Quadratic forms with absolutely maximal splitting

Oleg Izhboldin and Alexander Vishik

Abstract. Let F be a field and ϕ be a quadratic form over F. The higher Witt indices of ϕ are defined recursively by the rule i_{k+1}(ϕ) = i_{k}((ϕ_{an})_{F (ϕ_{an})}), where i_{0}(ϕ) = i_{W} (ϕ) is the usual Witt index of the form ϕ. We say that ϕ has absolutely maximal splitting if i_{1}(ϕ) > i_{k}(ϕ) for all k > 1.

One of the main results of this paper claims that for all anisotropic forms ϕ satisfying the condition 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}, the following three conditions are equivalent: (i) the kernel of the natural homomorphism H^{n}(F, ℤ/2ℤ) → H^{n}(F (ϕ), ℤ/2ℤ) is nontrivial, (ii) ϕ has absolutely maximal splitting, (iii) ϕ has maximal splitting (i.e., i_{1}(ϕ) = dim ϕ − 2^{n−1}). Moreover, we show that if we assume additionally that dim ϕ ≥ 2^{n} − 7, then these three conditions hold if and only if ϕ is an anisotropic n-fold Pfister neighbor. In our proof we use the technique developed by V. Voevodsky in his proof of Milnor’s conjecture.

1. Introduction

Let F be a field of characteristic ≠ 2 and let H^{n}(F) be the Galois cohomology group of F with Z/2Z-coefficients. For a given extension L/F, we denote by H^{n}(L/F) the kernel of the natural homomorphism H^{n}(F) → H^{n}(L). Now, let ϕ be a quadratic form over F. An important part of the algebraic theory of quadratic forms deals with the behavior of the groups H^{n}(F) under the field extension F (ϕ)/F. Of particular interest is the group

H^{n}(F (ϕ)/F) = ker(H^{n}(F) → H^{n}(F (ϕ))).

The computation of this group is connected to Milnor’s conjecture and plays an important role in K-theory and in the theory of quadratic forms.

The first nontrivial result in this direction is due to J. K. Arason. In [1], he computed the group H^{n}(F (ϕ)/F) for the case n ≤ 3. The case n = 4 was completely studied by Kahn, Rost and Sujatha ([12]). In the cases where n ≥ 5, there are only partial results depending on Milnor’s conjecture: the group H^{n}(F (ϕ)/F) was computed for all Pfister neighbors ([20]) and for all 4-dimensional forms ([25]). All known results make natural the following conjecture.

Conjecture 1.1. Let F be a field, n be a positive integer, and let ϕ be an F -form of dimension > 2^{n−1}. Then the following conditions are equivalent:

(1) the group H^{n}(F (ϕ)/F) = ker(H^{n}(F) → H^{n}(F (ϕ))) is nonzero,

The first author was supported by Alexander von Humboldt Stiftung.

The second author was supported by Max-Planck Institut für Mathematik.

(2) the form ϕ is an anisotropic n-fold Pfister neighbor. Moreover, if these conditions hold, then the group H^{n}(F (ϕ)/F) is isomorphic to Z/2Z and is generated by e^{n}(π), where π is the n-fold Pfister form associated with ϕ.

The proof of the implication (2)⇒(1) follows from the fact that the Pfister quadric is isotropic if and only if the corresponding pure symbol α ∈ K_{n}^{M}(k)/2 is zero, and the fact that the norm-residue homomorphism is injective on α (see [28]). The implication (1)⇒(2) seems much more difficult. In this paper, we give only partial answer to the conjecture.

1.1. Forms with maximal splitting. It turns out that Conjecture 1.1 is closely related to a conjecture concerning so called forms with maximal splitting. Let us recall some basic definition and results. By i_{W} (ϕ) we denote the Witt index of ϕ. For anisotropic quadratic form ϕ, the first Witt invariant of ϕ is defined as follows: i_{1}(ϕ) = i_{W} (ϕ_{F (ϕ)}). Since ϕ_{F (ϕ)} is isotropic, we obviously have i_{1}(ϕ) ≥ 1. In [4] Hoffmann proved the following

Theorem 1.2. Let ϕ be an anisotropic quadratic form. Let n be such that 2^{n−1} < dim ϕ ≤ 2^{n} and m be such that dim ϕ = 2^{n−1} + m. Then

• i_{1}(ϕ) ≤ m,

• if ϕ is a Pfister neighbor, then i_{1}(ϕ) = m.

