} ∈ C̄h(X) if and only if 𝔦_{0}(X) > i (where 𝔦_{0}(X) = 𝔦_{0}(ϕ) is the Witt index).

Proof. The “if” part of the statement is trivial. Let us prove the “only if” part using an induction on i. The case of i = 0 is served by Lemma 2.1.

Now we assume that i > 0 and l_{i} ∈ C̄h(X). Since l_{i} · h = l_{i−1}, the element l_{i−1} is rational as well, and by the induction hypothesis 𝔦_{0}(X) ≥ i. If 𝔦_{0}(X) = i, then the image of l_{i} ∈ C̄h(X) under the map pr^{1} : Ch(X̄) → Ch(X̄_{0}) equals l_{0} and is rational by Corollary 2.4. Therefore, by Lemma 2.1, the quadric X_{0} is isotropic, a contradiction. □

We recall that the splitting pattern sp(ϕ) of an anisotropic quadratic form ϕ is defined as the set of integers

sp(ϕ) = {dim(ϕ_{E})_{0}| E/F is a field extension}

(here ϕ_{E} stays for the quadratic form over the field E obtained from ϕ by extending the scalars; (ϕ_{E})_{0} is the anisotropic part of ϕ_{E}).

The splitting pattern can be obtained using the generic splitting tower of M. Knebusch (arbitrary filed extensions of F are then replaced by concrete fields). To construct this tower, we put F_{1} = F (X), the function field of the projective quadric X given by ϕ. Then we put ϕ_{1} = (ϕ_{F_{1}})_{0} and write X_{1} for the projective quadric (over the field F_{1}) given by the quadratic form ϕ_{1}. We proceed by setting F_{2} = F_{1}(X_{1}) and so on until we can (we stop on F_{𝔥} such that dim(ϕ_{𝔥}) ≤ 1). The tower of fields F ⊂ F_{1} ⊂ ⋯ ⊂ F_{𝔥} obtained this way is called the generic splitting tower of ϕ and (see [15])

sp(ϕ) = {dim(ϕ), dim(ϕ_{F_{1}})_{0}, …, dim(ϕ_{F_{𝔥}})_{0}} = {dim(ϕ), dim(ϕ_{1}), …, dim(ϕ_{𝔥})}

(the integer 𝔥 = 𝔥(ϕ) is the height of ϕ; note that the elements of sp(ϕ) are written down in the descending order).

An equivalent invariant of ϕ is called the higher Witt indices of ϕ and defined as follows. Let us write the set of integers {𝔦_{0}(ϕ_{E})| E/F a field extension}, where 𝔦_{0}(ϕ_{E}) is the usual Witt index of ϕ_{E}, in the form

{0 = 𝔦_{0}(ϕ) < 𝔦_{1} < 𝔦_{1} + 𝔦_{2} < ⋯ < 𝔦_{1} + 𝔦_{2} + ⋯ + 𝔦_{𝔥}} .

The sequence of the positive integers 𝔦_{1}, …, 𝔦_{𝔥} is called the higher Witt indices of ϕ. For every q ∈ {0, 1, …, 𝔥}, we also set

𝔧_{q} = 𝔧_{q}(ϕ) = 𝔦_{0} + 𝔦_{1} + ⋯ + 𝔦_{q} = 𝔦_{0}(ϕ_{F_{q}}) .

Clearly, one has

{0 = 𝔧_{0}, 𝔧_{1}, …, 𝔧_{𝔥}} = {𝔦_{0}(ϕ_{E})| E/F is a field extension}

(this set of integers is sometimes also called the splitting pattern of ϕ in the literature). The following easy observation is crucial:

Theorem 2.6. The splitting pattern as well as the higher Witt indices of an anisotropic quadratic form ϕ (of some given dimension) are determined by the group

C̄h(X^{∗}) = ⨁_{r≥1} C̄h(X^{r}) .

Proof. The pull-back homomorphism g_{1}^{∗}: Ch(X^{r}) → Ch(X_{F (X)}^{r−1}) with respect to the morphism of schemes g_{1} : X_{F (X)}^{r−1} → X^{r} given by the generic point of, say, the first factor of X^{r}, is surjective. It induces an epimorphism C̄h(X^{r}) →→ C̄h(X_{F (X)}^{r−1}), which is a restriction of the epimorphism Ch(X̄^{r}) →→ Ch(X̄_{F (X)}^{r−1}) mapping each basis element of the shape h^{0} × β with β ∈ Ch(X̄^{r−1}) to β ∈ Ch(X̄_{F (X)}^{r−1}) and killing all other basis elements. Therefore the group C̄h(X^{∗}) determines the group C̄h(X_{F (X)}^{∗}). In particular, the group C̄h(X_{F (X)}) is determined, so that we have reconstructed 𝔦_{0}(X_{F (X)}) = 𝔦_{1}(X) (see Corollary 2.5). Moreover, by Corollary 2.4, the group C̄h(X_{F (X)}^{∗}) determines the group C̄h(X_{1}^{∗}) (via the surjection pr^{∗}; X_{1} staying for the anisotropic part of X_{F (X)}), and we can proceed by induction. □

Remark 2.7. The proof of Theorem 2.6 makes it clear that the statement of this theorem can be made more precise in the following way. If for some q ∈ {1, 2, …, 𝔥} the Witt indices 𝔦_{1}, …, 𝔦_{q−1} are already reconstructed, then one determines 𝔦_{q} = 𝔧_{q} − 𝔦_{q−1} − ⋯ − 𝔦_{1} by the formula

𝔧_{q} = max{j| the product h^{0} × h^{𝔧_{1}} × ⋯ × h^{𝔧_{q−1}} × l_{j−1} is contained in a rational cycle} .

Remark 2.8. Concluding this section, we would like to underline that the role of the algebraic closure F̄ in the definition of the group C̄h(X^{∗}) is secondary: the group Ch(X̄^{∗}) (used in the definition of C̄h(X^{∗})) has to be interpreted as the direct limit lim Ch(X_{E}^{∗}) taken over all field extensions E/F. The homomorphism Ch(X̄^{∗}) → lim Ch(X_{E}^{∗}) is an isomorphism. More generally, the homomorphism Ch(X_{E}^{∗}) → lim Ch(X_{E}^{∗}) for a given E/F is an isomorphism if and only if the quadratic form ϕ_{E} is completely split (in particular, for any E/F with completely split ϕ_{E}, there is a canonical isomorphism Ch(X_{E}^{∗}) = Ch(X̄^{∗}), coinciding with the composition res_{Ē/F̄}^{−1} ∘ res_{Ē/E}, where Ē is a field containing E and F̄).

3. Strategy of proof

As shown in Theorem 2.6, the group C̄h(X^{∗}) determines the splitting pattern of the quadratic form ϕ. In its turn, the splitting pattern of ϕ determines the powers of the fundamental ideal of the Witt ring containing the class of ϕ. At least it is easy to prove

Lemma 3.1. Let ϕ be an even-dimensional anisotropic quadratic form and let n ≥ 1 be an integer. We write p for the least positive integer of sp(ϕ) (note that p is a power of 2, [20, thm. 5.4(i)]). If [ϕ] ∈ I^{n}, then p ≥ 2^{n}.

Proof. We assume that [ϕ] ∈ I^{n}(F). Let E/F be a field extension such that dim(ϕ_{E})_{0} = p. Since 0 ≠ [(ϕ_{E})_{0}] ∈ I^{n}(E), we get that p ≥ 2^{n} by the Arason-Pfister theorem. □

Remark 3.2. It is not needed in this paper but nevertheless good to know that the converse statement to Lemma 3.1 is also true. This is a hard result, however. It is proved in [16, thm. 4.3].

Now we are able to describe the strategy of our proof of Conjecture 1.1. Let us consider a power I^{n} of the fundamental ideal. Let ϕ be an anisotropic quadratic form with [ϕ] ∈ I^{n}, having some dimension prohibited by Conjecture 1.1. The group C̄h(X^{∗}), where X is the projective quadric given by ϕ, should satisfy some restrictions listed bellow. This group is a subgroup of Ch(X̄^{∗}), the later one depends only on the dimension of ϕ. So, we prove Conjecture 1.1, if we check that every subgroup of Ch(X̄^{∗}), satisfying the list of restrictions, can not be C̄h(X^{∗}) for the form ϕ (see Lemma 3.1 with Theorem 2.6). This is the way we prove Conjecture 1.1.

And here is the list of restrictions on C̄h(X^{∗}) considered as a subset of Ch(X̄^{∗}) (a big part of this list is of course valid for an arbitrary smooth projective variety in place of the quadric X):

Proposition 3.3. Assuming that the quadric X is anisotropic,^{6} we have:

6Anisotropy is important only for (4), (8), (9), and (10).

(1) C̄h(X^{∗}) is closed under addition and multiplication;

(2) C̄h(X^{∗}) is closed under passing to the homogeneous components (with respect to the grading of the Chow group and to the ∗-grading);

(3) C̄h(X^{∗}) contains h^{0} = [X̄] and h^{1} = h (and therefore contains h^{i} for all i ≥ 0);

(4) [Springer-Satz] C̄h(X^{∗}) does not contain l_{0};

(5) for every r ≥ 1, C̄h(X^{r}) is closed under the automorphisms of Ch(X̄^{r}) given by the permutations of factors of X^{r};

(6) for every r ≥ 1, C̄h(X^{∗}) is closed under push-forwards and pull-backs with respect to all r projections X^{r} → X^{r−1} and to all r diagonals X^{r} → X^{r+1} (taking in account the previous restriction, it is enough to speak only about the first projection

x_{1} × x_{2} × ⋯ × x_{r} ↦ x_{2} × ⋯ × x_{r}

and the first diagonal

x_{1} × x_{2} × ⋯ × x_{r} ↦ x_{1} × x_{1} × x_{2} ⋯ × x_{r}

here);

(7) C̄h(X^{∗}) is closed under the total Steenrod operation

S: Ch(X̄^{∗}) → Ch(X̄^{∗}) ;

(8) [A. Vishik, “Size of binary correspondences”] if C̄h(X^{2}) ∋ h^{0} ×l_{i} +l_{i} ×h^{0} for some integer i ≥ 0, then the integer dim(X)−i+1 is a power of 2;

(9) [“Inductive restriction”] the image of C̄h(X^{∗+1}) under the composition

Ch(X̄^{∗+1}) ⟶^{g_{1}^{∗}} Ch(X̄_{F (X)}^{∗}) ⟶^{pr^{∗}} Ch(X̄_{1}^{∗})

(g_{1}^{∗} is introduced in the proof of Theorem 2.6, pr^{∗} in Corollary 2.4) should coincide with C̄h(X_{1}^{∗}) and therefore should satisfy all restrictions listed in this proposition (including the current one);

(10) [“Supplement to inductive restriction”] for any integer r ≥ 2, any integer s ∈ [1, r), and any projection pr_{i_{1}...i_{r}} of Proposition 2.3, the image of C̄h(X^{r}) under pr_{i_{1}...i_{r}}: Ch(X̄^{r}) → Ch(X̄_{1}^{s(i_{1},...,i_{r})}) is inside of C̄h(X_{1}^{s(i_{1},...,i_{r})}) (reconstructed by (9)).

