AROUND 16-DIMENSIONAL QUADRATIC FORMS IN I_{q}^{3}

NIKITA A. KARPENKO

Abstract. We determine the indexes of all orthogonal Grassmannians of a generic 16dimensional quadratic form in I_{q}^{3}. This is applied to show that the 3-Pfister number of the form is ≥ 4. Other consequences are: a new and characteristic-free proof of a recent result by Chernousov–Merkurjev on proper subforms in I_{q}^{2} (originally available in characteristic 0) as well as a new and characteristic-free proof of an old result by Hoffmann-Tignol and Izhboldin-Karpenko on 14-dimensional quadratic forms in I_{q}^{3} (originally available in characteristic ≠ 2). We also suggest an extension of the method, based on investigation of the topological filtration on the Grothendieck ring of a maximal orthogonal Grassmanian, which applies to quadratic forms of dimension higher than 16.

We work with non-degenerate quadratic forms over arbitrary fields. Recall that a quadratic form, similar to a Pfister form, is called a general Pfister form. We refer to [5] for general facts and terminology related to quadratic forms, especially for definition of a (quadratic) Pfister form in arbitrary characteristic. We write I_{q} = I_{q}(F) for the Witt group of classes of even-dimensional quadratic forms over a field F. Recall that I_{q}(F) is a module over the Witt ring W (F) of classes of non-degenerate symmetric bilinear forms. There is a filtration by submodules I_{q} = I_{q}^{1} ⊃ I_{q}^{2} ⊃ … defined as follows: for any d ≥ 1, I_{q}^{d} := I^{d−1}(F) · I_{q}(F), where I(F) ⊂ W (F) is the fundamental ideal and I^{d−1}(F) is its power.

Let φ be an even-dimensional non-degenerate quadratic form over a field F and let d ≥ 1 be an integer such that the Witt class [φ] ∈ I_{q}(F) is in I_{q}^{d}(F). Then [φ] can be written as a sum of classes of general d-fold Pfister forms. The minimal possible number of the summands is denoted Pf_{d}(φ) and called the d-Pfister number of φ, cf. [14, §9c].

Given a base field k and a positive even integer m, we are interested to determine

Pf_{d}(m) := sup_{φ} Pf_{d}(φ),

where φ runs over m-dimensional quadratic forms defined over some field F ⊃ k and satisfying [φ] ∈ I_{q}^{d}(F).

Trivially, Pf_{1}(m) = m/2 for any m. Also, it is known (and relatively easy to show) that Pf_{2}(m) = (m − 2)/2. In the present paper, we concentrate on the 3-Pfister number Pf_{3}(m) which is known to be finite. Finiteness of Pf_{d}(m) for d ≥ 4 is an open question.

Classical results from the theory of quadratic forms provide us with values of Pf_{3}(m) for m up to 12: Pf_{3}(m) is equal to 0 for m < 8, to 1 for m ∈ {8, 10}, and to 2 for m = 12.

Date: 29 Novermber 2015.

Key words and phrases. Quadratic forms over fields; algebraic groups; projective homogeneous varieties; Chow groups. Mathematical Subject Classification (2010): 11E04; 20G15; 14C25.

This work has been accomplished during author’s stay at the Universität Duisburg-Essen; it has been supported by a Discovery Grant from the National Science and Engineering Board of Canada.

For k with char k ≠ 2, the value Pf_{3}(14) = 3 has been determined independently in [6] and in [7]. More precisely, the lower bound Pf_{3}(14) ≥ 3 has been established in the papers. The upper bound Pf_{3}(14) ≤ 3 is a consequence of the classification of 14-dimensional quadratic forms in I_{q}^{3} obtained by M. Rost in [18], see [6, Proposition 2.3].

The main result of the present paper is Theorem 0.1 providing a lower bound for Pf_{3}(16). (It is worthy to mention that no upper bound for Pf_{3}(16) is available.) The lower bound for Pf_{3}(14) mentioned above, can also be obtained (this time in arbitrary characteristic) by the same method, see Corollary 3.2. Unlike with the previous approaches, this new method applies directly to the generic quadratic form of given dimension in I_{q}^{3} which a priori is expected to have the highest 3-Pfister number.

In view of recent [13, Conjecture 4.5] that the classifying spaces of spinor groups Spin_{16} and Spin_{15} are not retract rational (while the classifying space of Spin_{m} is retract rational for m ≤ 14 in characteristic ≠ 2, see [13, Theorem 4.4]), 16-dimensional forms in I_{q}^{3} and any new piece of information on them become particularly intriguing.

Theorem 0.1. For any base field k (of arbitrary characteristic) one has Pf_{3}(16) ≥ 4.

The proof is given in the beginning of §2.

Remark 0.2. A lower bound on Pf_{3}(m) for arbitrary m is obtained in [1] via essential dimension of spinor groups (in characteristic ≠ 2). Although this bound is very impressive for large m, it does not provide any non-trivial information for m = 16. This situation is not changed even if the bound on ed Spin_{16}, used in [1], is replaced by the precise value of ed Spin_{16}, obtained later in [3] (in characteristic 0), or if the precise value of the essential dimension of the functor of 16-dimensional quadratic forms in I_{q}^{3}, obtained in [3] as well (still in characteristic 0), is inserted into the computations of [1].