This theorem gives rise to the following

Definition 1.3 (see [4]). Let ϕ be an anisotropic quadratic form. Let us write dim ϕ in the form dim ϕ = 2^{n−1} + m, where 0 < m ≤ 2^{n−1}. We say that ϕ has maximal splitting if i_{1}(ϕ) = m.

Our interest in forms with maximal splitting is motivated (in particular) by the following easy observation (which depends on Milnor’s conjecture, see Proposition 7.5): Let ϕ and n be as in Conjecture 1.1. If H^{n}(F (ϕ)/F) ≠ 0, then ϕ has maximal splitting and H^{n}(F (ϕ)/F) ≃ Z/2Z. Therefore, the problem of classification of forms with maximal splitting is closely related to Conjecture 1.1. On the other hand, there are many other problems depending on classification of forms with maximal splitting.

Let us explain known results concerning classification. By Theorem 1.2, all Pfister neighbors and all forms of dimension 2^{n} + 1 have maximal splitting. By [5], these examples present exhaustive list of forms with maximal splitting of dimension ≤ 9. The case dim ϕ = 10 is much more complicated. In [9], it was proved that a 10-dimensional form ϕ has maximal splitting only in the following cases:

• ϕ is a Pfister neighbor,

• ϕ can be written in the form ϕ = ⟨ ; ⟨a⟩ ; ⟩ q, where q is 5-dimensional form. Structure of quadratic forms with maximal splitting of dimensions 11, 12, 13, 14, 15, and 16 is very simple: they are Pfister neighbors (see [5] or [7]). Since 17 = 2^{4} +1, it follows that any 17-dimensional form has maximal splitting. The previous discussion shows that for form of dimension ≤ 17, we have complete classification of forms with maximal splitting.

Conjecture 1.1 together with our previous discussion make natural the following problem:

Problem 1.4. Find the condition on the positive integer d such that each ddimensional form ϕ with maximal splitting is necessarily a Pfister neighbor.

The following example is due to Hoffmann. Let F be the field of rational function k(x_{1}, …, x_{n−3}, y_{1}, …, y_{5}) and let

q = ⟨ ; ⟨x_{1}, …, x_{n−3}⟩ ; ⟩ ⊗ ⟨y_{1}, y_{2}, y_{3}, y_{4}, y_{5}⟩ .

Then q has maximal splitting and is not a Pfister neighbor. We obviously have dim q = 2^{n−1} + 2^{n−3}. This example gives rise to the following

Proposition 1.5. Let d be an integer satisfying condition 2^{n−1} ≤ d ≤ 2^{n−1} + 2^{n−3} for some n ≥ 4. Then there exists a field F and d-dimensional F -from ϕ with maximal splitting which is not a Pfister neighbor.

For the proof, we can define ϕ as an arbitrary d-dimensional subform of the form q constructed above.

Let us return to Problem 1.4. By Proposition 1.5, it suffices to study Problem 1.4 only in the case where 2^{n−1} + 2^{n−3} < d ≤ 2^{n}. Here, we state the following

Conjecture 1.6. Let n ≥ 3 and F be an arbitrary field. For any anisotropic quadratic F -form with maximal splitting the condition 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n} implies that ϕ is a Pfister neighbor.

This conjecture is true in the cases n = 3 and n = 4 (see [5], [7]). At the time we cannot prove Conjecture 1.6 in the case n ≥ 5. However, we prove the following partial case of the conjecture.

Theorem 1.7. Let n ≥ 5 and q be an anisotropic form such that 2^{n} − 7 ≤ dim q ≤ 2^{n}. Then the following conditions are equivalent:

(i) q has maximal splitting, (ii) q is a Pfister neighbor.

Moreover, we show that for any form ϕ satisfying the condition 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}, Conjectures 1.1 and 1.6 are equivalent for all fields of characteristic zero (here we use Milnor’s conjecture). The equivalence of conjectures follows readily from the following theorem.

Theorem 1.8. Let n be an integer ≥ 4 and F be a field of characteristic 0. Let ϕ be an anisotropic form such that 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}. Then the following conditions are equivalent:

(1) ϕ has maximal splitting,

(2) H^{n}(F (ϕ)/F) ≠ 0.

On the other hand, Theorems 1.7 and 1.8 give rise to the proof of the following partial case of Conjecture 1.1.

Corollary 1.9. Let F be a field of characteristic zero and let n ≥ 5. Then for any F -form ϕ satisfying the condition dim ϕ ≥ 2^{n} − 7, the following conditions are equivalent:

(1) the group H^{n}(F (ϕ)/F) = ker(H^{n}(F) → H^{n}(F (ϕ))) is nonzero,

(2) the form ϕ is an anisotropic n-fold Pfister neighbor.