Proof. Only the property (8) needs a proof. We note that this property does not seem to be a consequence of the others. It is proved in [13, thm. 5.1] by some computation in the integral Chow group of X^{∗}, not in the modulo 2 Chow group (although involving modulo 2 Steenrod operations) (the original proof is in [7, thm. 6.1]; it makes use of higher motivic cohomology).

More precisely, the case of i = 0 is proved in [13, thm. 5.1]. In order to reduce the general case to the case of i = 0, we take an arbitrary subquadric Y ⊂ X of codimension i and pullback the cycle h^{0} × l_{i} + l_{i} × h^{0} with respect to the imbedding Y^{2} ↪ X^{2}. The result is h^{0} × l_{0} + l_{0} × h^{0} ∈ C̄h(Y^{2}). Therefore, dim(Y) + 1 is a power of 2 by [13, thm.

5.1]. Since dim(Y) = dim(X) − i, it follows that the integer dim(X) − i + 1 is the same power of 2. □

Remark 3.4. Obviously, one can write down some additional restrictions on C̄h(X^{∗}). However, all restrictions I know are consequences of the restrictions of Proposition 3.3. For instance, C̄h(X^{∗}) should be stable with respect to the exterior products; but this is a consequence of the stability with respect to the internal products (1) and the pull-backs with respect to projections (6). Another example: the image of the total Chern class c: K_{0}(X̄^{∗}) → Ch(X̄^{∗}) restricted to K_{0}(X^{∗}) (note that K_{0}(X^{∗}) is computed for quadrics [21] and, more generally, for all projective homogeneous varieties [17]) should be inside of C̄h(X^{∗}); but it is already guarantied by the fact that C̄h(X^{∗}) is closed under addition and multiplication (1) and contains h^{0} and h^{1} (3).^{7} One more example: C̄h(X^{2}) should be closed under the composition of correspondences (see [5, §16] for the definition of composition of correspondences); but the operation of composition of correspondences is produced by pull-backs and push-forwards with respect to projections together with the operation of multiplication of cycles.

Remark 3.5. Let us remark that all operations involved in the list of restrictions of Proposition 3.3 are easy to compute in terms of the basis elements. The multiplication in Ch(X̄^{∗}) was described in the previous §; a formula for the total Steenrod operation was given already as well. Also the operations used in the inductive restriction are computed (see Corollary 2.4 and the proof of Theorem 2.6). As to the pull-backs and push-forwards with respect to the first projection pr : X^{r+1} → X^{r} and to the first diagonal δ: X^{r} → X^{r+1}, they are computed for basis elements β_{0}, β_{1}, …, β_{r} ∈ Ch(X̄) as follows:

pr^{∗}(β_{1} × ⋯ × β_{r})  = h^{0} × β_{1} × ⋯ × β_{r} ;

pr_{∗}(β_{0} × β_{1} × ⋯ × β_{r}) = \begin{cases}β_{1} × ⋯ × β_{r} , & if β_{0} = l_{0}, \\ 0 , & otherwise; \end{cases}

δ^{∗}(β_{0} × β_{1} × ⋯ × β_{r}) = (β_{0} · β_{1}) × β_{2} × ⋯ × β_{r} ;
δ_{∗}(β_{1} × ⋯ × β_{r})  = ((β_{1} × h^{0}) · Δ) × β_{2} × ⋯ × β_{r} =
((h^{0} × β_{1}) · Δ) × β_{2} × ⋯ × β_{r} ;

where Δ ∈ C̄h(X^{2}) is the class of the diagonal computed in Corollary 3.9.

Remark 3.6. One can obviously simplify a little bit the list of restrictions of Proposition 3.3. For instance, instead of stability under the push-forwards with respect to the diagonals, it suffices to require that the cycle ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}), related to the diagonal, lies in C̄h(X^{2}) (see Remark 3.5 and Corollary 3.9). Also the inductive restriction is not so restrictive as it may seem to: the group C̄h(X_{1}^{∗}) automatically satisfies most of the required restrictions.

Remark 3.7. Looking at the list of restrictions, it is easy to see that every group C̄h(X^{r}) determines C̄h(X^{<r}). Moreover, one can show that C̄h(X^{d}) determines the whole group C̄h(X^{∗}).^{8} Since C̄h(X^{d}) is a subgroup of the finite group Ch(X̄^{d}), it follows, in particular, that the invariant C̄h(X^{d}) (of the quadratic forms ϕ of a given dimension) has only a finite number of different values (this way one also sees that Conjecture 1.1 can be checked for any concrete dimension by computer).

7However the property with the Chern class can be a good replacement for (3) when transfering this theory to other algebraic varieties in place of the quadric X.

We will often use the composition of correspondences, even for the cycles on bigger than 2 powers of X: this is a convenient way to handle the things. Namely, for α ∈ C̄h(X^{r}) and α′ ∈ C̄h(X^{r′}), we may consider α as a correspondence, say, from X^{r−1} to X, and we may consider α′ as a correspondence from X to X^{r′−1}; then the composition α′ ∘ α is a cycle in C̄h(X^{r+r′−2}), and here is the formula for composing the basis elements (we put here this obvious formula because it will be used many times in our computations):

Lemma 3.8. The composition β′ ∘ β ∈ Ch(X̄^{r+r′−2}) of two basis elements

β = β_{1} × ⋯ × β_{r} ∈ Ch(X̄^{r}) and β′ = β′_{1} × ⋯ × β′_{r′} ∈ Ch(X̄^{r′})

is equal to β_{1} × ⋯ × β_{r−1} × β′_{2} × ⋯ × β′_{r′}, if β_{r} · β′_{1} = l_{0}; otherwise the composition β′ ∘ β is 0. □

Corollary 3.9. For the diagonal class Δ ∈ C̄h(X^{2}), one has:

Δ = ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}) + (D + 1) · (d + 1) · (h^{d} × h^{d}) .

In particular, the sum ∑_{i=0}^{d}(h^{i} × l_{i} + l_{i} × h^{i}) is always rational.

Proof. Using Lemma 3.8, it is straight-forward to verify that the cycle given by the above formula acts (by composition) trivially on any basis cycle of Ch^{2}(X̄^{2}). □

4. Cycles on X^{2}

We are using the notation introduced in § 2. In particular, X is the projective quadric of an even dimension D = 2d or of an odd dimension D = 2d + 1 given by a quadratic form ϕ over the field F. In this § we assume that X is anisotropic. Let 𝔦_{1}, … 𝔦_{𝔥} be the higher Witt indices of ϕ (with 𝔥 being the height of ϕ). We write S for the set {1, 2, …, 𝔥} and we set 𝔧_{q} = 𝔦_{1} + 𝔦_{2} + ⋯ + 𝔦_{q} for every q ∈ S.

An “important” as well as the “first interesting” part of the group C̄h(X^{∗}) is the group C̄h(X^{2}) and especially C̄h_{D}(X^{2}) = C̄h^{D}(X^{2}) (note that, due to the Rost nilpotence theorem ([19], see also [3]), the latter group detects all motivic decompositions of X). The group C̄h_{D}(X^{2}) was studied intensively by A. Vishik (see, e.g., [23]). In the next § we reproduce most of his results concerning this group. Actually we give an extended version of these results describing the structure of a bigger group, namely of the group C̄h^{≤D}(X^{2}).

Originally, Vishik’s results are formulated in terms of motivic decompositions of X; by this reason, their proofs use the Rost nilpotence theorem for quadrics, which is not used in the present text at all. Here we simplify the formulation; we also give a different complete self-contained proof and show that all these results are consequences of the restrictions on C̄h(X^{∗}) listed in Proposition 3.3. More precisely, we start with some general results concerning an arbitrary anisotropic quadric X; the proofs of these results use neither the restriction provided by the Steenrod operation, nor the “size of binary correspondences” restriction; a summary of these results is given in Theorem 4.24. Proposition 4.28, appearing in the very end of the section, contains the result on the so called small quadrics (Definition 4.27), needed in the proof of Conjecture 1.1; its proof uses the “size of binary correspondences” restriction (the Steenrod operation does still not show up).

8It would be interesting to rewrite all restrictions of Proposition 3.3 in terms of the group C̄h(X^{d}).

We start by the following easy observation:

Lemma 4.1. For an even D = dim(X), the basis element l_{d} × l_{d} ∈ C̄h_{D}(X^{2}) does not appear in the decomposition of a rational cycle.

Proof. We assume the contrary. Let α be a cycle in C̄h_{D}(X^{2}) containing l_{d} × l_{d}. Then the push-forward with respect to the projection onto the second factor pr : X^{2} → X of the cycle α · (h^{d} × h^{0}) is rational and equals l_{d} or h^{d} + l_{d} (because pr_{∗} (β · (h^{d} × h^{0})) is l_{d} for β = (l_{d}×l_{d}), h^{d} for β = l_{d}×h^{d}, and 0 for every other basis cycle β ∈ Ch_{D}(X̄^{2})). Therefore the cycle l_{d} is rational, showing that X is hyperbolic (Corollary 2.5), a contradiction. □

Lemma 4.2. If α_{1}, α_{2} ∈ C̄h^{≤D}(X^{2}), then the cycle α_{1} ∩α_{2} is rational (where the notation α_{1} ∩ α_{2} means the sum of the basis cycles contained simultaneously in α_{1} and in α_{2}).

Proof. Clearly, we may assume that α_{1} and α_{2} are homogeneous of the same dimension D + i and do not contain any non-essential basis element. Then the intersection (modulo the non-essential elements) is computed as α_{1} ∩ α_{2} = α_{2} ∘ (α_{1} · (h^{0} × h^{i})) (see Lemma 3.8). □

Definition 4.3. We write Che(X̄^{∗}) for the subgroup of Ch(X̄^{∗}) generated by the essential basis elements. We set Che(X^{∗}) = Che(X̄^{∗}) ∩ C̄h(X^{∗}) . Note that the group Che(X^{∗}) is a subgroup of C̄h(X^{∗}) isomorphic to the quotient of C̄h(X^{∗}) by the subgroup of the non-essential elements.

