Minimal Canonical Dimensions

of Quadratic Forms

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Nikita A. Karpenko^{1}

Received: September 11, 2013 Revised: January 7, 2014

Abstract. Canonical dimension of a smooth complete connected variety is the minimal dimension of image of its rational endomorphism. The i-th canonical dimension of a non-degenerate quadratic form is the canonical dimension of its i-th orthogonal grassmannian. The maximum of a canonical dimension for quadratic forms of a fixed dimension is known to be equal to the dimension of the corresponding grassmannian. This article is about the minima of the canonical dimensions of an anisotropic quadratic form. We conjecture that they equal the canonical dimensions of an excellent anisotropic quadratic form of the same dimension and we prove it in a wide range of cases.

2010 Mathematics Subject Classification: 14L17; 14C25 Keywords and Phrases: Algebraic groups, quadratic forms, projective homogeneous varieties, Chow groups and motives.

Contents

1. Introduction  368
2. Excellent forms  370
3. Upper motives  371
4. Some tools  372
4a. Outer excellent connections  372
4b. A motivic decomposition  372
4c. Maximal orthogonal grassmannian  373
4d. Values of first Witt index  373
4e. Dimensions of forms in I^{n}  373
5. Level 1  373
6. Level 2  374

1The author acknowledges a partial support of the French Agence Nationale de la Recherche (ANR) under reference ANR-12-BL01-0005; his work has been also supported by the start-up grant of the University of Alberta.

Documenta Mathematica · Extra Volume Merkurjev (2015) 367–385

7. Third canonical dimension  375
8. Height 3  376
9. “Counter-example” with maximal grassmannian  378
10. Fourth canonical dimension  378
11. Fifth canonical dimension  380
12. Final comments  382
References  384

1. Introduction

The canonical dimension cd(X) of a smooth complete connected algebraic variety X over a field F is the minimum of dimension of the image of a rational map X −−→ X. This integer depends only on the class of field extensions L/F with X(L) ≠ ∅. We refer to [9] and [16] for interpretations and basic properties of cd(X). We will also use a 2-local version cd_{2}(X) of cd(X) called canonical 2-dimension. All fields here are of characteristic ≠ 2. (The questions we are discussing can be raised in characteristic 2 as well, but all results we get are for characteristic ≠ 2 mainly because their proofs need the Steenrod operations on Chow groups modulo 2 which are not available in characteristic 2.) Let φ be a non-degenerate quadratic form over a field F. (Our general reference for quadratic forms is [3].) For any integer i lying in the interval [1, (dim φ)/2], the i-th canonical dimension cd[i](φ) is defined as the canonical dimension of the orthogonal grassmannian of i-dimensional totally isotropic subspaces of φ (i-grassmannian of φ for short). A little care should be given to the case of i = (dim φ)/2 because the corresponding i-grassmannian is not connected if the discriminant of φ is trivial. However, the (two) connected components it has are isomorphic to each other so that we can define the canonical dimension by taking any of them. For arbitrary i and a given field extension L/F, the i-grassmannian of φ has an L-point if and only if the Witt index 𝔦_{0}(φ_{L}) is at least i. Therefore, cd[i](φ) is an invariant of the class of field extensions L/F satisfying 𝔦_{0}(φ_{L}) ≥ i. Similarly, the i-th canonical 2-dimension cd_{2}[i](φ) is the canonical 2-dimension of the i-grassmannian. Since in general, canonical 2-dimension is a lower bound for canonical dimension, we have cd[i](φ) ≥ cd_{2}[i](φ) for any i. This is known to be equality for i = 1 (see Section 5) and no example when this inequality is not an equality (for some i > 1) is known. The study of canonical dimensions of quadratic forms naturally commences with the question about the range of their possible values for anisotropic quadratic forms of a fixed dimension (over all fields or over all field extensions of a given field). It has been shown in [12] (see also [13]) that the evident upper bound on cd[i](φ) and cd_{2}[i](φ), given by the dimension of the i-grassmannian, is sharp. Here is a formula for this dimension:

i(i − 1)/2 + i(dim φ − 2i).

The question on the sharp upper bound being therefore closed, the present paper addresses the question about the sharp lower bound. Natural candidates are canonical dimensions of excellent quadratic forms. We do not really have a strong evidence supporting this, but we may, for instance, recall [3, Theorem 84.1] where the excellent forms appear in the answer to the question about the minimal height of quadratic forms. For any n ≥ 1 and any i ∈ [1, n/2], we write cd[i](n) (resp., cd_{2}[i](n)) for the ith canonical (2-)dimension of an anisotropic excellent n-dimensional quadratic form over some field. Note that cd[i](n) depends only on i, n and coincides with cd_{2}[i](n) (see Section 2). The following conjecture therefore gives a complete answer to the question about the sharp lower bound on canonical dimension and canonical 2-dimension of anisotropic quadratic forms:

Conjecture 1.1. Let φ be an anisotropic quadratic form over a field F satisfying dim φ > 2i for some i ≥ 1. Then cd_{2}[i](φ) ≥ cd[i](dim φ).

The reason of excluding the case 2i = dim φ in the statement is that in this case cd_{2}[i](φ) = cd_{2}[i − 1](φ_{E}) and cd[i](φ) = cd[i − 1](φ_{E}), where E/F is the discriminant field extension of φ (E = F if the discriminant of φ is trivial) and i ≥ 2. So, understanding of cd_{2}[i](φ) and cd[i](φ) for i < (dim φ)/2 would provide their understanding for i = (dim φ)/2 and, on the other hand, using these relations it is easy to get counter-examples to the formula of Conjecture 1.1 with i = (dim φ)/2 (see Section 9). In this paper we prove Conjecture 1.1 for “small” values of i, namely, for i not exceeding the 2-nd absolute Witt index of φ (see Theorem 6.1) as well as for i ≤ 5 (see Theorems 7.1, 10.1 and 11.1). Finally, we prove Conjecture 1.1 with arbitrary i for all quadratic forms of height ≤ 3 (see Theorem 8.2). The proofs make use of a wide spectrum of modern results on quadratic forms and Chow motives (the question seems to be a good testing ground for them). However most of the results under use already became “classical” at least in the sense that they have been exposed in a book (in [3] in most of the cases). For instance, we are using only a part of Excellent Connections Theorem [20, Theorem 1.3], called Outer, which was available already before the whole result and is exposed in [3, Corollary 80.13]. The most recent (and certainly yet non-classical) tool is a kind of going down principle for Chow motives due to Charles De Clercq [2], used in the proofs of Theorem 3.2 and (in a slightly different situation) Theorem 8.2. Applications of some particular cases of this principle exist already in the literature (see, e.g., [4]). We are using it here (in the proof of Theorem 3.2) in a new situation (still not in its full generality but in the biggest generality which may occur in the case of projective homogeneous varieties). This principle generalizes [10, Proposition 4.6], this older result is not sufficient for our purposes here. Those methods can certainly be used to prove a bit more of Conjecture 1.1, but it seems that something is missing for a complete solution.