Actually, Theorem 0.1 is an immediate consequence of Theorem 2.1 – a stronger result which also has another application: it allows one to recover – now in characteristic-free context – a recent result of [3] on absence of proper even-dimensional subforms of trivial discriminant inside of generic 16-dimensional quadratic forms in I_{q}^{3}. Basically, Theorem 2.1 determines the indexes of all orthogonal Grassmannians of a generic 16-dimensional quadratic form in I_{q}^{3}, or, equivalently, the maximal possible indexes of orthogonal Grassmannians of arbitrary 16-dimensional quadratic form in I_{q}^{3}, see Remark 2.2. Note that the index of the maximal orthogonal Grassmannian of a generic form in I_{q}^{3} of arbitrary dimension m is the torsion index of the spinor group Spin_{m} and has been computed by B. Totaro in [19]. We show that our method does not apply directly to dimensions higher than 16 but we suggest its enhanced modification which does. This is based on the observation that the topological filtration on the Grothendieck group of the maximal orthogonal Grassmannian (as well as of any flag variety projecting to the maximal orthogonal Grassmannian) of a generic quadratic form in I_{q}^{3} coincides with the gamma filtration, see Theorem 4.3 and Remark 4.5. In general, gamma filtration provides a computable lower bound for a much less accessible topological filtration. Usually, this lower bound is very far from being sharp, see, e.g., [10], where the computations for projective quadrics are performed. One class of projective homogeneous varieties for which this bound is sharp has been previously found in [12, Theorem 3.7] (these were the Severi-Brauer varieties of, in certain sense, generic central simple algebras).

The starting point here is matching of gamma and topological filtrations in the case of a generic quadratic form in I_{q}^{2}, or, equivalently, of a generic quadratic form of odd dimension, see Proposition 4.2 (the only condition satisfied by the generic form that matters is maximality of the index of the Clifford invariant). This is achieved due to the fact that the orders of cokernels of the change of field homomorphisms to an algebraic closure for the Grothendieck group and the Chow group of the variety, turn out to be equal to each other, which seems to be a lucky coincidence. The remaining step consisting in killing the Clifford invariant generically (thus getting a quadratc form in I_{q}^{3} called generic here) is standard and similar to [12, Theorem 3.7].

1. Chow groups of maximal orthogonal Grassmannians

In this section, φ is a non-degenerate quadratic of dimension 2n + 2 (for some n ≥ 0) and of trivial discriminant over a field F. The maximal orthogonal Grassmannian of φ is a smooth projective variety consisting of two isomorphic connected components; let X be one of them. We refer to [5, Chapter XVI] for general information on maximal orthogonal Grassmannians of quadratic forms. Note that X can be identified with the maximal orthogonal Grassmannian of an arbitrary non-degenerate 2n + 1-dimensional subform of φ, [5, Propositions 85.2 and 86.17]. We switch to odd-dimensional forms in §4, but we stay with even-dimensional φ in the present section.

First we assume that the quadratic form φ is hyperbolic. By [5, §86] (see [20] for original proofs), the Chow ring CH X in this case is generated by the special Schubert classes (i.e., classes of the special Schubert varieties) e_{i} ∈ CH^{i} X with i = 1, …, n subject to the relations

(1.1)  e_{i}^{2} − 2e_{i−1}e_{i+1} + 2e_{i−2}e_{i+2} + ⋯ + 2(−1)^{i−1}e_{1}e_{2i−1} + (−1)^{i}e_{2i} = 0,

where e_{i} := 0 for i > n. Let Y be the projective quadric of φ and let f : Z → X be the projective bundle given by the tautological vector bundle on X. Note that Z is a closed subvariety of Y × X. Let g : Z → Y be the first projection. For any i = 1, …, n, the element e_{i} satisfy (and can be defined by) the formula

(1.2)  e_{i} := f_{∗}g^{∗}(l_{n−i}),

where l_{n−i} ∈ CH_{n−i} Y is the class of an n−i-dimensional linear subspace lying on Y (given by an n − i + 1-dimensional totally isotropic subspace of φ). Note that the class l_{n−i} does not depend on the choice of the linear subspace. By [5, Theorem 86.12], the additive group of the Chow ring CH X is free with a basis given by the products e_{I} := ∏_{i∈I} e_{i}, where I runs over the subsets of the set {1, …, n}. The empty product e_{∅} is the unity [X] of the ring, while e_{{1,...,n}} is the class of a rational point.

For arbitrary (i.e., not necessarily hyperbolic) φ, we fix a field extension F̄/F with hyperbolic φF̄ and write X̄ for XF̄. Let \overline{CH} X be the image of the change of field homomorphism CH X → CH X̄. This is clearly a subring in CH X̄. Moreover, 2e_{i} ∈ \overline{CH} X for all i by [5, Proposition 86.13 and Remark 86.14]. Consequently, 2^{|I|}e_{I} ∈ \overline{CH} X for any I, where |I| is the cardinality.

The even Clifford algebra C_{0}(φ) is a direct product of two copies of a central simple F -algebra of degree 2^{n} which we, following [5, Remark 13.9], denote by C^{+}(φ). We have C^{+}(φ) ≃ C_{0}(φ′) for any non-degenerate subform φ′ ⊂ φ of dimension 2n + 1. Besides, the Clifford algebra C(φ) is isomorphic to the algebra of 2 × 2-matrices over C^{+}(φ).