Definition 4.4. A non-zero cycle of C̄h^{≤D}(X^{2}) is called minimal, if it does not contain any proper rational subcycle. Note that a minimal cycle does not contain non-essential basis elements, i.e., lies in Che(X^{2}). Also note that a minimal cycle is always homogeneous.

A very first structure result on C̄h^{≤D}(X^{2}) reads as follows:

Proposition 4.5. The minimal cycles form a basis of the group Che^{≤D}(X^{2}). Two different minimal cycles do not intersect each other (here we speak about the notion of intersection of cycles introduced in Lemma 4.2). The sum of the minimal cycles of dimension D is equal to the sum ∑_{i=0}^{d} h^{i} × l_{i} + l_{i} × h^{i} of all D-dimensional essential basis elements (excluding l_{d} × l_{d} in the case of even D).

Proof. The first two statements of Proposition 4.5 follow from Lemma 4.2. The third statement follows from previous ones together with the rationality of the diagonal cycle (see Corollary 3.9). □

Definition 4.6. Let α be a homogeneous cycle in Ch^{≤D}(X̄^{2}). For every i with 0 ≤ i ≤ dim(α) − D, the products α · (h^{0} × h^{i}), α · (h^{1} × h^{i−1}), and so on up to α · (h^{i} × h^{0}) will be called the (i-th order) derivatives of α. Note that all derivatives are still in Ch^{≤D}(X̄^{2}) and that all derivatives of a rational cycle are also rational.

Lemma 4.7. (1) Each derivative of any essential basis element β ∈ Che^{≤D}(X̄^{2}) is

an essential basis element.

(2) For any r ≥ 0 and any essential basis cycles β_{1}, β_{2} ∈ Che_{D+r}(X̄^{2}), two derivatives β_{1} · (h^{i_{1}} × h^{j_{1}}) and β_{2} · (h^{i_{2}} × h^{j_{2}}) of β_{1} and β_{2} coincide only if β_{1} = β_{2}, i_{1} = i_{2}, and j_{1} = j_{2}.

Proof. (1) If β ∈ Che_{D+r}(X̄^{2}) for some r ≥ 0, then, up to transposition, β = h^{i} × l_{i+r} with i ∈ [0, d − r]. An arbitrary derivative of β is equal to β · (h^{j_{1}} × h^{j_{2}}) = h^{i+j_{1}} × l_{i+r−j_{2}} with some j_{1}, j_{2} ≥ 0 such that j_{1} + j_{2} ≤ r. We have 0 ≤ i + j_{1} ≤ d and therefore h^{i+j_{1}} is a basis element. We also have d ≥ i + r − j_{2} ≥ 0 and therefore l_{i+r−j_{2}} is a basis element too.

Statement (2) is trivial. □

Remark 4.8. For the sake of visualization, it is good to think of the basis cycles of Ch^{≤D}(X̄^{2}) (with l_{D/2} × l_{D/2} excluded by the reason of Lemma 4.1) as of points of the

“pyramid” ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∗ ∗ ∗ ∗ ∗

(D = 8 on the picture; for an odd D the pyramid has no “step”, see, e.g., the picture of Remark 4.12), where the ∘-s stay for the non-essential basis elements while the ∗-s stay for the essential ones; the point on the top is h^{0} × h^{0}; for every i ∈ {0, 1, …, D}, the i-th row of the pyramid represents the basis of Ch^{i}(X̄^{2}) (in the case of even D, the D-th row is the basis without l_{d} × l_{d}) ordered by the codimension of the first factors (starting with h^{0}×?). For any α ∈ Che^{≤D}(X̄^{2}), we can put a mark on the points representing basis elements contained in the decomposition of α; the set of marked points is the diagram of α. If α is homogeneous, the marked points lie in the same row. Now it is easy to interpret the derivatives of α: the diagram of an i-th order derivative is a projection of the marked points of the diagram of α to the i-th row bellow along some direction. The diagram of every derivative of α has the same number of marked points as the diagram of α (Lemma 4.7). The diagrams of two different derivatives of the same order are shifts (to the right or to the left) of each other.

Lemma 4.9. The following conditions on a homogeneous cycle α ∈ C̄h^{≤D}(X^{2}) are equivalent:

(1) α is minimal;

(2) all derivatives of α are minimal;

(3) at least one derivative of α is minimal.

Proof. Derivatives of a proper subcycle of α are proper subcycles of the derivatives of α; therefore, (3) ⇒ (1).

In order to show that (1) ⇒ (2), it suffices to show that two first order derivatives α · (h^{0} × h^{1}) and α · (h^{1} × h^{0}) of a minimal cycle α are minimal. In the contrary case, replacing eventually α by its transposition, we come to the situation where the derivative α · (h^{0} × h^{1}) of a minimal α is not minimal. It follows that the cycle α · (h^{0} × h^{i}), where i = dim(α)−D, is not minimal too; let α′ be its proper subcycle. Taking the composition α ∘ α′ and removing the non-essential summands, we get a proper subcycle of α (see Lemma 3.8). □

Corollary 4.10. The derivatives of a minimal cycle are disjoint.

Proof. The derivatives of a minimal cycle are minimal (Lemma 4.9) and pairwise different (Lemma 4.7). Two different minimal cycles are disjoint by Lemma 4.2 (see also Proposition 4.5). □

Lemma 4.11. Let α be an element of C̄h_{D+k−1}(X^{2}) with some k ≥ 1. For any q ∈ {1, …, 𝔥} and for any non-negative i with 𝔦_{q} − k < i < 𝔦_{q}, the cycle α contains neither the product h^{𝔧_{q−1}+i} × l_{𝔧_{q−1}+i+k−1} nor the transposition of this product.

Proof. Let us assume the contrary: for some k ≥ 1, some q ∈ {1, …, 𝔥}, and some i with 𝔦_{q} − k < i < 𝔦_{q}, there exists a rational cycle α containing the product h^{𝔧_{q−1}+i} × l_{𝔧_{q−1}+i+k−1} or the transposition of this product. If α contains the transposition of the product, we replace α by the transposition of α. Passing to the (q − 1)-th field of the generic splitting tower and using the projection of Corollary 2.4, we come to the situation where q = 1 and α contains the product h^{i}×l_{i+k−1} such that 𝔦_{1}−k < i < 𝔦_{1}. The projection pr_{D−i,𝔦_{1}}(α_{F (X)}) is a rational cycle on X̄_{1} containing l_{i+k−1−𝔦_{1}} (note that i + k − 1 − 𝔦_{1} ≥ 0). We get a contradiction with Corollary 2.2. □

Remark 4.12. In order to “see” the statement of Lemma 4.11, it is helpful to mark by • the essential basis elements which are not “forbidden” by this lemma (we are speaking about the pyramid of basis cycles drawn in Remark 4.8). We will get isosceles triangles based on the lower row of the pyramid. For example, if X is a 29-dimensional quadric with the higher Witt indices 4, 3, 5, 2 (such a quadric X does not exist in reality, but is convenient for the illustration), then the picture looks as follows:

∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∘ ∘ ∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ • ∗ ∗ ∘ ∘ ∘ ∘ ∗ ∗ • ∗ ∗ ∗ ∗ ∗ ∗ ∗ • ∗ ∗ ∗ ∗ ∗ ∗ • • ∗ ∗ ∘ ∘ ∘ ∗ ∗ • • ∗ ∗ ∗ ∗ ∗ ∗ • • • ∗ ∗ • ∗ ∗ • • • ∗ ∗ ∘ ∘ ∗ ∗ • • • ∗ ∗ • ∗ ∗ • • • • • ∗ • • ∗ • • • • ∗ • ∘ • ∗ • • • • ∗ • • ∗ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Definition 4.13. The triangles of Remark 4.12 will be called the shell triangles (their bases are shells in the sense of A. Vishik). The shell triangles in the left half of the pyramid are counted from the left starting by 1. The shells triangles in the right half of the pyramid are counted from the right starting by 1 as well (so that the symmetric triangles have the same number; for any q ∈ S, the bases of the q-th triangles have (each) 𝔦_{q} points). The rows of the shell triangles are counted from bellow starting by 0. The points of rows of the shell triangles (of the left ones as well as of the right one) are counted from the left starting by 1.

Lemma 4.14. For every rational cycle α ∈ C̄h^{≤D}(X^{2}), the number of the essential basis cycles contained in α is even (that is, the number of the marked points in the diagram of any α ∈ Che^{≤D}(X^{2}) is even).

Proof. We may assume that α is homogeneous, say, α ∈ C̄h_{D+k}(X^{2}), k ≥ 0. Let n be the number of the essential basis cycles contained in α. The pull-back δ^{∗}(α) of α with respect to the diagonal δ: X → X^{2} produces n · l_{k} ∈ C̄h(X). By Corollary 2.2, it follows that n is even. □

Lemma 4.15. Let α ∈ C̄h(X^{2}) be a cycle containing β = h^{𝔧_{q−1}} × l_{𝔧_{q}−1} for some q ∈ S = {1, 2, …, 𝔥} (this β is the top of the q-th left shell triangle). Then α also contains the transposition of β.

Proof. Replacing F by the field F_{q−1} of the generic splitting tower of F, X by X_{q−1}, and α by pr^{2}(α_{F_{q−1}}), we come to the situation where q = 1.

Then we replace α by its homogeneous component containing β and apply to it Lemma 4.11 (with k = 𝔦_{1}). Let us assume that the transposition of β is not contained in α.

By Lemma 4.11 α does not contain any of the essential basis cycles having h^{i} with 0 < i < 𝔦_{1} as a factor; therefore the number of the essential basis elements contained in α and the number of the essential basis elements contained in pr^{2}(α_{F (X)}) ∈ C̄h(X_{1}^{2}) differ by 1. In particular, these two numbers have different parity. However, the number of the essential basis elements contained in α is even by Lemma 4.14. By the same lemma, the number of the essential basis elements contained in pr^{2}(α_{F (X)}) is even too. □

Definition 4.16. A minimal cycle α ∈ C̄h^{≤D}(X^{2}) is called primordial, if it is not a positive order derivative of another rational cycle.

Lemma 4.17. Let α ∈ C̄h(X^{2}) be a minimal cycle. Assume that for some q ∈ S, the cycle α contains h^{𝔧_{q−1}} × l_{𝔧_{q}−1}. Then α is symmetric and primordial.