One could expect that the case of maximal i should be more accessible because maximal orthogonal grassmannians are so well-understood (mainly due to results of [19] also exposed in [3, Chapter XVI]). Though in our approach we have to go through all values of i in order to get to the maximal one. This paper is an extended version of [6]. For more introduction see §12. Acknowledgements. This work has been initiated during my stay at the Fields Institute for Research in Mathematical Sciences in Spring 2013. I thank the Fields Institute for hospitality and perfect working conditions.

2. Excellent forms

Here we recall some standard facts about excellent forms needed to complete the statement of Conjecture 1.1. Proofs (along with a definition) can be found, e.g., in [3, §28]. Every positive integer n is uniquely representable in the form of an alternating sum of 2-powers:

n = 2^{p_{0}} − 2^{p_{1}} + 2^{p_{2}} − ⋯ + (−1)^{r−1}2^{p_{r−1}} + (−1)^{r}2^{p_{r}}

for some integers r ≥ 0 and p_{0}, p_{1}, …, p_{r} satisfying p_{0} > p_{1} > ⋯ > p_{r−1} > p_{r} + 1 > 0. For any integer i ∈ [1, n/2], we define an integer cd[i](n) as

cd[i](n) := 2^{p_{s−1}−1} − 1,

where s is the minimal positive integer with

n − 2i ≥ 2^{p_{s}} − 2^{p_{s+1}} + ⋯ + (−1)^{r−s}2^{p_{r}}.

Note that cd[i](n) ≥ cd[i + 1](n) (for any i, n such that both sides are defined).

Lemma 2.1. For any field k and any positive integer n, there exists an ndimensional anisotropic quadratic form φ over an appropriate extension field F/k such that

cd[i](φ) = cd_{2}[i](φ) = cd[i](n)

for any i ∈ [1, n/2].

Proof. One may take as F a field extension of k generated by p_{0} algebraically independent elements. (For k ⊂ ℝ one may simply take F = ℝ.) Then there exists an anisotropic p_{0}-fold Pfister form over F and therefore an anisotropic excellent quadratic form φ of dimension n. (For F = ℝ, the unique up to isomorphism anisotropic n-dimensional quadratic form is excellent.) We claim that canonical dimensions of such φ are as required. Indeed, for i ∈ [1, n/2] let s be the defined above integer. Then by [3, Theorem 28.3], there exists a p_{s−1}-fold Pfister form ρ over F such that for any field extension L/F the condition 𝔦_{0}(φ_{L}) ≥ i is equivalent to isotropy of ρ_{L}. It follows that cd_{2}[i](φ) = cd[i](φ) = 2^{p_{s−1}−1} − 1. □

3. Upper motives

By motives we always mean the Chow motives with coefficients in 𝔽_{2} := ℤ/2ℤ; we use related terminology and notation as in [3, Chapter XII]. In particular, M(X) is the motive of a variety X; the motive M(Spec F) and all its shifts M(Spec F)(i), i ∈ ℤ, are called Tate motives. If M is a motive over F, M̄ is the corresponding motive over an algebraic closure of F. Let φ be a non-degenerate quadratic form over a field F. For an integer i with 0 ≤ i < dim φ/2, let X_{i} = X_{i}(φ) be the i-grassmannian of φ. In particular, X_{0} is the point and X := X_{1} is the projective quadric of φ. According to the general notion of upper motive, introduced in [14] and [11], the upper motive U(X_{i}) of the variety X_{i} is the unique summand in the complete motivic decomposition of X with the property that Ū(X_{i}) contains a Tate summand with no shift (i.e., with the shift 0). According to the general criterion of isomorphism for upper motives, U(X_{i}) ≃ U(X_{j}) if and only if

𝔦_{0}(φ_{L}) ≥ i ⟺ 𝔦_{0}(φ_{L}) ≥ j

for any extension field L/F. This means that i and j are in the same semi-open interval (𝔧_{r−1}, 𝔧_{r}] for some r ≥ 0, where 𝔧_{r} is the r-th absolute Witt index of φ and 𝔧_{−1} := −∞. According to the general [11, Theorem 1.1], applied to quadrics, any summand of the complete motivic decomposition of X is a shift of U(X_{i}) for some i or – in the case of even-dimensional φ with non-trivial discriminant – U(Spec E), where E/F is the quadratic discriminant field extension. Shifts of U((X_{i})_{E}), which may a priori appear by [11, Theorem 1.1], aren’t possible because for any j ≠ (dim X)/2 the motive M̄(X) contains at most one Tate summand with the shift j while Ū((X_{i})_{E}) contains two Tate summands without shift and two Tate summands with the shift dim U((X_{i})_{E}). A more precise information can be derived from [18, §4] (see also [3, §73]): if a shift of U(X_{i}) for some i ∈ (𝔧_{r−1}, 𝔧_{r}] with r ≥ 1 really appears in the decomposition (note that this is always the case for r = 1), then it appears precisely 𝔦_{r} := 𝔧_{r}−𝔧_{r−1} times and the shifting numbers are 𝔧_{r−1}, 𝔧_{r−1}+1, …, 𝔧_{r}− 1. A shift of U(Spec E) appears if and only if φ_{E} is hyperbolic in which case it appears only once and with the shifting number (dim X)/2. Note that U(X_{i}) for i ≤ 𝔧_{0} is just the motive of a point (= the Tate summand with no shift), it appears precisely 2𝔧_{0} times and the shifting numbers are 0, …, 𝔧_{0} − 1 and dim X, …, dim X − (𝔧_{0} − 1). Given any i and setting Y := X_{i}, one can answer the question, whether a shift of U(Y) does appear, in terms of canonical dimension. First of all we have

Theorem 3.1 ([9, Theorem 5.1]). cd_{2}(Y) = dim U(Y).

The following result is new. It provides a criterion of appearance of U(Y) and is proved with a help of the going down principle of [2].

Theorem 3.2. Assume that i ∈ (𝔧_{r−1}, 𝔧_{r}] for some r ≥ 1 and set T := X_{𝔧_{r−1}}, Y := X_{i}. A shift of U(Y) appears in the complete motivic decomposition of X if and only if

cd_{2}(Y) = cd_{2}(Y_{F (T)}).

Remark 3.3 (cf. §5). cd_{2}(Y_{F (T)}) = dim φ − 2𝔧_{r−1} − 𝔦_{r} − 1.

Remark 3.4. Note that cd_{2}(Y) ≥ cd_{2}(Y_{F (T)}) in general, [16].

Remark 3.5. As already mentioned, for i = 𝔧_{1}, the 𝔦_{1} shifts of U(X_{i}) appear always.

Remark 3.6. Sufficient criteria of appearance given in [18, Theorems 4.15 and 4.17] are easily derived from Theorem 3.2.

Proof of Theorem 3.2. By Theorem 3.1, we may replace cd_{2}(Y) with dim U(Y) as well as cd_{2}(Y_{F (T)}) with dim U(Y_{F (T)}) in the statement. If a shift of U(Y) does appear, then dim U(Y) = dim U(Y_{F (T)}) by [18, §4] (see also [3, §73]). This proves one (“easy”) direction of Theorem 3.2. Let us concentrate on the opposite direction. Note that a shift of U(Y_{F (T)}) is a summand in M(X_{F (T)}) (see Remark 3.5). If dim U(Y) = dim U(Y_{F (T)}), then we conclude by [2, Theorem 1.1] that the same shift of U(Y) is a summand in M(X). □

4. Some tools

In this section we recall some results which appear most frequently in the proofs below.