Proposition 1.3. If C^{+}(φ) is a division algebra, then the subgroup \overline{CH} X ⊂ CH X̄ is generated by the elements 2^{|I|}e_{I}, I ⊂ \{1, …, n\}. In particular, the subring \overline{CH} X ⊂ CH X̄ is generated by the elements 2e_{i}, i = 1, …, n.

Proof. Let us take an arbitrary element x ∈ \overline{CH} X and write it as a linear combination of the basic elements: x = ∑_{I} a_{I}e_{I} with some a_{I} ∈ ℤ. We want to show that for any I, the coefficient a_{I} is divisible by 2^{|I|}.

Assume that this is not the case. Multiplying x by an appropriate power of 2, we come to the case with min_{I}(v_{2}(a_{I})−|I|) = −1, where v_{2} is the 2-adic valuation. Let us choose a set I with v_{2}(a_{I}) − |I| = −1 and minimal |I|. Let J be the compliment of I. The product y := x · 2^{|J|}e_{J} is clearly in \overline{CH} X.

Let us consider the degree homomorphism deg : CH X̄ → ℤ. By [5, Corollary 86.10], deg(e_{I} · e_{J}) = 1. And deg(e_{I′} · e_{J}) = 0 (mod 2) for I′ ≠ I by [5, Lemma 87.6 with Propositions 85.2 and 86.17]. It follows that v_{2}(deg(y)) = n − 1. Indeed,

v_{2}(deg(a_{I}e_{I}2^{|J|}e_{J})) = v_{2}(a_{I}) + |J| = |I| − 1 + |J| = n − 1.

At the same time, for any I′ ≠ I, we have v_{2}(deg(a_{I′}e_{I′}2^{|J|}e_{J})) ≥ n because v_{2}(a_{I′}) ≥ |I′| − 1 ≥ |I| − 1 and deg(e_{I′} · e_{J}) = 0 (mod 2).

On the other hand, since the residue field of any point on X splits the algebra C^{+}(φ), the degree of any element in \overline{CH} X is divisible by ind C^{+}(φ) = 2^{n}. In particular, v_{2}(deg(y)) cannot be n − 1. □

Corollary 1.4. For φ and X as in Proposition 1.3, the index [CH X̄ : \overline{CH} X] of the subgroup \overline{CH} X in CH X̄ is [CH X̄ : \overline{CH} X] = 2^{∑_{i=0}^{n} i\binom{n}{i}} = 2^{n2^{n−1}}. □

Let now S be the Severi-Brauer variety of the central simple F -algebra C^{+}(φ). From now on, we write X̄ for X_{F̄(S)} and \overline{CH} X is the image of CH X in CH X̄. We also consider an intermediate ring \overline{CH} X ⊂ \overline{CH} X_{F (S)} ⊂ CH X̄ defined as the image of CH X_{F (S)} in CH X̄. The variety X_{F (S)} is a component of the maximal orthogonal Grassmannian of the quadratic form φ_{F (S)} whose Clifford invariant is trivial so that [φ_{F (S)}] ∈ I_{q}^{3}(F (S)), see [5, §16] for fields of characteristic 2 and [5, Chapter VIII] for field of characteristic ≠ 2.

Proposition 1.5. The \overline{CH} X-algebra \overline{CH} X_{F (S)} is generated by e_{1}.

Proof. By [7, Proposition 4.3], the projection X × S → X is a projective bundle. Therefore, by the Projective Bundle Theorem for Chow groups (see, e.g., [5, Theorem 57.14]), the CH X-algebra CH(X × S) is generated by an element of CH^{1}(X × S). The epimorphism of CH X-algebras CH(X × S) →→ CH X_{F (S)} given by pull-back with respect to the morphism of schemes X_{F (S)} → X × S induced by the generic point of S (for surjectivity of the pull-back see [5, Proposition 57.10]), shows that the CH X-algebra CH X_{F (S)} is generated by an element of CH^{1} X_{F (S)}. In particular, the \overline{CH} X-algebra \overline{CH} X_{F (S)} is generated by an element of \overline{CH}^{1} X_{F (S)}.

The group CH^{1} X̄ is generated by e_{1}. Since e_{1} ∈ \overline{CH} X_{F (S)} by [5, Exercise 88.14(1)], the group \overline{CH}^{1} X_{F (S)} is also generated by e_{1} and we are done.

Here is a solution of the part of [5, Exercise 88.14(1)] that has been used. Since the even Clifford algebra of φ_{F (S)} is split, for any field extension L/F (S) the Witt index of φ_{L} differs from n − 1. It follows by [5, Proposition 88.8] that an odd multiple of e_{1} is in the group \overline{CH} X_{F (S)}. Since 2e_{1} is there as well, we get that e_{1} is there. □

Corollary 1.6. If C^{+}(φ) is a division algebra, then the ring \overline{CH} X_{F (S)} is generated by the elements e_{1} and 2e_{i}, i = 2, …, n. □

The quadratic form φ_{F (S)} as in Corollary 1.6 will be called a generic 2n+2-dimensional quadratic form in I_{q}^{3}. This is a slight abuse of terminology convenient for our purposes. For any given field k, such a form can be constructed over an appropriate field extension of k. An example is given by a generic (in a proper sense) form in I_{q}^{3} which is obtained out of a generic form φ in I_{q}^{2} living over a purely transcendental extension F/k of sufficiently large transcendence degree, by passing to the function field of the Severi-Brauer variety of the division algebra C^{+}(φ).