Proof. The cycle α ∩ t(α) (where t(α) is the transposition of α; intersection of cycles is defined in Lemma 4.2) is symmetric, rational (Lemma 4.2), contained in α, and, by Lemma 4.15, still contains h^{𝔧_{q−1}} × l_{𝔧_{q}−1} (in particular, α ∩ t(α) ≠ 0). It coincides with α by the minimality of α.

It is easy to “see” that α is primordial looking at the picture of Remark 4.12 (because α contains the top point of some shell triangle). Nevertheless, let us do the proof by formulae. If there exists a rational cycle β ≠ α such that α is a derivative of β, then there exists a rational cycle β′ such that α is an order one derivative of β′, that is, α = β′ · (h^{0} × h^{1}) or α = β′ · (h^{1} × h^{0}). In the first case β′ should contain the basis cycle h^{𝔧_{q−1}} × l_{𝔧_{q}}, while in the second case β′ contains h^{𝔧_{q−1}−1} × l_{𝔧_{q}−1}. However, these both cases are not possible by Lemma 4.11 (take k = 𝔦_{q} + 1 with i = 0 for the first case and i = 𝔦_{q−1} − 1 for the second case). □

It is easy to see that a cycle α with the property of Lemma 4.17 exists at least for q = 1:

Lemma 4.18. There exists a cycle in C̄h_{D+𝔦_{1}−1}(X^{2}) containing h^{0} × l_{𝔦_{1}−1}.

Proof. Take a preimage of l_{𝔦_{1}−1} ∈ C̄h(X_{F (X)}) under the surjection C̄h(X^{2}) →→ C̄h(X_{F (X)}) given by the pull-back with respect to the morphism X_{F (X)} → X^{2} produced by the generic point of the first factor of X^{2}. □

The following lemma is proved already in [11] (under the name of “Vishik’s principle”), but only for odd-dimensional quadrics and by a different as here method.

Lemma 4.19. For any cycle ρ ∈ C̄h_{D}(X^{2}), any q ∈ S, and any i ∈ [1, 𝔦_{q}], the element h^{𝔧_{q−1}+i−1} × l_{𝔧_{q−1}+i−1} is contained in ρ if and only if the element l_{𝔧_{q}−i} × h_{𝔧_{q}−i} is contained in ρ.

Proof. Clearly, it is enough to prove Lemma 4.19 for q = 1 only. Let ρ be a counterexample to Lemma 4.19. Replacing ρ by its transposition, if necessary, we can come to the situation where ρ ∋ l_{𝔦_{1}−i} × h^{𝔦_{1}−i} and ρ ∌ h^{i−1} × l_{i−1} for some i ∈ [1, 𝔦_{1}].

Let α be the cycle of Lemma 4.18. The composition β = (t(α) · (h^{i−1} × h^{0})) ∘ ρ (where t(α) is the transposition of α) is a rational homogeneous cycle containing l_{𝔦_{1}−i} × h^{0} and not containing h^{i−1} × l_{𝔦_{1}−1}. Therefore γ = t(β · (h^{𝔦_{1}−i} × h^{0})) is a rational cycle containing h^{0} × l_{0} and not containing l_{𝔦_{1}−1} × h^{𝔦_{1}−1}. It follows that the composition α ∘ γ is a rational cycle containing h^{0} × l_{𝔦_{1}−1} and not containing l_{𝔦_{1}−1} × h^{0}, a contradiction with Lemma 4.15. □

To announce the result which follow, we prefer to use the language of picture rather then the language of formulae:

Corollary 4.20. The diagram of an arbitrary α ∈ C̄h^{≤D}(X^{2}) has the following property: for any q ∈ S and any integers i ≥ 1 and k ≥ 0, the i-th point of the k-th row of the q-th left shell triangle is marked if and only if the i-th point of the k-th row of the q-th right shell triangle is marked (see Definition 4.13 for the agreement on counting the rows and the points of the shell triangles).

Proof. The case of k = 0 is treated in Lemma 4.19 (while Lemma 4.15 treats the case of “maximal” k). The case of an arbitrary k is reduced to the case of k = 0 by taking a k-th order derivative of α. □

Remark 4.21. By Corollary 4.20, it follows that the diagram of a cycle in C̄h^{≤D}(X^{2}) is determined by, say, the left half of itself.

Example 4.22. As an application of the results on X^{2} obtained by now (first of all, of Corollary 4.20), we give a short (simpler as the original) proof of the main result of [11], which can be stated as follows: if ϕ is an anisotropic quadratic form and 2^{r} is the biggest power of 2 dividing the difference dim(ϕ) − 𝔦_{1}(ϕ), then 𝔦_{1}(ϕ) ≤ 2^{r}. For the proof, assume that 𝔦_{1} = 𝔦_{1}(ϕ) > 2^{r} and consider the Steenrod operation S^{2^{r}}(α) of a homogeneous cycle α ∈ C̄h^{≤D}(X^{2}) containing h^{0} × l_{𝔦_{1}−1} (for the existence of α see Lemma 4.18; note that S^{2^{r}}(α) is still inside of C̄h^{≤D}(X^{2}) just because of the inequality 𝔦_{1} > 2^{r}). Since

S^{2^{r}}(h^{0} × l_{𝔦_{1}−1}) = h^{0} × S^{2^{r}}(l_{𝔦_{1}−1}) = \binom{dim(ϕ) − 𝔦_{1}}{2^{r}} · (h^{0} × l_{𝔦_{1}−1−2^{r}})

and the binomial coefficient is odd, we get that S^{2^{r}}(α) ∋ h^{0} ×l_{𝔦_{1}−1−2^{r}}. On the other hand, α ∌ l_{𝔦_{1}−1+i}×h^{i} for any i ∈ [1, 𝔦_{1} −1] by Lemma 4.11; consequently, S^{2^{r}}(α) ∌ l_{𝔦_{1}−1−2^{r}+i}×h^{i} for these i; in particular, this is so for i = 𝔦_{1} − 1. Now, applying Corollary 4.20, we get that S^{2^{r}}(α) ∌ h^{0} × l_{𝔦_{1}−1−2^{r}}, a contradiction.

The following lemma generalizes Lemma 4.18:

Lemma 4.23. Let q ∈ S. Assume that the group C̄h_{D}(X^{2}) contains a cycle γ such that (1) γ does not contain any h^{i} × l_{i} with i < 𝔧_{q−1};

(2) γ contains h^{i} × l_{i} for some integer i ∈ [𝔧_{q−1}, 𝔧_{q}) (note that the interval is semi-

open). Then the group C̄h_{D+𝔦_{q}−1}(X^{2}) contains a cycle α such that α ∋ h^{𝔧_{q−1}} × l_{𝔧_{q}−1} and α ∌ h^{i} × l_{i+𝔦_{q}−1} for any i < 𝔧_{q−1}.

Proof. We use an induction on q. In the case of q = 1, the assumption of Lemma 4.23 is always true (think of γ = Δ); the cycle α is constructed in Lemma 4.18. In the remaining part of the proof we assume that q > 1.

Let i be the smallest integer such that γ ∋ h^{𝔧_{q−1}+i} × l_{𝔧_{q−1}+i}. As a first step, we proof that the group C̄h^{≤D}(X^{2}) contains a cycle α′ containing h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1} and none of h^{j} × l_{?} with j < 𝔧_{q−1} + i (if i = 0 then we can take α = α′ and finish the proof).

Applying the induction hypothesis to the quadric X_{1} with the cycle pr^{2}(γ_{F (X)}) ∈ C̄h(X_{1}^{2}) (and using the inclusion homomorphism of Corollary 2.4), we get a cycle in C̄h_{D+𝔦_{q}−1}(X_{F (X)}^{2}) containing h^{𝔧_{q−1}} × l_{𝔧_{q}−1} . One of its derivatives is a homogeneous cycle in C̄h(X_{F (X)}^{2}) containing h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1} . Therefore the group C̄h(X^{3}) contains a homogeneous cycle containing h^{0} × h^{𝔧_{q−1}+i} × l_{𝔧_{q}−1} . Considering it as a correspondence of the middle factor of X^{3} into the product of two outer factors, composing it with γ, and taking the pull-back with respect to the first diagonal X^{2} → X^{3}, we get the required cycle α′.

The highest order derivative α′ · (h^{𝔦_{q}−1−i} × h^{0}) of α′ contains h^{𝔧_{q}−1} × l_{𝔧_{q}−1} . By Lemma 4.19, it also contains l_{𝔧_{q−1}} × h^{𝔧_{q−1}} . Therefore its transposition contains h^{𝔧_{q−1}} × l_{𝔧_{q−1}} . Replacing γ by the constructed rational cycle, we come to the situation with i = 0 (see the second paragraph of the proof), finishing the proof. □

We come to the main result of on the structure of C̄h^{≤D}(X^{2}) for an arbitrary anisotropic projective quadric X:

Theorem 4.24. The set of the primordial (see Definition 4.16) cycles Π ⊂ Che^{≤D}(X^{2}) has the following properties.

(1) All derivatives of all cycles of Π are minimal and pairwise different; they form a basis of Che^{≤D}(X^{2}).

(2) Every cycle in Π is symmetric.

(3) For every π ∈ Π, there exists one and only one q = f(π) ∈ S = {1, 2, …, 𝔥} such that

(a) dim(π) = D + 𝔦_{q} − 1;

(b) π ∌ h^{i} × l_{i+𝔦_{q}−1} for any i < 𝔧_{q−1};

(c) π ∋ h^{𝔧_{q−1}} × l_{𝔧_{q}−1}.

(4) The map f: Π → S thus obtained is injective, its image consists of q ∈ S such that there exists a cycle α ∈ C̄h^{≤D}(X^{2}) satisfying α ∋ h^{i} × l_{?} for some i ∈ [𝔧_{q−1}, 𝔧_{q}) and α ∌ h^{i} × l_{?} for any i ∈ [0, 𝔧_{q−1}) (in particular, f(Π) ∋ 1).