4a. Outer excellent connections. The following statement is a part of [20, Theorem 1.3]. It is also proved in [3, Corollary 80.13].

Theorem 4.1 (Outer Excellent Connections). Let X be the quadric of an anisotropic quadratic form of dimension 2^{n} + m with n ≥ 1 and m ∈ [1, 2^{n}]. Let M be a summand of the complete motivic decomposition of X. If M̄ contains a Tate summand with a shift i < m, then it also contains a Tate summand with the shift 2^{n} − 1 + i = dim X − (m − 1) + i.

Using Theorem 4.1, we will be able to see that no shift of U(Y) is a summand of M(X) for certain concrete X and Y as in Theorem 3.2. The latter theorem will then tell us that cd_{2}(Y) > cd_{2}(Y_{F (T)}) (see Remark 3.4). Afterwards, we usually get even a sharper lower bound on cd_{2}(Y) using the motivic decomposition described right below.

4b. A motivic decomposition. Let φ be a non-degenerate quadratic form over F of dimension n and let Y be the 𝔦_{0}-grassmannian of φ. A variety is called anisotropic if all its closed points are of even degree.

Lemma 4.2 ([7, Theorem 15.8 and Corollary 15.14] or [1]). The motive of Y decomposes in a sum of shifts of motives of some anisotropic varieties plus

⨁_{i=0}^{𝔦_{0}} M(Γ_{i})(i(i − 1)/2 + i(n − 2𝔦_{0})),

where Γ_{i} is the i-grassmannian of an 𝔦_{0}-dimensional vector space (Γ_{0} and Γ_{𝔦_{0}} are points, Γ_{1} and Γ_{𝔦_{0}−1} – projective spaces).

Corollary 4.3. The motive of Y does not contain any Tate summand with a positive shift strictly below n − 2𝔦_{0}.

Proof. By preceding Lemma, the motive of Y decomposes in a sum of shifts of motives of certain varieties. Those summands of this motivic decomposition which are motives of isotropic varieties^{2} (and therefore can contain Tate summands while the motives of anisotropic varieties cannot, see, e.g., [14, Lemma 2.21]) come with shifts i(i − 1)/2 + i(n − 2𝔦_{0}), i ≥ 0. For i = 0 the shifting number is 0 and the corresponding variety is just the point. For i ≥ 1 the shifting numbers are at least n − 2𝔦_{0}. □

4c. Maximal orthogonal grassmannian. Let φ be a non-degenerate quadratic form of dimension 2n + 1 and let Y = X_{n}(φ) be the maximal orthogonal grassmannian of φ. Let e_{i} ∈ Ch^{i}(Ȳ), i = 0, 1, …, e_{2^{n−1}+1}, be the standard generators of the modulo 2 Chow ring Ch(Ȳ) defined as in [3, §86]. We say that e_{i} is rational if it is in the image of the change of field homomorphism Ch^{i}(Y) → Ch^{i}(Ȳ); otherwise is irrational. We recall [3, Theorem 90.3] stating that cd_{2}(Y) is equal to the sum of all j such that e_{j} is irrational.

4d. Values of first Witt index. By [3, Proposition 79.4 and Remark 79.5], the first Witt index 𝔦_{1} of an anisotropic quadratic form of dimension d ≥ 2 satisfies the relations

𝔦_{1} ≡ d (mod 2^{r}) and 1 ≤ 𝔦_{1} ≤ 2^{r}

for some integer r ≥ 0 with 2^{r} < d.

4e. Dimensions of forms in I^{n}. By [3, Proposition 82.1], dimension d of an anisotropic quadratic form in I^{n} (the n-th power of the fundamental ideal in the Witt ring of the base field), where n ≥ 1, is either ≥ 2^{n+1} or equals 2^{n+1} − 2^{i} with 1 ≤ i ≤ n + 1. Actually, apart from the old Arason-Pfister Hauptsatz (saying that d ∉ (0, 2^{n})), we are only using the statement about the “first hole”, saying that d is outside of the open interval (2^{n}, 2^{n} + 2^{n−1}) and proved earlier ([18, Theorem 6.4]).

5. Level 1

We explain here that Conjecture 1.1 is actually already known in “level 1”, that is, for i not exceeding the first Witt index of φ. It is well-known that cd[1](φ) = cd_{2}[1](φ) ≥ cd[1](dim φ) for any anisotropic φ. This is a consequence of the formula cd[1](φ) = cd_{2}[1](φ) = dim φ − 𝔦_{1}(φ) − 1 ([3, Theorem 90.2]) and the fact that the first Witt index of an excellent form is maximal among the first Witt indexes of quadratic forms of a given dimension ([5, Corollary 1]).

2A variety is isotropic here if it has a closed point of odd degree.

Documenta Mathematica · Extra Volume Merkurjev (2015) 367–385

As an immediate consequence, we get the following, formally more general statement – (a bit more than) the “level 1” case of Conjecture 1.1:

Proposition 5.1. Let φ be an anisotropic quadratic form over F of height ≥ 1. For any i ≤ 𝔦_{1}(φ) one has cd[i](φ) = cd_{2}[i](φ) ≥ cd[i](dim φ).

Proof. cd[i](φ) = cd_{2}[i](φ) = cd_{2}[1](φ) ≥ cd[1](dim φ) ≥ cd[i](dim φ). □

6. Level 2

In this Section we prove (a bit more than) the “level 2” case of Conjecture 1.1:

Theorem 6.1. Let φ be an anisotropic quadratic form over F of height ≥ 2. For any positive integer i ≤ 𝔦_{1}(φ) + 𝔦_{2}(φ) one has cd_{2}[i](φ) ≥ cd[i](dim φ).

Corollary 6.2. Let φ be an anisotropic quadratic form over F of dimension dim φ ≥ 4. Then cd_{2}[2](φ) ≥ cd[2](dim φ). □

Corollary 6.3. Let φ be an anisotropic quadratic form over F of height ≤ 2. Then cd_{2}[i](φ) ≥ cd[i](dim φ) for any i ∈ [1, (dim φ)/2]. □

Proof of Theorem 6.1. We write 𝔦_{1} for 𝔦_{1}(φ) and 𝔦_{2} for 𝔦_{2}(φ). By Proposition 5.1, we may assume that i ∈ (𝔦_{1}, 𝔦_{1} + 𝔦_{2}]. Let us write dim φ = 2^{n} + m with n ≥ 1 and m ∈ [1, 2^{n}]. In the case of 𝔦_{1} = m we have

cd_{2}[i](φ) ≥ cd_{2}[i − m](φ_{1}) ≥ cd[i − m](dim φ_{1}) = cd[i](dim φ),

where φ_{1} is the 1-st anisotropic kernel of φ, [3, §25]. The first inequality here is a particular case of the general principle saying that cd_{2}(T_{L}) ≤ cd_{2}(T) for a variety T over F and a field extension L/F, [16]. The second inequality holds by Proposition 5.1. Below we are assuming that 𝔦_{1} < m and we have to show that cd_{2}[i](φ) ≥ 2^{n}−1. In the case of 𝔦_{1} < m/2 we have

cd_{2}[i](φ) ≥ cd_{2}[1](φ_{1}) ≥ cd[1](2^{n} + m − 2𝔦_{1}(φ)) = 2^{n} − 1.