2. Dimension 16

Proof of Theorem 0.1. Let φ be a generic 16-dimensional quadratic form in I_{q}^{3} and let F be its field of definition. Assume that [φ] = [π_{1}] + [π_{2}] + [π_{3}] for some general 3-Pfister forms π_{1}, π_{2}, π_{3} over F. Since non-zero Witt class of any general Pfister form vanishes over a quadratic field extension, there exists a finite field extension L/F of degree dividing 4 such that [π_{2}]_{L} = 0 = [π_{3}]_{L}. It follows that [φ_{L}] = [π_{1}]_{L} so that the Witt index i_{W} (φ_{L}) of φ_{L} is at least 4 = (dim φ − dim π)/2. This contradicts Theorem 2.1 below. □

Theorem 2.1. Let φ/F be a generic 16-dimensional quadratic form in I_{q}^{3}. Then the degree [L : F] of any finite field extension L/F with Witt index i_{W} (φ_{L}) ≥ 3 is divisible by 8.

Remark 2.2. Theorem 2.1 completes computation of maximal possible indexes of the orthogonal Grassmannians of a 16-dimensional quadratic form in I_{q}^{3}. (The answer to the similar question in dimensions ≤ 14 is known; for dimension 14 Theorem 3.1 below can be used.) By index i(X) of a variety X we mean the greatest common divisor of the degrees of closed points on X.

Namely, writing i_{n} for the maximal index of the n-th Grassmannian (the numbering is such that the 1-st Grassmannian is the quadric), Proposition 2.1 says that 8 | i_{3}. This implies that i_{3} = 8. For the remaining Grassmannians one has: i_{1} = 2 (trivial), i_{2} = 4 (a consequence of [3, Theorem 4.2] as well as of [19, Theorem 0.1]), i_{3} | i_{4} | i_{5} = i_{6} = i_{7} = i_{8}, and i_{8} = 8 (again by [19, Theorem 0.1]).

Proof of Theorem 2.1. Let X be a component of the maximal orthogonal Grassmannian of φ and let Y be the projective quadric of φ. We write X̄ for XF̄ and Ȳ for YF̄, where F̄ is an algebraic closure of F.

By Corollary 1.6, the image \overline{CH} X of CH X in CH X̄ coincides with the subring generated by e_{1}, 2e_{2}, …, 2e_{7}.

Assume that there exists a finite field extension L/F with i_{W} (φ_{L}) ≥ 3 and with [L : F] not divisible by 8. Let us fix an F -imbedding L ↪ F̄. Then the elements l_{2}, l_{1}, l_{0} ∈ CH Ȳ are in the image of CH Y_{L} and therefore, as follows from (1.2), e_{5}, e_{6}, e_{7} belong to the ring \overline{CH} X_{L} := Im(CH X_{L} → CH X̄). In particular, the product e_{5}e_{6}e_{7} is in \overline{CH} X_{L}. Applying the norm homomorphism CH X_{L} → CH X, we get that [L : F]e_{5}e_{6}e_{7} ∈ \overline{CH} X. Since 8e_{5}e_{6}e_{7} = (2e_{5})(2e_{6})(2e_{7}) ∈ \overline{CH} X, it follows that 4e_{5}e_{6}e_{7} ∈ \overline{CH} X. This contradicts Lemma 2.3 (a purely algebraic statement) below. □

Lemma 2.3. Let R be the ring defined by generators and relations: the generators are e_{1}, …, e_{7} and relations are as in (1.1) with n = 7. Then the subring of R generated by e_{1}, 2e_{2}, …, 2e_{7} does not contain 4e_{5}e_{6}e_{7}.

Proof. Since relations (1.1) are homogeneous (with deg e_{i} := i), the ring R is graded. By the shape of (1.1), the 2^{7} products of distinct generators additively generate R. Moreover, as we known, these products form a ℤ-basis of R, cf. [5, Theorem 86.12 and Proposition 86.16].

It follows that the product (e_{2}e_{4}) · 4(e_{5}e_{6}e_{7}) is not divisible by 8 in R. To show that 4e_{5}e_{6}e_{7} is not a polynomial in e_{1}, 2e_{2}, …, 2e_{7}, it suffices to show that every monomial M in e_{1}, 2e_{2}, …, 2e_{7} of degree 5 + 6 + 7 = 18, multiplied by e_{2}e_{4}, is divisible by 8 in R.

If M contains at least 3 of 2e_{2}, …, 2e_{7} as factors, then already M itself is divisible by 8.

Note that e_{2} = e_{1}^{2} and e_{4} = 2e_{1}e_{3} − e_{1}^{4} so that

(2.4)  e_{2}e_{4} = 2e_{1}^{3}e_{3} − e_{1}^{6}.

If M contains precisely 2 of 2e_{2}, …, 2e_{7} as factors, the remaining factors being copies of e_{1}, it contains 4e_{1}^{4}. Multiplying by e_{2}e_{4} and taking into account formula (2.4) for e_{2}e_{4}, we get a sum of two monomials, one of which contains 8 as a factor and the other contains 4e_{1}^{10} as a factor.