Proof. We construct a chain of subsets

∅ = Π_{0} ⊂ Π_{1} ⊂ ⋯ ⊂ Π_{𝔥}

of the set Π such that for every q ∈ S, all highest derivatives of all cycles of Π_{q} are minimal and pairwise different, and their sum contains h^{i} × l_{i} for all i < 𝔧_{q}. The procedure looks as follows. If for some q ∈ S the set Π_{q−1} is already constructed, we decide whether we set Π_{q} = Π_{q−1} ∪ {π} with certain cycle π or we set Π_{q} = Π_{q−1}. To make this decision, we consider the sum α of all highest derivatives of all cycles of Π_{q−1}. We know that α contains h^{i} × l_{i} for all i ∈ [0, 𝔧_{q−1}). If α also contains h^{i} × l_{i} for all i ∈ [𝔧_{q−1}, 𝔧_{q}), then we set Π_{q} = Π_{q−1}; otherwise the cycle γ = α + Δ satisfies the hypothesis of Lemma 4.23, and we set Π_{q} = Π_{q−1} ∪ {π} with π being the minimal cycle containing h^{𝔧_{q−1}} × l_{𝔧_{q}−1} (π exists and has Property (3b) by Lemma 4.23; π is primordial by Lemma 4.17).

The set Π_{𝔥} thus constructed has all properties claimed for Π in Theorem 4.24. Indeed the elements of Π_{𝔥} are symmetric by Lemma 4.17. The sum of all highest derivatives of all elements of Π_{𝔥} contains h^{i} × l_{i} for all i; therefore this sum also contains the remaining basis elements l_{i} × h^{i} for all i (see Lemma 4.19). It follows that every D-dimensional minimal cycle is a derivative of an element of Π_{𝔥}. Consequently, every minimal cycle in C̄h^{≤D}(X^{2}) is a derivative of of a cycle of Π_{𝔥}. It follows that Π_{𝔥} = Π. All minimal cycles form a basis according to Proposition 4.5. □

As easy as important information on relations between the primordial cycles on X^{2} and on X_{1}^{2} is as follows:

Proposition 4.25. Let Π be the set of all primordial cycles for X; let Π_{1} be the set of all primordial cycles for X_{1}. As usual we set 𝔦_{1} = 𝔦_{1}(X). One has:

(1) #Π − 1 ≤ #Π_{1};

(2) if Π ∌ h^{0} × l_{𝔦_{1}−1} + l_{𝔦_{1}−1} × h^{0}, then #Π ≤ #Π_{1}.

Proof. Let us extend the function f: Π → S on the set of all non-zero cycles in Che^{≤D}(X^{2}), defining f(α) as the minimal q ∈ S such that α ∋ h^{i} × l_{?} for some i ∈ [𝔧_{q−1}, 𝔧_{q}) and α ∌ h^{i} × l_{?} for any i ∈ [0, 𝔧_{q−1}). By Item 4 of Theorem 4.24 (which is a consequence of Lemma 4.23), the image of the extended f coincides with f(Π). Let f_{1} : Che^{≤D}(X_{1}^{2}) → S_{1} be the same map for the quadric X_{1}. We denote as Π′ the set Π without the primordial cycle containing h^{0} × l_{𝔦_{1}−1} (this is the primordial cycle whose image under f is 1). For any π ∈ Π′ the cycle pr^{2}(π) ∈ Che(X_{1}^{2}) is non-zero and f_{1}(pr^{2}(π)) = f(π) − 1. It follows that #Π_{1} = #f_{1}(Π_{1}) = # Im(f_{1}) ≥ #f(Π′) = #Π′ = #Π − 1, and the first statement of Proposition 4.25 is proved.

If now Π ∌ h^{0} ×l_{𝔦_{1}−1}+l_{𝔦_{1}−1}×h^{0}, then pr^{2}(π_{F (X)}) is non-zero for every π ∈ Π. Note that for the cycle π ∈ Π containing h^{0} × l_{𝔦_{1}−1}, one has f_{1}(pr^{2}(π)) ∉ f_{1}(pr^{2}(Π′)) (because π is disjoint with all derivatives of the cycles of Π′ and, consequently, pr^{2}(π) is disjoint with all derivatives of the cycles of pr^{2}(Π′)). Therefore #Π ≤ #Π_{1}, and the second statement of Proposition 4.25 is proved as well. □

We need some more notation.

Definition 4.26. For any r ≥ 1, the symmetric group S_{r} acts on the group Ch(X̄^{r}) by permutations of factors of X̄^{r}. If α ∈ Ch(X̄^{r}), we write Sym(α) for the “symmetrization” of α, that is,

Sym(α) = ∑_{s∈S_{r}} s(α) .

Definition 4.27. A non-zero anisotropic quadratic form ϕ over F is said to be small if for some positive integer n (which is uniquely determined by dim(ϕ) by the Arason-Pfister theorem) one has ϕ ∈ I^{n} while dim ϕ < 2^{n+1}. A projective quadric is small if so is the corresponding quadratic form.

The following result is an extended version of [22, thm. 4.1].

Proposition 4.28. Let X be a small 2d-dimensional quadric of the first Witt index a = 𝔦_{1}(X). Then

(1) the integer a divides all the higher Witt indices 𝔦_{1}, …, 𝔦_{𝔥} of X; in particular, it divides d + 1 = 𝔦_{1} + ⋯ + 𝔦_{𝔥};

(2) the cycle

π = Sym ( ∑_{i=1}^{(d+1)/a} h^{(i−1)a} × l_{ia−1}) ∈ Ch_{2d+a−1}(X̄^{2})

is rational;

(3) moreover, for every k ≥ 0 the Chow group C̄h_{2d+k}(X^{2}) is generated by the nonessential basis elements and the cycles

π · (h^{j−1} × h^{a−k−j}) , j = 1, 2, …, a − k

(in particular, for k ≥ a, this Chow group consists of the non-essential elements only).

Proof. Let Π be the set of primordial cycles. It suffices to show that #Π = 1 (then the unique element π ∈ Π automatically has the form and the property required). We prove it using an induction on 𝔥 = 𝔥(X). If 𝔥 = 1, then #Π = 1, since generally 1 ≤ #Π ≤ 𝔥.

Now we assume that 𝔥 ≥ 2. Let us consider the quadric X_{1} (over the field F (X)) and let Π_{1} be the set of primordial cycles for X_{1}. Then #Π_{1} = 1 by the induction hypothesis, and we get what we need by Item 2 of Proposition 4.25, if we check that the cycle Sym(h^{0} × l_{a−1}) is not rational. By Item 8 of Proposition 3.3, this cycle can be rational only if the integer 2d − (a − 1) + 1 = 2d − a + 2 is a power of 2. Since however

2^{n} ≤ dim(ϕ_{1}) = 2d + 2 − 2a < \underline{\underline{2d − a + 2}} < 2d + 2 = dim(ϕ) ≤ 2^{n+1} ,

the integer 2d − a + 2 is not a power of 2. □

Remark 4.29. Proposition 4.28 holds also for anisotropic ϕ with [ϕ] ∈ I^{n} and dim(ϕ) = 2^{n+1} (the same proof is valid for such ϕ as well).

5. Cycles on X^{3}

Let ϕ be a small quadratic form and let n be the positive integer such that [ϕ] ∈ I^{n} while dim(ϕ) < 2^{n+1}. We recall that X stays for the projective quadric given by ϕ. Let us write down the dimension of ϕ as a sum of powers of 2:

dim(ϕ) = 2^{n} + 2^{n_{1}} + ⋯ + 2^{n_{m}} , n > n_{1} > ⋯ > n_{m} ≥ 1 .

In this section we assume that m ≥ 2, that the height of ϕ is at least 3, and that the first two higher Witt indices of ϕ are as follows: 𝔦_{1}(ϕ) = 2^{n_{m}−1} and 𝔦_{2}(ϕ) = 2^{n_{m−1}−1}. To simplify the formulae which follow, we introduce the notation

a = 𝔦_{1}(ϕ); b = 𝔦_{2}(ϕ); c = 𝔦_{3}(ϕ)

and

d = dim(X)/2 = 2^{n−1} + 2^{n_{1}−1} + ⋯ + 2^{n_{m}−1} − 1 .

Here is our main construction:

Proposition 5.1 (cf. [12, prop. 2.7]). The group C̄h(X^{3}) contains a homogeneous cycle

μ = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{0} × h^{(i−1)b+a} × l_{ib+a−1}) + μ′ ,

where μ′ is a sum of only those essential basis elements which contain neither h^{0} nor h^{i} with a |̸ i.

Proof. Let X_{1} be the projective quadric (over the field F (X)) given by the anisotropic part of the form ϕ_{F (X)}. Applying Item 2 of Proposition 4.28 to the quadric X_{1} (taking in account that 𝔦_{1}(X_{1}) = 𝔦_{2}(X) = b and dim(X_{1}) = 2(d − a)), we , in particular, get that the group C̄h(X_{1}^{2}) contains the cycle

β′ = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b} × l_{ib−1}) .

Therefore (see Corollary 2.4), the group C̄h(X_{F (X)}^{2}) contains the cycle

β = in^{2}(β′) = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1}) .

The pull-back homomorphism g_{1}^{∗}: C̄h(X^{3}) → C̄h(X_{F (X)}^{2}) with respect to the morphism g_{1} : X_{F (X)}^{2} → X^{3}, given by the generic point of the first factor of X^{3}, is surjective. Therefore, there exists a homogeneous cycle μ ∈ C̄h(X^{3}) such that g_{1}^{∗}(μ) = β. Note that g_{1}^{∗} sends every basis cycle of the type h^{0} × ζ × ξ to ζ × ξ while killing the other basis elements. Consequently we have

μ = h^{0} × Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1}) + ϵ ,

where ϵ is a sum of some basis cycles which do not have h^{0} on the first factor place.

We now proceed by transforming the cycle μ in such a way that μ does not leave the group C̄h(X^{3}) and g_{1}^{∗}(μ) remains the same.

By Proposition 4.28 (now applied to X itself), the cycle

γ = Sym ( ∑_{i=1}^{(d+1)/a} h^{(i−1)a} × l_{ia−1}) · (h^{0} × h^{a−1}) =

∑_{i=1}^{(d+1)/a}(h^{(i−1)a} × l_{(i−1)a} + l_{ia−1} × h^{ia−1})

is in C̄h(X^{2}). Considering it as a correspondence, we replace μ by the composition μ ∘ γ, where μ ∈ C̄h(X_{1} × X_{2} × X_{3}) is considered as a correspondence from X_{1} to X_{2} × X_{3} (all X_{i} are copies of X). Now a basis element h^{i}×?×? occurs in the decomposition of μ only if i is divisible by a (see Lemma 3.8).

Considering μ as a correspondence from X_{2} to X_{1} × X_{3} and replacing it by the composition μ ∘ γ, ^{9} we come to the situation where a basis element ? × h^{i}×? occurs in the decomposition of μ only if i is divisible by a (while all previously established properties of μ still hold).