Below we are assuming that m/2 ≤ 𝔦_{1} < m. It follows by §4d that 𝔦_{1} = m/2 (in particular, m is ≥ 2 and even). This implies that 𝔦_{2} ≤ 2^{n−1}. If 𝔦_{1} + 𝔦_{2} < m, then 𝔦_{1} + 𝔦_{2} ≤ m − 𝔦_{1} by [17, Theorem 1.2] which is impossible with 𝔦_{1} = m/2. Therefore 𝔦_{1} + 𝔦_{2} ≥ m and it follows by Theorem 4.1 that U(Y)(𝔦_{1}) is not a direct summand of the motive of X, where X is the quadric of φ and Y is the (𝔧_{2} = 𝔦_{1} + 𝔦_{2})-th grassmannian of φ. Since cd_{2}[i](φ) = cd_{2}(Y), all we need to show is cd_{2}(Y) ≥ 2^{n} − 1. First of all we have cd_{2}(Y) > cd_{2}(Y_{F (X)}) by Theorem 3.2 and Remark 3.4. Now we claim that the complete decomposition of M(Y_{F (X)}) does not contain a summand U(Y_{F (X)})(j) with j inside of the open interval

(0, 2^{n} + m − 2(𝔦_{1} + 𝔦_{2})).

Indeed, if U(Y_{F (X)})(j) with some j is there, then M(Y_{F (Y)}) contains a Tate summand with the shift j. By Corollary 4.3 we necessarily have j = 0 or j ≥ 2^{n} + m − 2(𝔦_{1} + 𝔦_{2}), and the claim is proved.

By [9, Proposition 5.2], the complete decomposition of U(Y)_{F (X)} ends with a summand U(Y_{F (X)})(j) with some j ≥ 0. (We say “ends” meaning that dim U(Y)_{F (X)} = dim U(Y_{F (X)}) + j.) By the first claim, j ≠ 0. It follows by the second claim that j ≥ 2^{n} + m − 2(𝔦_{1} + 𝔦_{2}). Thus

cd_{2}(Y) = dim U(Y) = dim U(Y)_{F (X)} = dim U(Y_{F (X)})+j = cd_{2}[1](φ_{1})+j =

(2^{n} + m − 𝔦_{1} − 1) + j ≥ (2^{n} + m − 𝔦_{1} − 1) + (2^{n} + m − 2(𝔦_{1} + 𝔦_{2})) =

2^{n+1} + 2m − 3𝔦_{1} − 2𝔦_{2} − 1 = 2^{n+1} + m/2 − 2𝔦_{2} − 1 ≥ 2^{n}.

The last inequality here holds because 𝔦_{2} ≤ 2^{n−1} and m ≥ 2 (see above). The very first equality holds by Theorem 3.1. □

7. Third canonical dimension

Theorem 7.1. For any positive integer i ≤ 3 and any anisotropic quadratic form φ of dimension ≥ 2i, one has cd_{2}[i](φ) ≥ cd[i](dim φ).

Proposition 7.2. In order to prove Theorem 7.1, one only needs to show that cd_{2}[3](φ) ≥ 2^{n}−1 for φ satisfying dim φ = 2^{n}+3 (n ≥ 2) and 𝔦_{1}(φ) = 𝔦_{2}(φ) = 1.

Proof. We are reduced to the case of i = 3 and of φ of height ≥ 3 with 𝔦_{1}(φ) = 𝔦_{2}(φ) = 1 by Theorem 6.1. So, we assume that dim φ ≥ 6. Having written dim φ = 2^{n}+m with m ∈ [1, 2^{n}] (where n ≥ 2), we get

cd_{2}[3](φ) ≥ cd_{2}[2](φ_{1}) ≥ cd[2](2^{n} + m − 2) =

2^{n} − 1 = cd[3](dim φ)  provided that m ≥ 4;
2^{n−1} − 1 ≥ cd[3](dim φ)  for m = 1, 2 and
2^{n−1} − 1 < 2^{n} − 1 = cd[3](dim φ) for m = 3.

So, the only problematic value of m is 3. □

Proof of Theorem 7.1. We are showing that cd_{2}[i](φ) ≥ 2^{n} − 1 for φ as in Proposition 7.2. Let X be the quadric of φ, T the 2-grassmannian of φ, and Y its (2 + 𝔦_{3})-grassmannian, where 𝔦_{3} = 𝔦_{3}(φ) is the third Witt index of φ. We have to show that cd_{2}(Y) ≥ 2^{n} − 1. We claim that cd_{2}(Y) > cd_{2}(Y_{F (T)}). We get the claim as a consequence of Theorem 3.2 because by Theorem 4.1, U(Y)(2) is not a summand of M(X). By §4b, the complete motivic decomposition of M(Y_{F (Y)}) does not contain a Tate summand with a positive shift strictly below

dim φ − 4 − 2𝔦_{3} = 2^{n} − 1 − 2𝔦_{3}.

Since cd_{2}(Y_{F (T)}) = dim φ − 4 − 𝔦_{3} − 1 = 2^{n} − 2 − 𝔦_{3}, it follows that

cd_{2}(Y) ≥ (2^{n} − 2 − 𝔦_{3}) + (2^{n} − 1 − 2𝔦_{3}).

Therefore cd_{2}(Y) ≥ 2^{n} − 1 provided that 3𝔦_{3} ≤ 2^{n} − 2. The integer 𝔦_{3} is the first Witt index 𝔦_{1}(φ_{2}) of the anisotropic quadratic form φ_{2} (the 2-nd anisotropic kernel of φ) of dimension 2^{n} − 1. It follows by §4d that 𝔦_{3} = 2^{n−1} − 1 or 𝔦_{3} ≤ 2^{n−2} − 1. In the second case we are done and we are considering the first case below. The equality 𝔦_{3} = 2^{n−1} − 1 we are assuming now means that φ is a (2^{n} + 3)- dimensional anisotropic quadratic form of height 3 with the splitting pattern (𝔦_{1}, 𝔦_{2}, 𝔦_{3}) = (1, 1, 2^{n−1} − 1). This is actually possible only for n = 2 and n = 3 (see [18, §7.2] for n ≤ 4), but we will not use this fact because our argument will work for arbitrary n. Note that the variety Y is now the maximal grassmannian of φ. Therefore cd_{2}(Y) can be computed as in §4c in terms of the generators e_{i} ∈ Ch^{i}(Ȳ), i = 0, 1, …, e_{2^{n−1}+1}. Note that φ_{2} is a (2^{n} − 1)-dimensional form of height 1. So, φ_{2} is similar to a 1-codimensional subform of an anisotropic n-fold Pfister form. It follows by [3, Example 88.10] that e_{2^{n−1}−1} is irrational. As can be easily deduced from [3, Corollary 88.6], the homomorphism Ch(Y) → Ch(Y_{F (T)}) is surjective in codimensions ≤ 2^{n−1}−1. Consequently, if both e_{2^{n−1}} and e_{2^{n−1}+1} are rational, then cd_{2}(Y_{F (T)}) = cd_{2}(Y) contradicting the proved above claim. So, at least one of these two standard generators is irrational and it follows that cd_{2}(Y) ≥ (2^{n−1} − 1) + 2^{n−1} = 2^{n} − 1. □