1.2. Plan of works. In section 3, we prove Theorem 1.7. Our proof is based on the following ideas of Bruno Kahn ([10]): First of all, we recall some result of M. Knebusch: let q be an anisotropic F -form. If the F (q)-form (q_{F (q)})_{an} is defined over F, then q is a Pfister neighbor. Now, let q be an F -form satisfying hypotheses of Theorem 1.7 (in particular, dim q > 16). Let us consider the F (q)- form ϕ = (q_{F (q)}))_{an}. By the definition of forms with maximal splitting, we obviously have dim ϕ ≤ 7. By the construction, the form ϕ belongs to the image of the homomorphism W (F) → W (F (q)). This implies that ϕ belongs to the unramified part W_{nr}(F (q)) of the Witt group W (F (q)). Using some deep results concerning the group W_{nr}(F (q)), we prove that all forms of dimension ≤ 7 belonging to W_{nr}(F (q)) are necessarily defined over F (provided that dim q > 16). In particular, this implies that ϕ = (q_{F (q)})_{an} is defined over F. Then Knebusch’s theorem says that q is a Pfister neighbor. This completes the proof of Theorem 1.7.

To prove Theorem 1.8, we need Milnor’s conjecture. The implication (1)⇒(2) is the most difficult part of the theorem. To explain the plan, we introduce the notion of “forms with absolutely maximal splitting”. First, we recall that for any F -form ϕ, we can define the higher Witt indices by the following recursive rule: i_{s+1}(ϕ) = i_{s}((ϕ_{an})_{F (ϕ_{an})}).

Definition 1.10. Let ϕ be an anisotropic quadratic form. We say that ϕ has absolutely maximal splitting ^{1} if i_{1}(ϕ) > i_{r}(ϕ) for all r > 1.

It is not difficult to show, that if the form ϕ has maximal splitting and satisfies the condition 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}, then ϕ is an AMS-form. This shows, that the form ϕ satisfying the condition (1) of Theorem 1.8, is necessarily an AMS-form. Therefore, it suffices to prove the following theorem.

Theorem 1.11. Let F be a field of characteristic zero. Let ϕ be an AMS-form satisfying the condition 2^{n−1} < dim ϕ ≤ 2^{n}. Then H^{n}(F (ϕ)/F) ≠ 0.

To prove this theorem, we study the motive of the projective quadric Q corresponding to a subform q of ϕ of codimension i_{1}(ϕ) − 1. It is well known that the function fields of the forms ϕ and q are stably equivalent. Hence, it suffices to prove that H^{n}(F (q)/F) ≠ 0. In §5, we show that the motive M(Q) of the quadric Q has some specific endomorphism ω : M(Q) → M(Q) which we call the Rost projector. Let us give the definition of the later. First, we recall that the set of endomorphisms M(Q) → M(Q) is defined as CH^{d}(Q × Q), where d = dim Q. We say that ω ∈ End(M(Q)) is a Rost projector, if ω is an idempotent (ω ∘ ω = ω), and the identity ωF̄ = pt × QF̄ + QF̄ × pt holds over the algebraic closure F̄ of F. The existence of the Rost projector means that M(Q) contains a direct summand N such that N\overline{k} is isomorphic to the direct sum of the two so-called Tate-motives ℤ ⊕ ℤ(d)[2d]. The final step in the proof of theorem 1.8 is based on the following theorem.

Theorem 1.12. Let F be a field of characteristic zero. Let Q be the projective quadric corresponding to an anisotropic F -form q. Assume that Q admits a Rost projector. Then dim q = 2^{m−1} + 1 for suitable m. Moreover, H^{m}(F (q)/F) ≠ 0.

Now, it is very easy to complete the proof of the implication (1)⇒(2) of Theorem 1.8. Since 2^{m−1} < dim q ≤ 2^{m}, 2^{n−1} < dim ϕ ≤ 2^{n}, and the extensions F (ϕ)/F and F (q)/F are stably equivalent, it follows that n = m (this follows easily from Hoffmann’s theorem [4]). Therefore, H^{n}(F (ϕ)/F) = H^{m}(F (q)/F) ≠ 0. This completes the proof of the implication (1)⇒(2).

1To simplify notation, we will say that ϕ is an AMS-form

The proof of Theorem 1.12 is given in section 6. It is based on the technique developed by V.Voevodsky for the proof of Milnor’s conjecture (see [28]). All needed results of Voevodsky’s preprints are collected in Appendix A.