Finally, considering μ as a correspondence from X_{3} to X_{1} × X_{2} and replacing it by the composition μ ∘ γ, we come to the situation where a basis element ?×? × h^{i} occurs in the decomposition of μ only if i is divisible by a (while all the previously established properties of μ still hold).

The last change we apply to μ is as follows: we remove all non-essential basis cycles in the decomposition of μ. We claim that now our cycle μ has the required shape.

Let us write μ_{0} for the sum of those summands in the decomposition of μ which have h^{0} as at least one factor. To finish the proof of the proposition, it suffices to check that

μ_{0} = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{0} × h^{(i−1)b+a} × l_{ib+a−1}) .

First of all let us check that none of the 3 basis cycles obtained from h^{0} × h^{0} × l_{b−1} by a permutation of factors appears in the decomposition of μ_{0}. We assume that the cycle h^{0} × h^{0} × l_{b−1} does appear and we pull-back μ with respect to the morphism g_{12} : X_{F (X×X)} → X^{3} given by the generic point of the product of the first two factors of X^{3}. We get

C̄h(X_{F (X×X)}) ∋ g_{12}^{∗}(μ) = g_{12}^{∗}(h^{0} × h^{0} × l_{b−1}) = l_{b−1}

showing that the Witt index of the quadric X_{F (X×X)} is at least b (see Corollary 2.5). However this Witt index coincides with 𝔦_{1}(X) = a and a is smaller than b (actually a ≤ b/2). The contradiction obtained shows that the cycle h^{0} × h^{0} × l_{b−1} does not appear in the decomposition of μ_{0}. For every permutation of h^{0} ×h^{0} ×l_{b−1} we get the same result simply by changing in the appropriate way the numeration of the factors.

It follows that μ_{0} = μ_{1}+μ_{2}+μ_{3}, where μ_{i} is the sum of summands in the decomposition of μ such that h^{0} is their i-th factor. By the construction of μ we known that

μ_{1} = h^{0} × Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1}) ,

9Strictly speaking, this is t_{12}(t_{12}(μ) ∘ γ), where t_{12} is the automorphism of C̄h(X^{3}) induced by the transposition of the first two factors of X^{3}.

and it suffices to check that μ_{2} = t_{12}(μ_{1}) and μ_{3} = t_{13}(μ_{1}) with t_{1i} staying for the automorphism of the Chow group Ch(X̄^{3}) given by the transposition of the first and i-th factor of X̄^{3}.

In order to see that μ_{2} = t_{12}(μ_{1}), we pull-back the cycle μ to X^{2} with respect to the morphism

δ_{1} : X^{2} → X^{3} , x_{1} × x_{2} ↦ x_{1} × x_{1} × x_{2}

given by the diagonal map of the first factor of X^{2} into the product of the first two factors of X^{3}. The decomposition of the homogeneous cycle δ_{1}^{∗}(μ) ∈ C̄h_{2d+b−1}(X^{2}) does not contain any non-essential cycle. Therefore, since b > a, δ_{1}^{∗}(μ) = 0 by Proposition 4.28. On the other hand, δ_{1}^{∗}(μ_{1}) contains h^{a} × l_{b+a−1} while neither δ_{1}^{∗}(μ_{3}) nor δ_{1}^{∗}(μ − μ_{0}) do. It follows that δ_{1}^{∗}(μ_{2}) contains h^{a} × l_{b+a−1} as well and consequently μ_{2} contains the basis cycle h^{a} × h^{0} × l_{b+a−1}. Now we use the pull-back with respect to the morphism g_{2} : X_{F (X)}^{2} → X^{3} given by the generic point of the second factor of X^{3}. The homogeneous cycle g_{2}^{∗}(μ) = g_{2}^{∗}(μ_{2}) lies in C̄h(X_{F (X)}^{2}), contains the basis cycle h^{a} × l_{b+a−1}, and does not contain any non-essential basis element. Passing to the anisotropic part X_{1} of X_{F (X)} and using Corollary 2.4, we get a homogeneous cycle η in C̄h(X_{1}^{2}), namely η = pr^{2}(g_{2}^{∗}(μ)), which contains h^{0} × l_{b−1} and does not contain any non-essential cycle. Note that g_{2}^{∗}(μ_{2}) is in the image of in^{2} : Ch(X̄_{1}^{2}) → Ch(X̄_{F (X)}^{2}), so that μ_{2} can be reconstructed from η.

By Proposition 4.28 it follows that

η = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b} × l_{ib−1}) .

Consequently

g_{2}^{∗}(μ_{2}) = in^{2}(η) = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1})

and μ_{2} = t_{12}(μ_{1}).

The equality μ_{3} = t_{13}(μ_{1}) is checked similarly. □

We remark that the “defect part” μ′ of the cycle μ does not appear in [12, prop. 2.7] when working with a small quadric of height 2. In our case here, the height of X is at least 3, μ′ does really exist and represents an additional difficulty. The main observation which is crucial to overcome this difficulty is as follows:

Lemma 5.2. Let μ′ be as in Proposition 5.1. In the decomposition of μ′ we consider the basis elements with h^{a} on the i-th factor place and write μ′_{i} for their sum. Then each of the cycles μ′_{1}, t_{12}(μ′_{2}), and t_{13}(μ′_{3}) is the sum of \underline{some} of the following (c − b)/a elements

χ_{j} = h^{a} × (Sym ( ∑_{i=1}^{(d−b−a+1)/c} h^{(i−1)c+b+a} × l_{ic+b+a−1}) · (h^{(j−1)a} × h^{c−b−ja}))

where j ∈ {1, 2, …, (c − b)/a}.

Proof. Clearly, it suffices to prove the statement on μ′_{1} (the statements on μ′_{2} and μ′_{3} are proved in the same way interchanging the roles of the three factors of X^{3}).

Let us go over the function field F (X). We still have μ ∈ C̄h(X_{F (X)}^{3}). Therefore pr_{X}^{3}(μ) ∈ C̄h(X_{1}^{3}), where X_{1} is the anisotropic part of X_{F (X)} and pr_{X}^{3} : Ch(X̄_{F (X)}^{3}) → Ch(X̄_{1}^{3}) is the projection of Corollary 2.4. We note that pr_{X}^{3}(μ) = pr_{X}^{3}(μ′). Moreover, μ′ is in the image of the inclusion in_{X}^{3} : Ch(X̄_{1}^{3}) → Ch(X̄_{F (X)}^{3}) because every h^{i} which is a factor of a basis element in the decomposition of μ′ has i ≥ a and every l_{i} which is a factor of a basis element in the decomposition of μ′ has i ≥ a as well (just look at the dimension of μ′). Therefore μ′ can be reconstructed from its image under pr_{X}^{3}, namely, μ′ = in_{X}^{3}(pr_{X}^{3}(μ′)).

Now we move from C̄h(X_{1}^{3}) to C̄h ((X_{1})_{F (X)(X_{1})}^{2}) using g_{1}^{∗} (the pull-back with respect to the morphism given by the generic point of the first factor of X_{1}^{3}). Note that g_{1}^{∗}(pr_{X}^{3}(μ′)) = g_{1}^{∗}(pr_{X}^{3}(μ′_{1})) and the cycle pr_{X}^{3}(μ′_{1}) can be reconstructed from its image under g_{1}^{∗}. Moreover the cycle g_{1}^{∗}(pr_{X}^{3}(μ′_{1})) is in the image of the inclusion in_{X_{1}}^{2} : Ch(X̄_{2}^{2}) → Ch ((X̄_{1})_{F (X_{1})}^{2}), where X_{2} is the anisotropic part of (X_{1})_{F (X)(X_{1})}. In order to see it, we note that every basis cycle in the decomposition of g_{1}^{∗}(pr_{X}^{3}(μ′_{1})) is equal to h^{(i−1)a} ×l_{b+ia−1} for some i ≥ 1. Clearly, such a basis cycle is in the image of in_{X_{1}}^{2} if and only if (i−1)a ≥ b. So, if the cycle g_{1}^{∗}(pr_{X}^{3}(μ′_{1})) is not in the image of in_{X_{1}}^{2}, then its decomposition contains the basis element h^{(i−1)a} × l_{b+ia−1} with some i such that (i − 1)a < b. It follows that the decomposition of the rational cycle

pr_{2d−2a−(i−1)a,a}(g_{1}^{∗}(pr_{X}^{3}(μ′_{1}))) ∈ C̄h(X_{2})

contains l_{ia−1}. This is a contradiction because X_{2} is anisotropic and therefore the group C̄h(X_{2}) does not contains essential elements (Corollary 2.2).

So, one can reconstruct the cycle g_{1}^{∗}(pr_{X}^{3}(μ′_{1})) from its image under the projection pr_{X_{1}}^{2} : C̄h ((X_{1})_{F (X)(X_{1})}^{2}) → C̄h(X_{2}^{2}) and for our purposes it is sufficient to determine this image. To do so, we apply Proposition 4.28 to the quadric X_{2} getting that the cycle

(pr_{X_{1}}^{2} ∘g_{1}^{∗} ∘ pr_{X}^{3})(μ′_{1}) ∈ C̄h_{2d−b−a−1}(X_{2}^{2})

is the sum of some essential generators of the group C̄h_{2d−b−a−1}(X_{2}^{2}) indicated in Item 3 of Proposition 4.28 (we note that dim(X_{2}) = 2(d − b − a) so that 2d − b − a − 1 = dim(X_{2}) + b + a − 1 ≤ dim(X_{2}) + c − 1). Finally, taking in account that h^{i} for a given i can be a factor of a basis element appearing in the decomposition of (pr_{X_{1}}^{2} ∘g_{1}^{∗} ∘ pr_{X}^{3})(μ′_{1}) only if i is divisible by a, we get the desired description of the cycle μ′_{1}. □

6. Proof of Conjecture 1.1

In this § we prove Conjecture 1.1. Suppose that this conjecture is not true, that is, over some field F and for some positive integer n, there exists a quadratic form ϕ over F with [ϕ] ∈ I^{n} and with dim(ϕ) prohibited by Conjecture 1.1. Note that n is at least 4 (see §1). In the splitting pattern of the form ϕ, let us choose the smallest number dim(ϕ_{E})_{0} prohibited by Conjecture 1.1. Let us replace the form ϕ by this (ϕ_{E})_{0} (and F by E) and write X for the projective quadric given by the new ϕ. Note that dim(ϕ) > 2^{n} + 2^{n−1} ([23] (the original proof), or [14, thm. 4.4], or [12]). Moreover,

dim(ϕ_{F (X)})_{0} = 2^{n} + 2^{n−1} + ⋯ + 2^{m} = 2^{n+1} − 2^{m}

for some m with 3 ≤ m ≤ n − 1 and dim(ϕ) = 2^{n} + 2^{n−1} + ⋯ + 2^{m} + 2𝔦_{1} where 𝔦_{1} = 𝔦_{1}(ϕ) is the first Witt index of ϕ. Note that 𝔦_{1} < 2^{m−1} simply because dim(ϕ) < 2^{n+1}. Now it follows by [11, conject. 0.1] or by Item 1 of Proposition 4.28 (take in account that the highest Witt index of ϕ is 2^{n−1}) that 𝔦_{1} = 2^{p−1} for some integer p satisfying 1 ≤ p ≤ m−1, and dim(ϕ) = 2^{n} + 2^{n−1} + ⋯ + 2^{m} + 2^{p}. Since ϕ is a counter-example, p is not m − 1, so that 1 ≤ p ≤ m − 2 in fact.