8. Height 3

We prove (a bit more than) Conjecture 1.1 for all forms φ of height ≤ 3 in this Section. We recall the classification of splitting patterns of quadratic forms of height 2 first (for reader’s convenience, we include a proof):

Theorem 8.1 ([21, Theorem 2]). Let φ be a non-zero anisotropic quadratic form of height ≤ 2 over a field of characteristic ≠ 2 with a non-excellent splitting pattern. Then

(1) either dim φ = 2^{n+1} and 𝔦_{1}(φ) = 2^{n−1} = 𝔦_{2}(φ) for some n > 0 or

(2) dim φ = 2^{n} + 2^{n−1}, 𝔦_{1}(φ) = 2^{n−2}, and 𝔦_{2}(φ) = 2^{n−1} for some n > 1.

Proof. By [3, Theorem 84.1], the height of φ is at least the height of an anisotropic excellent form of dimension dim φ. Moreover, for odd dim φ this is an equality by [3, Remark 84.6]. It follows that either dim φ = 2^{n} for some n ≥ 0, or dim φ = 2^{m} − 2^{n−1} for some m > n > 1, or dim φ = 2^{m} − 2^{n} + 1 for some m > n > 1. To finish, it suffices to look at the possible values of 𝔦_{1}(φ) satisfying the condition of §4d together with the condition that dim φ − 2𝔦_{1}(φ) is 2^{r} or 2^{r+1} − 1 for some r ≥ 1. The latter condition comes from the classical [15, Theorem 5.8] giving the list of possible dimensions of height 1 anisotropic quadratic forms. □

Theorem 8.2. Let φ be an anisotropic quadratic form over F of height ≤ 3. For any positive integer i ≤ (dim φ)/2 one has cd_{2}[i](φ) ≥ cd[i](dim φ). In particular, Conjecture 1.1 holds for all φ of height ≤ 3.

Proof. By Theorem 6.1, we only need to consider φ of precisely height 3. Let n := v_{2}(dim φ).

Even-dimensional φ. We assume that n ≥ 1 here. We have to show that

cd_{2}[m](φ) ≥ 2^{n−1} − 1,

where m = (dim φ)/2. If 2^{n−1}| 𝔦_{1}, then 2^{n}| dim φ_{1} and we are done. Otherwise, by §4d, 𝔦_{1} = 2^{r} for some 0 ≤ r ≤ n − 2. Since φ_{1} is of height 2, it follows by Theorem 8.1 that dim φ = 2^{n}. If r = n − 2 then 𝔦_{1} = 2^{n−2} and 𝔦_{2} = 𝔦_{3} = 2^{n−3}. It follows by [3, Corollary 83.4] that dim φ−𝔦_{1} is a 2-power which is false. Therefore r ≤ n−3 and we have 𝔦_{2} = 2^{n−1} − 2^{r+1}, 𝔦_{3} = 2^{r}; or r = n − 3 and 𝔦_{2} = 2^{n−3}, 𝔦_{3} = 2^{n−2}. In the first case, it follows by [18, Theorem 7.7] as well as by [3, Theorem 83.3] that U(Y_{F (X)})(𝔦_{1} + 𝔦_{2}) is a summand of M(X_{F (X)}), where X is the projective quadric and Y the mgrassmannian of φ. On the other hand, U(Y)(𝔦_{1}+𝔦_{2}) is not a summand of M(X) by Theorem 4.1. It follows by [2, Theorem 1.1] that cd_{2}(Y_{F (X)}) < cd_{2}(Y). Therefore the standard generator of maximal codimension e_{2^{n−1}−1} ∈ Ch(Ȳ) is irrational and it follows that cd_{2}(Y) ≥ 2^{n−1} − 1. So, cd_{2}[m](φ) ≥ 2^{n−1} − 1 as required. In the second case, we simply have

cd_{2}(Y) = cd_{2}[2^{n−3} + 1](φ_{1}) ≥ cd[2^{n−3} + 1](2^{n−1} + 2^{n−2}) = 2^{n−1} − 1.

Odd-dimensional φ. Here we assume that n = 0. By [3, Theorem 84.1 and Remark 84.6], the height of an anisotropic excellent quadratic form of dimension dim φ is 1 or 3. In the first case we have dim φ = 2^{n} − 1 for some n ≥ 2 and we need to show that cd_{2}[2^{n−1} − 1](φ) ≥ 2^{n−1} − 1. By §4d, 𝔦_{1} = 2^{r} − 1 for some 1 ≤ r ≤ n − 1. Moreover, r ≤ n − 2 because height of φ is 3. It follows that dim φ_{1} = 2^{n}−2^{r+1}+1. Since φ_{1} is of height 2, it has an excellent splitting pattern by Theorem 8.1 so that we have 𝔦_{2} = 2^{n−1} −2^{r+1} +1 and 𝔦_{3} = 2^{r} − 1. Note that n ≥ 3 at this stage. If n = 3 then we are done by Theorem 7.1. Assuming that n ≥ 4, we claim that U(Y_{F (X)})(𝔦_{1} + 𝔦_{2}) is a summand of M(X_{F (X)}), where X is the quadric and Y the maximal grassmannian of φ. For r ≤ n − 3, this is a consequence of the inequality 𝔦_{2} > 𝔦_{3} and [18, Theorem 7.7]. For the remaining case of r = n − 2 we have 𝔦_{2} = 1 and the above argument does not work. However, Theorem 4.1 ensures that the first shell of φ is connected with the third one. Since 𝔦_{1} = 2^{r} − 1 > 𝔦_{2} = 1, the first shell is not connected with the second one, and the claim follows. Using the claim, we finish the proof of the current case the way we did it above for even-dimensional φ. It remains to consider the case when the height of an anisotropic excellent quadratic form of dimension dim φ is 3. This means that dim φ = 2^{n_{0}} − 2^{n_{1}} + 2^{n_{2}} − 1 for some integers n_{0} > n_{1} > n_{2} ≥ 2. The first Witt index 𝔦_{1} should satisfy §4d and in the same time be such that the height of the integer^{3} dim φ_{1} = dim φ − 2𝔦_{1} is 2. It follows that dim φ_{1} = 2^{n_{1}} − 2^{n_{2}} + 1 or dim φ_{1} = 2^{n_{0}} − 2^{n_{1}} + 1. In both cases we have

3As in [3, §84], by the height of a positive integer we mean the height of an anisotropic excellent quadratic form of dimension equal this integer.