However, relations (1.1) modulo 2 show that

e_{1}^{2} = e_{2}, e_{1}^{4} = e_{2}^{2} ≡ e_{4}, and e_{1}^{8} ≡ e_{4}^{2} ≡ e_{8} = 0.

In particular, e_{1}^{8} is divisible by 2 in R. Therefore 4e_{1}^{10} is divisible by 8.

Next we consider the case where M contains precisely 1 of 2e_{2}, …, 2e_{7}. Then M contains 2e_{1}^{11}. Multiplying M by e_{2}e_{4} and taking into account formula (2.4), we get a sum of two monomials, one of which contains 4e_{1}^{14} as a factor and the other contains 2e_{1}^{17} as a factor. Since e_{1}^{14} is divisible by 2, the first monomial is divisible by 8. Since e_{1}^{17} = e_{1}^{8} ·e_{1}^{8} ·e_{1} is divisible by 4, the second monomial is also divisible by 8.

Finally, when M is a power of e_{1}, then M is a multiple of e_{1}^{18}. Multiplying M by e_{2}e_{4}, we also get divisibility by 8. □

Using Theorem 2.1, we can recover

Corollary 2.5 ([3]). Generic 16-dimensional quadratic form φ/F in I_{q}^{3} does not contain (2- or) 4- or 6-dimensional subforms of trivial discriminant. Moreover, the same property holds for φ_{E} with any finite field extension E/F of odd degree.

Proof. Assume that φ_{E} does contain a 6-dimensional subform φ′ of trivial discriminant. The form φ′_{L} is hyperbolic for some finite extension L/E of degree dividing 4. We therefore have i_{W} (φ_{L}) ≥ 3, contradicting Proposition 2.1.

If φ_{E} contains 4- (or 2)-dimensional subform of trivial discriminant, we can construct a finite field extension L/E of degree dividing 4 with hyperbolic φ_{L}. This contradicts not only Proposition 2.1, but also an earlier [19, Theorem 0.1]. □

Let us recall an open question (cf. [3, Theorem 4.2]): does any 16-dimensional quadratic form of trivial discriminant and Clifford invariant contain an 8-dimensional subform of trivial discriminant? The positive answer would provide an upper bound on Pf_{3}(16) (at least in characteristic ≠ 2):

Proposition 2.6. Let φ be a 16-dimensional quadratic form with trivial discriminant and Clifford invariant over a field F of characteristic ≠ 2. Assume that φ contains a proper even-dimensional subform φ_{1} of trivial discriminant. Then the Witt class of φ is a sum of classes of six general 3-Pfister forms.

Proof. We have φ = φ_{1}⊥φ_{2} for some φ_{2}. We may assume that dim φ_{1} ≤ dim φ_{2} so that dim φ_{1} ≤ 8. Choose a ∈ F^{×} such that the form φ′ := aφ_{1}⊥φ_{2} is isotropic. We have [φ] = [φ′] + [⟨ ; ⟨a⟩ ; ⟩] · [φ_{1}] ∈ I_{q}(F), where ⟨ ; ⟨a⟩ ; ⟩ is the diagonal form ⟨1, −a⟩. As per [6, Proposition 2.3], [φ′] is a sum of classes of three general 3-Pfister forms. Since dim φ_{1} ≤ 8, [φ_{1}] is a sum of classes of three general 2-Pfister forms, [5, Lemma 38.1]. □

Unfortunately, the way of proving Corollary 2.5 does not work for 8-dimensional subforms in φ of trivial discriminant. Absence of such subforms would follow from absence of finite field extensions L/F of degree dividing 8 with i_{W} (φ_{L}) ≥ 4. However such a field extension does exist. More than that, φ becomes hyperbolic over certain field extension of degree 8.

3. Dimensions 14 and 18

As already mentioned, the following result has been obtained (in characteristic ≠ 2) independently in [6] and in [7]. The proof presented here is different; it is parallel to the proof of Theorem 2.1.

Theorem 3.1. Let φ be a generic 14-dimensional quadratic form in I_{q}^{3} over a field F (of arbitrary characteristic). The degree of any finite field extension L/F with i_{W} (φ_{L}) ≥ 2 is divisible by 4. Equivalently, for any finite extension E/F of odd degree, the quadratic form φ_{L} does not contain a 4-dimensional subform of trivial discriminant.

Proof. Assuming the contrary, we get that 2e_{5}e_{6} ∈ \overline{CH} X, where X is a component of the maximal orthogonal Grassmannian of φ. On the other hand, \overline{CH} X, being the subring of CH X̄ generated by e_{1} and 2e_{i} with i = 2, …, 6, does not contain 2e_{5}e_{6}. □

Note that any 14-dimensional quadratic form in I_{q}^{3} over a field of characteristic ≠ 2 does contain a 6-dimensional subform of trivial discriminant as a consequence of M. Rost’s classification result.

Corollary 3.2. For any base field k (of arbitrary characteristic), one has Pf_{3}(14) ≥ 3.