Finally, [11, conject. 0.1] (or Item 1 of Proposition 4.28) and the fact that all dimensions dim(ϕ_{E})_{0} < dim(ϕ) are allowed by Conjecture 1.1, allows one to determine all further higher Witt indices of ϕ (compare with the proof of Corollary 6.2). They are as follows (starting from 𝔦_{2}): 2^{m−1}, 2^{m−2}, …, 2^{n−1}, meaning that the splitting pattern of ϕ consists of the partial sums of the sum 2^{n} + 2^{n−1} + ⋯ + 2^{m} + 2^{p}. Therefore the hypothesis of the preceding § (and, in particular, the hypothesis of Proposition 5.1 and Lemma 5.2) are satisfied.

To simplify the formulae which follow, like we did in the previous §, we introduce the notation

a = 𝔦_{1}(ϕ) = 2^{p−1}; b = 𝔦_{2}(ϕ) = 2^{m−1}; c = 𝔦_{3}(ϕ) = 2^{m}

and

d = dim(X)/2 = 2^{n−1} + 2^{n−2} + ⋯ + 2^{m−1} + 2^{p−1} − 1 .

Let us consider the cycle μ ∈ C̄h(X^{3}) of Proposition 5.1 as a correspondence from X̄ to X̄^{2}; let us consider the cycle S^{2a}(μ) · (h^{0} × h^{0} × h^{b−1}) ∈ C̄h(X^{3}) as a correspondence from X̄^{2} to X̄ (where S^{i} stays for the degree i component of the total Steenrod operation S). Then we may take the composition of correspondences

μ ∘ (S^{2a}(μ) · (h^{0} × h^{0} × h^{b−1})) ∈ C̄h(X^{4}) .

Let us additionally consider the morphism

δ: X^{2} = X_{1} × X_{2} → X_{1} × X_{2} × X_{3} × X_{4} = X^{4} , x_{1} × x_{2} ↦ x_{1} × x_{2} × x_{1} × x_{2}

(all X_{i} are copies of X) given by the product of the diagonals X_{1} → X_{1} × X_{3} and X_{2} → X_{2} × X_{4} (that is, δ is the diagonal morphism of X^{2}).

The following proposition contradicts to Proposition 4.28 (note that dim(ξ) = 2d + b − 2a − 1 ≥ 2d + a) and proves therefore Conjecture 1.1.

Proposition 6.1. Let μ ∈ C̄h(X^{3}) be the cycle of Proposition 5.1. Then the decomposition of the cycle

ξ = δ^{∗}(μ ∘ (S^{2a}(μ) · (h^{0} × h^{0} × h^{b−1}))) ∈ C̄h_{2d+b−2a−1}(X^{2})

contains the basis cycle h^{a} × l_{b−a−1}.

Proof. It is easy to see that each power of h which is a factor of a basis element involved in the decomposition of the cycle μ ∘ (S^{2a}(μ) · (h^{0} × h^{0} × h^{b−1})) is a multiple of a. Therefore the same is true for the cycle ξ.

As before, we set μ_{0} = μ − μ′. We have:

ξ = δ^{∗}(μ_{0} ∘ (S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1})))+

δ^{∗}(μ′ ∘ (S^{2a}(μ′) · (h^{0} × h^{0} × h^{b−1})))+
δ^{∗}(μ′ ∘ (S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1})))+
δ^{∗}(μ_{0} ∘ (S^{2a}(μ′) · (h^{0} × h^{0} × h^{b−1}))) ,

and we consider each of these four summands separately, one by one.

First summand. First of all we compute S^{2a}(μ_{0}). For every summand h^{0} × h^{(i−1)b+a} × l_{ib+a−1} in the decomposition of μ_{0}, and for any r ≠ a with 1 ≤ r ≤ 2a, we have: S^{r}(h^{(i−1)b+a}) = S^{r}(l_{ib+a−1}) = 0 (for r = 2a the relation 4a|b, that is, b ≠ 2a, is used) while S^{a}(h^{(i−1)b+a}) = h^{(i−1)b+2a} and S^{a}(l_{ib+a−1}) = l_{ib−1}. Therefore

S^{2a}(h^{0} × h^{(i−1)b+a} × l_{ib+a−1}) = h^{0} × h^{(i−1)b+2a} × l_{ib−1}

and

S^{2a}(μ_{0}) = Sym ( ∑_{i=1}^{(d−a+1)/b} h^{0} × h^{(i−1)b+2a} × l_{ib−1}) .

Now we can calculate the composition

μ_{0} ∘ (S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1}))

(see Lemma 3.8). The basis cycles which appear in the decomposition of S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1}) have on the third factor place the following elements:

(∗)  h^{b−1}, h^{ib+2a−1}, l_{(i−1)b} .

On the other hand, the basis elements which appear in the decomposition of μ_{0} itself have on the first factor place the following:

(∗∗)  h^{0}, h^{(i−1)b+a}, l_{ib+a−1} .

It is straight-forward to see that the only pair of elements, one from (∗), one from (∗∗), with the product l_{0} is (l_{0}, h^{0}) (look at the indices modulo 2a). Therefore

μ_{0} ∘ (S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1})) =

(h^{0} × Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1})) ∘ ( Sym(h^{0} × h^{2a}) × l_{0}) =

Sym(h^{0} × h^{2a}) × Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1}) .

Applying δ^{∗} to the cycle obtained, we get

Sym ( ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib−a−1} + h^{(i−1)b+3a} × l_{ib+a−1}) = h^{a} × l_{b−a−1} + ….

It remains to show that the “remaining part” of ξ does not contain the basis cycle h^{a} × l_{b−a−1}.

Second summand. A basis cycle of the shape h^{x}×? × h^{y}×? can be involved in the decomposition of

μ′ ∘ (S^{2a}(μ′) · (h^{0} × h^{0} × h^{b−1}))

only if x, y ≥ a. In this case δ^{∗}(h^{x}×? × h^{y}×?) = h^{x+y}×? with x + y > a, therefore the cycle

δ^{∗}(μ′ ∘ (S^{2a}(μ′) · (h^{0} × h^{0} × h^{b−1})))

does not contain the basis element h^{a} × l_{b−a−1}.

Third summand. In order to check that the cycle

δ^{∗}(μ′ ∘ (S^{2a}(μ_{0}) · (h^{0} × h^{0} × h^{b−1})))

does not contain the basis element h^{a} × l_{b−a−1}, it suffices to check that the number of basis elements in the decomposition of the composition

μ′_{2} ∘ (S^{2a}(μ_{1}) · (h^{0} × h^{0} × h^{b−1}))

is even (note that we replaced μ′ by μ′_{2}, the notation being introduced in Lemma 5.2, and we replaced μ_{0} by μ_{1}, the notation μ_{1} being introduced in the proof of Proposition 5.1 for the sum of the basis elements contained in the decomposition of μ having h^{0} on the first factor place). For this, due to Lemma 5.2, it suffices to check that every of the (c − b)/a compositions

t_{12}(χ_{j}) ∘ (S^{2a}(μ_{1}) · (h^{0} × h^{0} × h^{b−1})) , j ∈ [1, (c − b)/a]

contains an even number of basis elements. We show this by a straight-forward computation. The point is that the number of summands in the decomposition of every χ_{j} is even and either each or none of the summands “produces” a basis element in the composition (moreover, in the first case, precisely one basis element is produced by each summand of the cycle χ_{j}).

Let us do the computation. The cycles S^{2a}(μ_{1}) · (h^{0} × h^{0} × h^{b−1}) and χ_{j} (for a fixed j) are equal respectively to h^{0} × α and to h^{a} × β, where

α = ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+2a} × l_{(i−1)b} + l_{ib−1} × h^{ib+2a−1} ,

while

β = ∑_{i=1}^{(d−b−a+1)/c} h^{(i−1)c+b+ja} × l_{(i−1)c+2b+(j+1)a−1} + l_{ic+b−(j−2)a−1} × h^{ic−(j−1)a} ,

and we just need to check that the composition β ∘ α is a sum of an even number of basis cycles. There are two different cases depending on the value of j. If the product ja is not 0 modulo b, then every product of every second factor of the basis cycles appearing in the decomposition of α (namely, l_{(i−1)b} and h^{ib+2a−1}) by every first factor of the basis cycles appearing in the decomposition of β (namely, h^{(i−1)c+b+ja} and l_{ic+b−(j−2)a−1}) is different from l_{0} (to see this, look at the indices modulo b). Therefore, the composition of every basis cycle appearing in the decomposition of α with every basis cycles appearing in the decomposition of β is 0, and so, β ∘ α = 0 in this case.

In the contrary case — the case with ja ≡ 0 (mod b) — for every basis cycle y in the decomposition of β there is precisely one basis cycle x in the decomposition of α such that y ∘ x ≠ 0 (note that y ∘ x is a basis cycle in this case). Since the number of basis cycles in the decomposition of β is even (equal to the integer (d − b − a + 1)/c doubled), the composition β ∘ α is the sum of an even number of basis cycles.

Fourth summand. We finish the proof of Proposition 6.1 considering the cycle

δ^{∗}(μ_{0} ∘ (S^{2a}(μ′) · (h^{0} × h^{0} × h^{b−1}))) .