Documenta Mathematica · Extra Volume Merkurjev (2015) 367–385

cd_{2}[m](φ) ≥ cd_{2}[m_{1}](φ_{1}) ≥ cd[m_{1}](dim φ_{1}) ≥ cd[m](dim φ),

where m := (dim φ − 1)/2 and m_{1} := (dim φ_{1} − 1)/2. □

9. “Counter-example” with maximal grassmannian

Surprisingly, we didn’t exclude i = (dim φ)/2 in any case of Conjecture 1.1 proved so far. So, let us produce a “counter-example” to the case i = (dim φ)/2 of Conjecture 1.1. By Theorem 7.1, i should be at least 4 and therefore dim φ should be at least 8. We produce it in dimension 8. Let us find a field F and quadratic forms q and ψ such that q is 4-dimensional of discriminant a, q_{F (\sqrt{a})} is anisotropic, ψ is 4-dimensional and divisible by ⟨ ; ⟨a⟩ ; ⟩, and, finally, φ := q⊥ψ is anisotropic. For instance, taking F := k(a, b, c, d, e) with any field k and variables a, b, c, d, e, we can take ψ = ⟨ ; ⟨a, b⟩ ; ⟩ and q = ⟨c, d, e, acde⟩. Then

cd[4](φ) = cd_{2}[4](φ) = cd[2](q_{F (\sqrt{a})}) = 1 < 3 = cd[4](dim φ).

10. Fourth canonical dimension

Theorem 10.1. Conjecture 1.1 holds for i = 4.

Proposition 10.2. It suffices to prove Theorem 10.1 only for φ of dimension 2^{n} + 4 (n ≥ 3), of height at least 4, and of Witt indexes satisfying either 𝔦_{1} = 𝔦_{2} = 𝔦_{3} = 1; or 𝔦_{1} = 1, 𝔦_{2} = 2; or 𝔦_{1} = 2, 𝔦_{2} = 1. More precisely, it suffices to prove that cd_{2}[4](φ) ≥ 2^{n} − 1 for such φ.

Proof. Note that Conjecture 1.1 for i = 4 is only about quadratic forms φ of dimension ≥ 9. We may assume that 𝔦_{1} ≤ 2 (Theorem 6.1) and that the height of φ is at least 4 (Theorem 8.2). Moreover, we may assume that 𝔦_{1} + 𝔦_{2} + 𝔦_{3} = 3 or 𝔦_{1} + 𝔦_{2} = 3 (Theorem 7.1). Therefore, we have either 𝔦_{1} = 𝔦_{2} = 𝔦_{3} = 1; or 𝔦_{1} = 1, 𝔦_{2} = 2; or 𝔦_{1} = 2, 𝔦_{2} = 1. Let us write dim φ = 2^{n} + m with n ≥ 3 and 1 ≤ m ≤ 2^{n}. Assuming that 𝔦_{1} = 1, we have

cd_{2}[4](φ) ≥ cd[3](2^{n} + m − 2) = 2^{n} − 1 = cd[4](dim φ)

for m ≥ 5. On the other hand,

cd_{2}[4](φ) ≥ cd[3](2^{n} + m − 2) = 2^{n−1} − 1 = cd[4](dim φ)

for m ≤ 3. So, the only problematic value of m is 4. Assuming that 𝔦_{1} = 2, we have

cd_{2}[4](φ) ≥ cd[2](2^{n} + m − 4) = 2^{n} − 1 = cd[4](dim φ)

for m ≥ 6. On the other hand,

cd_{2}[4](φ) ≥ cd[2](2^{n} + m − 4) = 2^{n−1} − 1 = cd[4](dim φ)

for m ≤ 3. Moreover, since 𝔦_{1} = 2, m is necessarily even (§4d). So, the only problematic value of m is again 4. □

Proof of Theorem 10.1. Let φ be a quadratic form as in Proposition 10.2. Let r be the integer ∈ {3, 4} such that 𝔦_{1} + ⋯ + 𝔦_{r−1} = 3 (more concretely, r := 3 if 𝔦_{1}+𝔦_{2} = 3, r := 4 if 𝔦_{1}+𝔦_{2}+𝔦_{3} = 3). Let X be the quadric, T the 3-grassmannian, and Y the (3 + 𝔦_{r})-grassmannian of φ. Since cd_{2}[4](φ) = cd_{2}(Y), it suffices to prove that cd_{2}(Y) ≥ 2^{n} − 1. By Theorem 4.1, the motive U(Y)(3) is not a summand of M(X). It follows by Theorem 3.2 that cd_{2}(Y) > cd_{2}(Y_{F (T)}). Now, using §4b in the standard way, we get that

cd_{2}(Y) ≥ cd_{2}(Y_{F (T)}) + (dim φ − 2(𝔦_{1} + ⋯ + 𝔦_{r})) =

(2^{n} − 3 − 𝔦_{r}) + (2^{n} − 2 − 2𝔦_{r}) = 2^{n+1} − 5 − 3𝔦_{r}.

So, the inequality cd_{2}(Y) ≥ 2^{n} − 1 holds if 2^{n+1} − 5 − 3𝔦_{r} ≥ 2^{n} − 1, or, equivalently, if

(10.3)  2^{n} ≥ 3𝔦_{r} + 4.

Since the integer 𝔦_{r} is the first Witt index of the quadratic form φ_{r−1} of dimension dim φ_{r−1} = dim φ − 6 = 2^{n} − 2, we have 𝔦_{r} = 2^{n−1} − 2 or 𝔦_{r} ≤ 2^{n−2} − 2 or 𝔦_{r} = 1 (the last case is not included in the previous one if n = 3). The inequality (10.3) does not hold only in the case of 𝔦_{r} = 2^{n−1} − 2 which we consider now. Recall that now our anisotropic quadratic form φ is of dimension 2^{n} +4 (n ≥ 3) and has the splitting pattern

either (1, 1, 1, 2^{n−1} − 2, 1), or (1, 2, 2^{n−1} − 2, 1), or (2, 1, 2^{n−1} − 2, 1).

Let d ∈ F^{×} represents the discriminant of φ. We evidently have φ_{F (\sqrt{d})} ∈ I^{n}. It follows that the Clifford algebra C(φ) is Brauer-equivalent to a quaternion algebra (c, d) with some c ∈ F^{×}. Let ψ := φ⊥c ⟨ ; ⟨d⟩ ; ⟩. Then disc(ψ) is trivial and it follows by [3, Lemma 14.2] that the Clifford invariant of ψ is trivial as well, so that ψ ∈ I^{3}. Let us show that ψ ∈ I^{n}. We know this already for n = 3. To show this for n ≥ 4, it suffices to show that ψ_{L} is hyperbolic for any extension field L/F such that dim(ψ_{L})_{an} ≤ 2^{n−1}. Since dim ψ = 2^{n} + 6, the condition on L ensures that 𝔦_{0}(ψ_{L}) ≥ 2^{n−2} + 3. Since φ is a subform in ψ of codimension 2, 𝔦_{0}(φ_{L}) ≥ 2^{n−2} + 1 which is ≥ 4 because n ≥ 4. It follows that 𝔦_{0}(φ_{L}) ≥ 4 and therefore ≥ 2^{n−1} + 1 so that dim(φ_{L})_{an} ≤ 2 and dim(ψ_{L})_{an} ≤ 4. Since the discriminant and the Clifford invariant of ψ_{L} are trivial, it follows that ψ_{L} is hyperbolic. We have shown that ψ ∈ I^{n}. On the other hand, 2^{n} + 2 ≤ dim ψ_{an} ≤ 2^{n} + 6 so that for n ≥ 4 we get a contradiction with §4e. We proved that none of the above splitting patterns of φ is possible in the case of n ≥ 4. It remains to consider the case of n = 3, that is, of dim φ = 12. The splitting patterns of 12-dimensional anisotropic quadratic forms have been classified in [18, §7.3]. In particular, it has been shown there that only the first of our three splitting patterns is possible. For φ of this possible splitting pattern (1, 1, 1, 2, 1), the above procedure provides us with an anisotropic quadratic form ψ′ := ψ_{an} ∈ I^{3} of dimension 14 or 12 such that for any extension field L/F the condition 𝔦_{0}(φ_{L}) ≥ 4 holds if and only if 𝔦_{0}(ψ′_{L}) ≥ 4 is hyperbolic. It follows that cd_{2}[4](φ) = cd_{2}[4](ψ′). Since the height of ψ′ is ≤ 3, it follows by Theorem 8.2 that cd_{2}[4](ψ′) ≥ cd[4](dim ψ′) = 7 = 2^{n} − 1. □