Proof. Let φ/F ⊃ k be a generic 14-dimensional quadratic form in I_{q}^{3}. If the Witt class [φ] is the sum of two classes of general Pfister forms, then i_{W} (φ_{L}) ≥ 3 for some finite field extension L/F of degree dividing 2. This contradicts Theorem 3.1. □

Now we look at dimension 18 and higher. It has been shown in [3, Theorem 4.2] that for any field k of characteristic 0 and any even m ≥ 18, there exists a field F ⊃ k and an m-dimensional quadratic form φ over F with Witt class in I_{q}^{3}(F) such that for any finite field extension E/F of odd degree, φ_{E} does not contain any proper even-dimensional subform of trivial discriminant. We suggest the following conjecture implying this result:

Conjecture 3.3. For any even m ≥ 18, the index of the [m/4]-th orthogonal Grassmannian of any generic m-dimensional quadratic form φ/F in I_{q}^{3} is equal to 2^{[m/4]}.

Conjecture 3.3 actually implies that any generic m-dimensional quadratic form φ/F in I_{q}^{3}, for any even m ≥ 18, has the property of [3, Theorem 4.2]: if this is not the case, i.e., if φ_{E} does contain a proper even-dimensional subform ψ of trivial discriminant for some finite field extension E/F of odd degree, then, possibly replacing ψ by its complement, we have d := (dim ψ)/2 ≤ [m/4] and ψ becomes hyperbolic over an extension of degree dividing 2^{d−1}. Therefore φ_{E} acquires Witt index ≥ [m/4] over an extension of L of degree dividing 2^{[m/4]−1}, i.e., the index of the [m/4]-th orthogonal Grassmannian of φ divides 2^{[m/4]−1}.

Unfortunately, the method of proving Theorems 2.1 and 3.1 does not work for Conjecture 3.3. For instance, for m = 18 one needs to show that 2^{3}e_{5}e_{6}e_{7}e_{8} ∉ \overline{CH} X which is false – see next paragraph. This provides a motivation to develop a modification of the method; we do it in the next section.

Let us show that 2^{3}e_{5}e_{6}e_{7}e_{8} ∈ \overline{CH} X, where X is a component of the maximal orthogonal Grassmannian of a generic 18-dimensional quadratic form in I_{q}^{3}. We have e_{2} = e_{1}^{2} ∈ \overline{CH} X, e_{4} = e_{1}(2e_{3}) − e_{2}^{2} ∈ \overline{CH} X, so that e_{8} − 2e_{3}e_{5} = e_{1}(2e_{7}) − e_{2}(2e_{6}) − e_{4}^{2} ∈ \overline{CH} X. Multiplying by (2e_{5})(2e_{6})(2e_{7}) ∈ \overline{CH} X, we get that 2^{3}e_{5}e_{6}e_{7}e_{8} −2^{4}e_{3}e_{5}^{2}e_{6}e_{7} ∈ \overline{CH} X. The second summand is in \overline{CH} X as well because e_{5}^{2} = 2e_{4}e_{6} − 2e_{3}e_{7} + 2e_{2}e_{8}. Therefore the first summand 2^{3}e_{5}e_{6}e_{7}e_{8} is in \overline{CH} X.

4. Grothendieck ring of maximal orthogonal Grassmannian

For an integer n ≥ 1, let φ : V → F be a non-degenerate quadratic form of dimension 2n+1 over a field F (of arbitrary characteristic). The vector space V of definition of φ is a vector space over F of dimension 2n+1. Let X be the maximal orthogonal Grassmannian of φ, i.e., X is the F -variety of n-dimensional totally isotropic subspaces in V.

We are going to collect information on the Grothendieck group K(X). From Panin’s computation of K-theory of projective homogeneous varieties [16], we deduce

Lemma 4.1. There is a natural (with respect to field extensions of F) group isomorphism of K(X) onto the direct sum of 2^{n−1} copies of K(F) = ℤ and 2^{n−1} copies of K(C_{0}(φ)) = ind C_{0}(φ) · ℤ, where C_{0}(φ) is the even Clifford algebra of φ.

Proof. Consider the category of K-correspondences as in [15, §1.8] (where it is called motivic category). Objects of this category are pairs (Y, A), where Y is a smooth projective F -variety and A is a separable F -algebra. To simplify notation, one writes Y for the object (Y, F) and A for (Spec F, A).

As proved in [16], the object in the above category given by the maximal orthogonal Grassmannian X (as well as by any other projective homogeneous variety under the orthogonal group O^{+}(φ)) is isomorphic to a separable algebra A which is a finite direct product of a copies of F and of b copies of C_{0}(φ) for some integers a, b ≥ 0. As a consequence, we get a natural group isomorphism K(X) ≃ K(F)^{⊕a} ⊕ K(C_{0}(φ))^{⊕b}. The rank of the group K(X) coincides with the rank of CH(X) which, as we know, is equal to 2^{n}. Therefore a + b = 2^{n} and in order to finish the proof of Lemma 4.1 we only need to show that a = b provided that the central simple algebra C_{0}(φ) is not split (i.e., has index > 1).

Let S be the Severi-Brauer variety of C_{0}(φ). The projection X ×S → X is a projective bundle of rank 2^{n} − 1. By Projective Bundle Theorem for K-theory [17, Proposition 4.3], there is a natural isomorphism K(X × S) ≃ K(X)^{⊕2^{n}}. Therefore, K(X) is naturally isomorphic to the direct sum of 2^{n}a copies of K(F) and 2^{n}b copies of K(C_{0}(φ)).