We replace μ′ by μ′_{1} and, furthermore, μ′_{1} by χ_{j} with some j ∈ [1, (c − b)/a]. Also, we replace μ_{0} by μ_{2}. We are going to show that the number of the basis elements of the shape h^{a}×? × h^{0}×? appearing in the decomposition of the composition

μ_{2} ∘ (S^{2a}(χ_{j}) · (h^{0} × h^{0} × h^{b−1}))

is even (this will finish the proof of Proposition 6.1). We have χ_{j} = h^{a} × β and μ_{2} = t_{12}(h^{0} × α) with

α = ∑_{i=1}^{(d−a+1)/b} h^{(i−1)b+a} × l_{ib+a−1} + l_{ib+a−1} × h^{(i−1)b+a} ,

β = ∑_{i=1}^{(d−b−a+1)/c} h^{(i−1)c+b+ja} × l_{(i−1)c+2b+(j+1)a−1} + l_{ic+b−(j−2)a−1} × h^{ic−(j−1)a} .

Therefore the number we are looking for is the number of summands in the decomposition of α ∘ (S^{2a}(β) · (h^{0} × h^{b−1})).

We can compute the Steenrod operation S^{2a} on the summands of the decomposition of β. The formula depends on the value of j modulo 4 because of the rules (here S^{≤2a} stays for ∑_{k≤2a} S^{k}):

S^{≤2a}(h^{ia}) = \begin{cases}h^{ia} & if i ≡ 0 (mod 4); \\ h^{ia} + h^{(i+1)a} & if i ≡ 1 (mod 4); \\ h^{ia} + h^{(i+2)a} & if i ≡ 2 (mod 4); \\ h^{ia} + h^{(i+1)a} + h^{(i+2)a} & if i ≡ 3 (mod 4), \end{cases}

while (here recall that S(l_{i}) = l_{i} · (1 + h)^{2d−i+1}, d + 1 ≡ a (mod b), and 4a divides b):

S^{≤2a}(l_{ia−1}) = \begin{cases}l_{ia−1} + l_{(i−2)a−1} & if i ≡ 0 (mod 4); \\ l_{ia−1} + l_{(i−1)a−1} & if i ≡ 1 (mod 4); \\ l_{ia−1} & if i ≡ 2 (mod 4); \\ l_{ia−1} + l_{(i−1)a−1} + l_{(i−2)a−1} & if i ≡ 3 (mod 4). \end{cases}

Assume that j ≡ 0 (mod 4) or j ≡ 1 (mod 4). Then, applying the above formulae, we get that S^{2a}(β) = 0, and there is nothing more to prove in this case.

Now we assume that j ≡ 2 (mod 4) or j ≡ 3 (mod 4). Then

S^{2a}(β) · (h^{0} × h^{b−1}) = β_{1} + β_{2} ,

where

β_{1} = β · (h^{2a} × h^{b−1}) =

∑_{i=1}^{(d−b−a+1)/c} h^{(i−1)c+b+(j+2)a} × l_{(i−1)c+b+(j+1)a} + l_{ic+b−ja−1} × h^{ic+b−(j−1)a−1} ,

while

β_{2} = β · (h^{0} × h^{b+2a−1}) =

∑_{i=1}^{(d−b−a+1)/c} h^{(i−1)c+b+ja} × l_{(i−1)c+b+(j−1)a} + l_{ic+b−(j−2)a−1} × h^{ic+b−(j−3)a−1} .

If j ≡ 3 (mod 4), then the compositions α ∘ β_{1} and α ∘ β_{2} are 0 because so are the compositions of any basis cycles included in α with any basis cycle included in β_{1} or β_{2} (look at the indices modulo 2a). If j ≡ 2 (mod 4), then α ∘ β_{1} = 0 too and by the same reason (look at the indices modulo 4a). Finally, assume that j ≡ 2 (mod 4) and consider the composition α∘β_{2}. For every basis cycle y in the decomposition of β_{2} there is precisely one basis cycle x in the decomposition of α such that x ∘ y ≠ 0 (note that x ∘ y is a basis cycle in this case). Since the number of basis cycles in the decomposition of β_{2} is even (equal to the integer (d − b − a + 1)/c doubled), the composition α ∘ β_{2} is the sum of an even number of basis cycles. □

Conjecture 1.1 is proved. The following supplement is now easy to get:

Corollary 6.2. Let ϕ be a small quadratic form with dim(ϕ) = 2^{n+1} −2^{m}, m ∈ [1, n+1]. Then the splitting pattern {dim(ϕ_{E})_{0}| E/F is a field extension} of the form ϕ coincides with the set {2^{n+1} − 2^{i}}_{i=m}^{n+1} (in particular, the height of ϕ is equal to n + 1 − m).

Proof. By Conjecture 1.1 proved above, dim(ϕ_{F (X)})_{0} = 2^{n+1}−2^{r} for some r ∈ [m+1, n+1]. But by [11, conject. 0.1] (or by Item 1 of Proposition 4.28 taking in account that the highest Witt index of ϕ is 2^{n−1}), it follows that only the value r = m + 1 is possible. Proceeding this way (with the form (ϕ_{F (X)})_{0} and so on), we get the result. □

7. Possible dimensions

Let us recall some standard notation concerning quadratic forms: one writes ⟨a_{1}, …, a_{n}⟩, where a_{1}, …, a_{n} ∈ F, for the quadratic form

F^{n} → F , (x_{1}, …, x_{n}) ↦ a_{1}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2} ;

⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ for the n-fold Pfister form

⟨1, −a_{1}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩ ,

and ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩′ for the pure subform of the above Pfister form (see [20, def. 1.1 of ch. 4]).

The following result provides, in particular, examples for all dimensions which are not prohibited by Conjecture 1.1.

Theorem 7.1 (A. Vishik). Take any integers n ≥ 1 and m ≥ 2. Let k be a field (of char(k) ≠ 2), t_{i}, t_{ij} (i = 1, …, m, j = 1, …, n) variables, and

F = k(t_{i}, t_{ij})_{1≤i≤m, 1≤j≤n}

the field of rational functions in all these variables.

The splitting pattern of the quadratic form

ϕ = t_{1} · ⟨ ; ⟨t_{11}, …, t_{1n}⟩ ; ⟩ ⊥ … ⊥ t_{m} · ⟨ ; ⟨t_{m1}, …, t_{mn}⟩ ; ⟩

over F is

{2^{n+1} − 2^{i}| i = n + 1, n, …, 1} ∪ (2ℤ ∩ [2^{n+1}, m · 2^{n}]) .

Proof. First of all, it is easy to see that all the integers 2^{n+1}−2^{i} are in the splitting pattern of ϕ. Indeed, the anisotropic part of ϕ over the field E obtained from F by adjoining the square roots of t_{31}, t_{41} …, t_{m1}, of t_{1} and of −t_{2}, is isomorphic to the (generalized Albert) form

⟨ ; ⟨t_{11}, …, t_{1n}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨t_{21}, …, t_{2n}⟩ ; ⟩′

(the primes ′ mean the pure subforms of the Pfister forms) of dimension 2^{n+1} − 2; the splitting pattern of this form is {2^{n+1} − 2^{i}} because this set is the splitting pattern of any anisotropic (2^{n+1} − 2)-dimensional quadratic form whose class lies in I^{n} (Corollary 6.2).

Now let us assume that some (at least one) even integers of the interval [2^{n+1}, m · 2^{n}] are not in the splitting pattern of ϕ. Among all such integers we take the smallest one and call it a; let b be the biggest integer smaller that a and lying in the splitting pattern; let c be the smallest integer greater that a and lying in the splitting pattern. Let E be the field of the generic splitting tower of ϕ such that dim(ψ) = c for ψ = (ϕ_{E})_{0}. Let Y be the projective quadric given by the quadratic form ψ. Let π ∈ C̄h(Y^{2}) be the cycle of the set Π of Theorem 4.24 with f(π) = 1. We claim that π = Sym(h^{0} × l_{𝔦_{1}−1}) for 𝔦_{1} = 𝔦_{1}(Y). Indeed, since 𝔦_{1} = (c − b)/2 > 1 and 𝔦_{q}(Y) = 1 for all q ∈ S = S(Y) = {1, 2, …, 𝔥} (𝔥 is the height of ψ) such that dim(ψ_{q}) ∈ [2^{n+1} − 2, b], the cycle π does not contain h^{𝔦_{1}+···+𝔦_{q−1}} × l_{𝔦_{1}+···+𝔦_{q−1}+𝔦_{1}−1} for such q (Lemma 4.11), and also π ∌ h^{i} × l_{i+𝔦_{1}−1} for any i ∈ {1, 2, …, 𝔦_{1} − 1} (Lemma 4.11 as well). Finally, for the integer q ∈ S such that dim(ψ_{q}) = 2^{n+1} − 2, the cycle pr^{2}(π_{E_{q}}) ∈ C̄h(Y_{q}^{2}) (the homomorphism pr^{2} : C̄h(Y_{E_{q}}^{2}) → C̄h(Y_{q}^{2}) is defined in Corollary 2.4) has the dimension

dim(Y_{q}) + 𝔦_{1} − 1 ≥ dim(Y_{q}) + 1 = dim(Y_{q}) + 𝔦_{1}(Y_{q})

and therefore is 0 by Item 3 of Proposition 4.28.

We have shown that π = Sym(h^{0} × l_{𝔦_{1}−1}). By Item 8 of Proposition 3.3, it follows that the integer dim(Y) − 𝔦_{1} + 1 is a power of 2, say 2^{p}. Since

dim(Y) − 𝔦_{1} + 1 = (c − 2) − (c − b)/2 + 1 = (b + c)/2 − 1 ,

the integer 2^{p} sits inside of the open interval (b, c); therefore, satisfying 2^{n+1} ≤ 2^{p} < m·2^{n}, the integer 2^{p} is not in the splitting pattern of the quadratic form ϕ. But all the integers ≤ m · 2^{n} divisible by 2^{n} are evidently in the splitting pattern of ϕ. The contradiction obtained proves Theorem 7.1. □

Remark 7.2. Of course, the dimensions 2^{n+1} − 2^{i} can be realized more directly by the tensor products of Pfister forms and generalized Albert forms (u_{.}, v_{.}, w_{.} are variables):

⟨ ; ⟨u_{1}, … u_{i−1}⟩ ; ⟩ ⊗ ( ⟨ ; ⟨v_{1}, …, v_{n+1−i}⟩ ; ⟩′ ⊥ − ⟨ ; ⟨w_{1}, …, w_{n+1−i}⟩ ; ⟩′ ) .

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Laboratoire des Mathématiques, Faculté des Sciences Jean Perrin, Université d’Artois, rue Jean Souvraz SP 18, 62307 Lens Cedex, France

E-mail address: karpenko@euler.univ-artois.fr