11. Fifth canonical dimension

Theorem 11.1. Conjecture 1.1 holds for i = 5.

Proposition 11.2. It suffices to prove Theorem 11.1 only for φ of height at least 4 and with 𝔦_{1} + ⋯ + 𝔦_{r} = 4 for some r, having one of the following types:

(1) dim φ = 2^{n} + 5 (n ≥ 3) and 𝔦_{1} = 1;

(2) dim φ = 2^{n} + 6 (n ≥ 3) and 𝔦_{1} = 2;
(3) dim φ = 2^{n} + 7 (n ≥ 3) and 𝔦_{1} = 3.

More precisely, it suffices to prove that cd_{2}[4](φ) ≥ 2^{n} − 1 for above φ.

Proof. Note that Conjecture 1.1 for i = 5 is only about quadratic forms φ of dimension ≥ 11. We may assume that 𝔦_{1} ≤ 3 (Theorem 6.1) and that the height of φ is at least 4 (Theorem 8.2). Also we may assume that 𝔦_{1} + ⋯ + 𝔦_{r} = 4 for some r (Theorem 10.1). Let us write dim φ = 2^{n} + m with n ≥ 3 and 1 ≤ m ≤ 2^{n}. Assuming that 𝔦_{1} = 1, we have

cd_{2}[5](φ) ≥ cd[4](2^{n} + m − 2) = 2^{n} − 1 = cd[5](dim φ)

for m ≥ 6. On the other hand,

cd_{2}[5](φ) ≥ cd[4](2^{n} + m − 2) = 2^{n−1} − 1 = cd[5](dim φ)

for m ≤ 4. So, the only problematic value of m is 5. Assuming that 𝔦_{1} = 2, we have

cd_{2}[5](φ) ≥ cd[3](2^{n} + m − 4) = 2^{n} − 1 = cd[5](dim φ)

for m ≥ 7. On the other hand,

cd_{2}[5](φ) ≥ cd[3](2^{n} + m − 4) = 2^{n−1} − 1 = cd[5](dim φ)

for m ≤ 4. Moreover, since 𝔦_{1} = 2, m is necessarily even (§4d). So, the only problematic value of m is 6. Finally, assuming that 𝔦_{1} = 3, we have

cd_{2}[5](φ) ≥ cd[2](2^{n} + m − 6) = 2^{n} − 1 = cd[5](dim φ)

for m ≥ 8. On the other hand,

cd_{2}[5](φ) ≥ cd[2](2^{n} + m − 6) = 2^{n−1} − 1 = cd[5](dim φ)

for m ≤ 4. Moreover, since 𝔦_{1} = 3, m is necessarily odd (§4d). So, the only problematic values of m are 5 and 7. Since 3 cannot be the first Witt index of an anisotropic quadratic form of dimension 2^{n} + 5 (§4d again), the value 5 is not possible for m. □

Proof of Theorem 11.1. Let φ be a quadratic form as in Proposition 11.2. Let r be the integer such that 𝔦_{1} + ⋯ + 𝔦_{r−1} = 4. Let X be the quadric, T the 4grassmannian, and Y the (4+𝔦_{r})-grassmannian of φ. Since cd_{2}[5](φ) = cd_{2}(Y), it suffices to prove that cd_{2}(Y) ≥ 2^{n} − 1. By Theorem 4.1, the motive U(Y)(4) is not a summand of M(X). It follows by Theorem 3.2 that cd_{2}(Y) > cd_{2}(Y_{F (T)}). Now, using §4b in the standard way, we get that

cd_{2}(Y) ≥ cd_{2}(Y_{F (T)}) + (dim φ − 2(𝔦_{1} + ⋯ + 𝔦_{r})) ≥

(2^{n} + m − 9 − 𝔦_{r}) + (2^{n} + m − 8 − 2𝔦_{r}) = 2^{n+1} + 2m − 17 − 3𝔦_{r}.

So, the inequality cd_{2}(Y) ≥ 2^{n} − 1 holds if 2^{n+1} + 2m − 17 − 3𝔦_{r} ≥ 2^{n} − 1, or, equivalently, if

(11.3)  2^{n} ≥ 3𝔦_{r} + 16 − 2m.

Since the integer 𝔦_{r} is the first Witt index of the quadratic form φ_{r−1} of dimension 2^{n} + m − 8, we have 𝔦_{r} = 2^{n−1} + m − 8 or 𝔦_{r} ≤ 2^{n−2} + m − 8. For n = 3 and m = 6, there is an additional case of 𝔦_{r} = 1. The inequality 11.3 does not hold only in the case of 𝔦_{r} = 2^{n−1} + m − 8 which we consider now. Let us start with the case of m = 5. So, φ is of dimension 2^{n} + 5 and has the splitting pattern (…, 2^{n−1} − 3, 1). First we consider the case of n = 3. In this case we have cd_{2}(Y_{F (T)}) = 3, cd_{2}(Y) ≥ 6, and §4b tells us that in the complete decomposition of M(Y_{F (Y)}) there is only one Tate summand with the shift 3. On the other hand, if cd_{2}(Y) = 6, then U(Y)_{F (T)} contains summands U(Y_{F (T)}) and U(Y_{F (T)})(3) so that there are two Tate summands with the shift 3 in the complete decomposition of M(Y_{F (Y)}). It follows that cd_{2}(Y) ≥ 7 and we are done in the case of n = 3 and m = 5. In the case of n ≥ 4 and m = 5, the splitting pattern of φ is impossible. Indeed, the anisotropic part of a (2^{n} + 6)-dimensional quadratic form of trivial discriminant containing φ is in I^{n} and has dimension 2^{n} + 6 or 2^{n} + 4. We go ahead to the case m = 7. Now φ is of dimension 2^{n} + 7 and has the splitting pattern (3, 1, 2^{n−1} − 1). This is only possible for n = 3, but anyway, the height of φ is 3 so that we don’t need to do anything more here. The remaining value of m is 6 so that dim φ = 2^{n} + 6 now. The splitting pattern of φ is either (2, 1, 1, 2^{n−1} − 2, 1) or (2, 2, 2^{n−1} − 2, 1). Adding to φ an appropriate binary quadratic form of discriminant disc(φ), we get a (2^{n} + 8)- dimensional quadratic form ψ lying in I^{3} and therefore in I^{n}. The anisotropic part of ψ has dimension 2^{n} + 8, 2^{n} + 6 or 2^{n} + 4 and it follows that n is 3 or 4. Note that for any field extension L/F, the condition 𝔦_{0}(φ_{L}) ≥ 5 is equivalent to 𝔦_{0}(ψ_{L}) ≥ 5 so that cd_{2}[5](φ) = cd_{2}[5](ψ). If n = 4, then ψ is anisotropic (of dimension 24) and of height 2. Therefore we have cd_{2}[5](ψ) ≥ cd[5](24) = 15 and the case is closed. If n = 3, then the anisotropic part ψ′ of ψ has dimension 12, 14, or 16. If dim ψ′ = 12, then cd_{2}[5](ψ) = cd_{2}[3](ψ′) ≥ cd[3](12) = 7. If dim ψ′ = 14, then cd_{2}[5](ψ) = cd_{2}[4](ψ′) ≥ cd[4](14) = 7. Finally, if dim ψ′ = 16, i.e., if ψ is anisotropic, then either the height of ψ is ≤ 3 or 𝔦_{1}(ψ) = 1. If the height is ≤ 3, then cd_{2}[5](ψ) ≥ cd[5](16) = 7. If the first Witt index is 1, then cd_{2}[5](ψ) ≥ cd_{2}[4](ψ_{1}) ≥ cd[4](14) = 7. □