On the other hand, S as an object in the category of K-correspondences is isomorphic to the product of 2^{n−1} copies of F and 2^{n−1} copies of C_{0}(φ). (Since the algebra C_{0}(φ) is of exponent 2, its higher tensor powers do not show up.) It follows that the object given by the direct product of varieties X × S, which is the tensor product in the category of K-correspondences of the objects given by X and by S, is isomorphic to the product of 2^{n−1}(a + b) copies of F and 2^{n−1}(a + b) copies of C_{0}(φ). This gives a natural isomorphism of K(X × S) with the direct sum of 2^{n−1}(a + b) copies of K(F) and 2^{n−1}(a + b) copies of K(C_{0}(φ)). Comparing the resulting computation for the order of cokernel of the change of field homomorphism for K(X × S) to an algebraic closure of F, we get that

(ind C_{0}(φ))^{2^{n}b} = (ind C_{0}(φ))^{2^{n−1}(a+b)}.

Since ind C_{0}(φ) ≠ 1, it follows that 2^{n}b = 2^{n−1}(a + b) giving a = b. □ Proposition 4.2. Assume that the index of the even Clifford algebra C_{0}(φ) is maximal: ind C_{0}(φ) = 2^{n}. Then the topological filtration on K(X) coincides with the gamma filtration. Proof. Let us consider the filtration of the ring K(X) generated by the K-theoretical Chern classes c_{i}(T ), i ≥ 0 of the tautological vector bundle T on X. By definition, the i-th term F^{i} of this filtration is additively generated by the products c_{i_{1}}(T ) · · · · · c_{i_{r}}(T ) with r ≥ 1 and i_{1} + ⋯ + i_{r} ≥ i.

Clearly, for any i, we have a chain of inclusions consisting of 4 terms: F^{i} is inside of the i-th term of the gamma filtration, which is inside of the i-term of the topological filtration on K(X), which is inside of the intersection with K(X) of the i-th term of the topological filtration on K(X̄). We are going to show that the smallest term in this chain coincides with the largest term meaning that all four terms are the same. This will prove Proposition 4.2.

Let F̄ be an algebraic closure of F and X̄ := XF̄. By Lemma 4.1, the order of the cokernel of the change of field homomorphism K(X) → K(X̄) is equal to 2^{n2^{n−1}}. On the other hand, the cokernel of the homomorphism of the associated graded rings GK(X) → GK(X̄) with respect to the filtration F on K(X) and the topological filtration on K(X̄), is identified with the cokernel of CH X → CH X̄ whose order also equals 2^{n2^{n−1}}, see Corollary 1.4. The formula

| Ker(GK(X) → GK(X̄))| = \frac{|Coker(GK(X) → GK(X̄))|}{|Coker(K(X) → K(X̄))|}

(proved as [11, Proposition 2]) implies that the homomorphism GK(X) → GK(X̄) is injective, i.e., that the filtration F on K(X) is induced by the topological filtration on K(X̄). □

Theorem 4.3. For φ as in Proposition 4.2, let S be the Severi-Brauer variety of the division algebra C_{0}(φ). Then the topological filtration on K(X_{F (S)}) coincides with the gamma filtration.

Proof. We proceed along the lines of [12, Theorem 3.7]. As we already observed in the proof of Proposition 1.5 and Lemma 4.1, the product X ×S is a projective bundle over X. This implies that CH(X ×S) and therefore CH X_{F (S)} is generated as CH X-algebra by the first Chern class of certain linear vector bundle (well, this Chern class is equal to ±e_{1}). The epimorphism CH X_{F (S)} →→ GK(X_{F (S)}), where GK is now the associated graded ring of the topological filtration, shows that also GK(X_{F (S)}) is generated as GK(X)-algebra by the first Chern class of a linear vector bundle. By Proposition 4.2, the ring GK(X) is generated by Chern classes of vector bundles. It follows that also the ring GK(X_{F (S)}) is generated by Chern classes of vector bundles. This precisely means that the gamma filtration on K(X_{F (S)}) coincides with the topological one. □

Remark 4.4. The proofs of Theorem 4.3 and Proposition 4.2 actually provide a simpler than a priori description of gamma filtration on K(X_{F (S)}): it is generated by e_{1} and the Chern classes of the tautological vector bundle. More precisely, for any i ≥ 0, the i-th term of the filtration is additively generated by the products e_{1}^{i_{0}}c_{i_{1}}(T ) … c_{i_{r}}(T ) with r ≥ 0 and i_{0} + i_{1} + ⋯ + i_{r} ≥ i.

Remark 4.5. Let X̃ be any flag variety of φ projecting to X, i.e., any variety of flags of totally isotropic subspaces in φ, where the flags under consideration include maximal totally isotropic subspaces. In particular, X̃ can be the variety of complete flags. Since the projection X̃ → X is a flag bundle of a vector bundle on X (namely, of the tautological vector bundle), Theorem 4.3 implies that the topological filtration on K(X̃_{F (S)}) (and on K(X̃)) coincides with the gamma filtration.

Using identification of the maximal orthogonal Grassmannian of an odd-dimensional quadratic form with a component of the maximal orthogonal Grassmannian of the corresponding even-dimensional form of trivial discriminant, we get the above statement as well for even-dimensional forms φ of trivial discriminant with division algebra C^{+}(φ).

Finally, using specialization arguments, one can prove coincidence of gamma and topological filtrations for K(T/P), where T is a generic principle homogeneous space under Spin_{m} (with any m: m = 2n + 1 or m = 2n + 2) and P ⊂ Spin_{m} is any special parabolic subgroup.

Remark 4.6. Instead of completely killing the division algebra C_{0}(φ) by passing to the function field of its Severi-Brauer variety S in the statement of Theorem 4.3, one may partially split C_{0}(φ) by passing to the function field of any of its generalized Severi-Brauer variety S′. The conclusion remains the same: gamma filtration on K(X_{F (S′)}) coincides with the topological filtration. The proof also remains basically the same, cf. [12, Theorem 3.7 and its proof].

It would be very interesting to completely understand the gamma filtration on K(X_{F (S)}). Note that for any field extension L/F (S), the change of field homomorphism K(X_{F (S)}) → K(X_{L}) is an isomorphism preserving the filtration. So, if this helps, it is enough to perform the computation, say, over an algebraically closed field. Actually, it is even enough to do it for a maximal orthogonal Grassmannian X̄ over ℂ. One may consider the additive basis of the group K(X̄) given by the Schubert classes; the ring structure is determined by the K-theoretical Littlewood-Richardson formulas obtained in [4]. Alternatively, one may describe the ring K(X̄) by generators and relations in the spirit of the discussed description of CH X̄, taking for generators the special Schubert classes e_{i} ∈ K(X̄). (Unfortunately, the relations on e_{i} in K(X̄) look more complicated than (1.1).)

Note that these K-theoretical special Schubert classes still satisfy relations (1.2), where now l_{n−i} stand for the K-theoretical classes of linear subspaces on the quadric. This can be shown using [2, Lemma 2.1].

Of particular interest is to understand the position of the special Schubert classes in the filtration. More specifically, let us consider the class e_{n} ∈ K(X_{F (S)}) of the special Schubert variety of the lowest dimension (corresponding to the class of a rational point on the quadric).

Conjecture 4.7. For n ≥ 8, the special Schubert class e_{n} ∈ K(X_{F (S)}) does not belong to the term number n + 1 − [(n + 1)/2] of the gamma filtration.

Conjecture 4.7 implies Conjecture 3.3. Indeed, assume that Conjecture 3.3 fails for some even m ≥ 18. This means that we can find a field F and a non-degenerate quadratic form φ over F of dimension m and trivial discriminant such that C^{+}(φ) is a division algebra and there exists a finite field extension L/F (S) of degree not divisible by 2^{r} with i_{W} (φ_{L}) ≥ r, where r := [m/4] and S is the Severi-Brauer variety of C^{+}(φ).

Since i_{W} (φ_{L}) ≥ r, the projective quadric Y_{L} of φ_{L} contains a linear subspace of dimension r −1. Its class l_{r−1} in K(Y_{L}) belongs to the term number dim Y −(r −1) = m−r −1 of the topological filtration on K(Y_{L}). Applying the norm homomorphism K(Y_{L}) → K(Y_{F (S)}), preserving the filtration and acting as multiplication by [L : F (S)], we get that the element 2^{r−1}l_{r−1} of the group K(Y_{F (S)}) is in the m − r − 1-th term of the topological filtration. Since 2^{r−1}l_{r−1} = l_{0} + h^{m−3} + 2h^{m−4} + ⋯ + 2^{r−2}h^{m−r−1} (see [9, §3.2]), where h is the class of a hyperplane section, we get that l_{0} is in the m − r − 1-th term as well. As in (1.2), we have e_{n} = f_{∗}g^{∗}(l_{0}) ∈ K(X_{F (S)}), where X is a component of the maximal orthogonal Grassmannian of φ. The pull-back homomorphism g^{∗} : K(Y_{F (S)}) → K(Z_{F (S)}) preserves the filtration. The push-forward homomorphism f_{∗} : K(Z_{F (S)}) → K(X_{F (S)}) lowers the number of the filtration term by dim Z − dim X = n := (m − 2)/2. It follows that e_{n} is in the term number m − r − 1 − n = n + 1 − [(n + 1)/2]. Finally, identifying X with the maximal orthogonal Grassmannian of a non-degenerate 2n + 1-dimensional subform of φ, we get a counter-example to Conjecture 4.7.

Remark 4.8. The number n + 1 − [(n + 1)/2] in Conjecture 4.7 is optimal for n = 8. Indeed, if φ is an 18-dimensional quadratic form of trivial discriminant and trivial Clifford invariant and Y is its projective quadric, since the Chow groups CH^{i}Y are torsion-free for i ≤ 3, [8], the class l_{8} ∈ K(Y) is in the 4-th term of the topological filtration, cf. [9, Theorem 3.10]. Since l_{0} = l_{8} · h^{8}, it follows that l_{0} is in the term number 4 + 8 = 12.

Therefore e_{8} is in the term number 4 = 12 − n. Conjecture 4.7 claims that e_{8} is not in the term number 5.

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Mathematical & Statistical Sciences, University of Alberta, Edmonton, CANADA

E-mail address: karpenko at ualberta.ca, web page:  www.ualberta.ca/~karpenko