Corollary 11.4. Conjecture 1.1 holds in full for φ of dimension ≤ 13.

Proof. We only need to consider cd_{2}[6](φ) for a 13-dimensional φ. But cd[6](13) = 1 so that the statement to prove is trivial. □

Remark 11.5. To prove Conjecture 1.1 for 14-dimensional φ, one “only” needs to check that cd_{2}[6](φ) ≥ 7.

12. Final comments

The material of this section has been added on the suggestion of the editors. The following proposition justifies appearance of excellent forms in the statement of Conjecture 1.1. It also answers a question raised by H. Bermudez during my talk at the International Conference on the Algebraic and Arithmetic Theory of Quadratic Forms (Puerto Natales, Patagonia, Chile) in December 2013.

Proposition 12.1. Let φ be an anisotropic quadratic form over F such that for any integer i with 1 ≤ i < (dim φ)/2, the i-th canonical dimension of φ is minimal among the i-th canonical dimensions of anisotropic quadratic forms (over field extensions of F) of dimension dim φ. Then

(1) the higher Witt indexes of φ are excellent , i.e., φ has the same height and the same higher Witt indexes as any anisotropic excellent quadratic form of the same dimension;

(2) cd[i](φ) = cd[i](dim φ), i.e. Conjecture 1.1 holds for quadratic forms of dimension dim φ;

(3) the quadric of φ has excellent motivic decomposition type;

(4) assuming an open [8, Conjecture 1.8], φ is excellent.

The statement of (3) will be explained in the proof. Since we do not know if such φ exists (in arbitrary dimension), (2) does not prove Conjecture 1.1. If the i-th canonical dimension cd[i](φ) of a given anisotropic quadratic form φ is minimal for some value of i, it is not necessarily minimal for other values of i. For instance, for any r ≥ 2 and any positive m < 2^{r−1}, we may find a field F and an m-dimensional quadratic form ψ over F such that the even Clifford algebra of ψ is a division algebra and ψ is a subform of an anisotropic r-fold Pfister form π. Then the i-th canonical dimension cd[i](φ) of the complement φ of ψ in π is minimal for i = 1, …, 𝔧_{1}(φ). For the remaining values of i however, cd[i](φ) coincides with cd[i − 𝔧_{1}(φ)](ψ) which is equal to dim X_{i−𝔧_{1}(φ)}(ψ) by [13]. In particular, cd[i](φ) is not minimal in general because cd[i](dim φ) = cd[i − 𝔧_{1}(φ)](dim ψ).

Proof of Proposition 12.1. We write 𝔧_{r} for 𝔧_{r}(φ). Since cd[1](φ) is minimal, the first Witt index of φ is excellent and (2) holds for i up to 𝔧_{1} by the results listed in §5. If we already know for some r ≥ 1 that the first r − 1 higher Witt indexes of φ are excellent and (2) holds for i up to 𝔧_{r−1}, the inequality cd[𝔧_{r−1} + 1](φ) ≥ cd[1](φ_{r−1}) (which is an equality for φ replaced by an anisotropic excellent form of the same dimension) tells us that 𝔧_{r} = 𝔧_{1}(φ_{r−1}) is excellent and (2) holds for i up to 𝔧_{r}. We proved (1) and (2) at this point. As a byproduct, we see that the above inequality is in fact an equality, which means by Theorem 3.2 that a shift (and therefore precisely 𝔧_{r} − 𝔧_{r−1} shifts) of U(X_{𝔧_{r}}) appear(s) in the complete motivic decomposition of the quadric. Having this for every r and counting the ranks of the motives over an algebraic closure, we see that each undecomposable summand of the motive of the quadric is binary, i.e. becomes over an algebraic closure a sum of two Tate motives. More precisely, every indecomposable summand looks over an algebraic closure precisely the same as the corresponding summand in the complete motivic decomposition of an anisotropic excellent quadric of the same dimension. This is what (3) means. Finally, [8, Conjecture 1.8] produces Pfister forms out of the binary motives and allows one to show that φ is excellent. In more details, since U(X_{1}) is binary, [8, Conjecture 1.8] implies that φ is a neighbor of a Pfister form π. By similar reason, the complement of φ in π is also a Pfister neighbor. Continuing this way, we eventually see that φ is excellent. □

Example 12.2. To visualize the statement of Conjecture 1.1, it is probably a good idea to draw the graph of the function i ↦ cd[i](n) for some concrete value of n. For n = 60 = 2^{6} − 2^{2}, the function is constantly 31 = 2^{6−1} − 1 on the interval [1, 28] and takes the value 1 = 2^{2−1} − 1 at 29. As for arbitrary n, it is piecewise constant (with values given by some powers of 2 minus 1) and decreasing. Conjecture 1.1 claims that for any 60-dimensional anisotropic φ, the graph of the function i ↦ cd[i](φ) is over the graph just described. We know that it is under the parabola i ↦ dim X_{i}(φ) = i(i−1)/2+i(60−2i). In contrast with the above lower bound, this piece of the parabola (constituting the upper bound for cd[i](φ)) is not monotone: it increases until 19 and decreases after 20.

One may view Conjecture 1.1 as an analogue of the Outer Excellent Connection Theorem for quadrics, where the quadrics are replaced by higher orthogonal grassmannian. Note that according to Theorem 3.1, Conjecture 1.1 is a statement about the structure of the Chow motives of higher orthogonal grassmannians. As such, it clearly affects our understanding of their Chow groups. Finally, orthogonal grassmannians constitute a special and important case of a flag variety under a semisimple algebraic group; Conjecture 1.1 is to consider in this general context.

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Nikita A. Karpenko

Mathematical & Statistical Sciences University of Alberta Edmonton, Alberta Canada 386

Documenta Mathematica · Extra Volume Merkurjev (2015)