Acknoledgements. An essential part of this work was done while the first author was visiting the Bielefeld University and the second author was visiting the Max-Plank Institute für Mathematik. We would like to express our gratitude to these institutes for their support and hospitality. The support of the Alexander von Humboldt Foundation for the first author is gratefully acknowledged. Finally, we would like to thank Professor Ulf Rehmann who made possible the visit of the second author to the Bielefeld University where this paper was completed.

2. Notation and background

In this article we use the standard quadratic form terminology from [17],[23]. We use the notation ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ for the Pfister form ⟨1, −a_{1}⟩⊗⋯⊗⟨1, −a_{n}⟩. The n-fold Pfister forms provide a system of generators for the abelian group I^{n}(F). We recall that the Arason–Pfister Hauptsatz (APH in what follows) states that: every quadratic form over F of dimension < 2^{n} which lies in I^{n}(F) is necessarily hyperbolic; if ϕ ∈ I^{n}(F) and dim ϕ = 2^{n}, then the form ϕ is necessarily similar to a Pfister form. We use the notation e^{n} for the generalized Arason invariant ^{2}

I^{n}(F)/I^{n+1}(F) → H^{n}(F), where ⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩ ↦ (a_{1}, …, a_{n}).

The following statements describe the relationship between the Witt-ring W (F) and the cohomology H^{n}(F). They will be used extensively in the next two sections.

Theorem 2.1. (Milnor, Arason, Merkurjev, Rost, Suslin, Jacob, Szyjevski) For n ≤ 4, we have canonical isomorphisms e^{n} : I^{n}(F)/I^{n+1}(F) → H^{n}(F)

Theorem 2.2. (Arason, Kahn-Rost-Sujatha) Let 0 ≤ m ≤ n ≤ 4 and π be an m-fold Pfister form over F. Then H^{n}(F (π)/F) = e^{m}(π)H^{n−m}(F).

Theorem 2.3. (Arason, Kahn-Rost-Sujatha) Let n ≤ 4 and ρ be a form over F of dimension > 2^{n}. Then H^{n}(F (ρ)/F) = 0.

The following statement is an evident corollary of the theorems above.

Corollary 2.4. Let ρ and n be a positive integer ≤ 5. Let ξ be a form such that ξ_{F (ρ)} ∈ I^{n}(F (ρ)). Then

• if ρ is a Pfister neighbor of Pfister form π, then ξ ∈ πW (F) + I^{n}(F),

• if dim ρ > 2^{n−1}, then ξ ∈ I^{n}(F).

Remark 2.5. Actually, the restriction on the degree n here is unnecessary - see Theorem 7.3.

In section 5 we use the notation ℤ for the trivial Tate-motive (which is just the motive of a point M(Spec(k))), and ℤ(m)[2m] for the tensor power ℤ(1)[2]^{⊗m} of the Tate-motive ℤ(1)[2], where the later is defined as a complimentary direct summand to ℤ in M(ℙ^{1}) (M(ℙ^{1}) = ℤ ⊕ ℤ(1)[2]). For this reason, we use the notation Z for all groups and rings ℤ throughout the text. In section 5 we work in the classical Chow-motivic category of Grothendieck (see [3],[18],[24],[22]). We remind that in this category the group Hom(M(P), M(Q)) is naturally identified with CH^{dim(Q)}(P ×Q), for any smooth connected projective varieties P and Q over k.

2The existence of e^{n} was proven by Arason for n ≤ 3, and by Jacob-Rost/Szyjevski for n = 4.

In this connection we should mention that the motive of a completely split quadric P (of dimension d) is a direct sum of Tate-motives:

M(P) = ⊕_{0≤i≤d}ℤ(i)[2i], if d is odd;

M(P) = (⊕_{0≤i≤d}ℤ(i)[2i]) ⊕ ℤ(d/2)[d], if d is even.

The corresponding mutually orthogonal projectors in End(M(P)) are given by h^{i} × l_{i}, and l_{i} × h^{i}, where 0 ≤ i < d/2, and h^{i} ⊂ P is a plane section of codimension i, and l_{i} ⊂ P is a projective subspace of dimension i (in the case d-even we also have l_{d/2}^{1} × l_{d/2}^{2} and l_{d/2}^{2} × l_{d/2}^{1}, where l_{d/2}^{1}, l_{d/2}^{2} ⊂ P are the projective subspaces of the middle dimension from the two different families).

In section 6 we work in the bigger triangulated category of motives DM_{−}^{eff}(k) constructed by V.Voevodsky (see [26]). This category contains the category of Chow-motives as a full additive subcategory closed with respect to direct summands. All the necessary facts and references are given in the Appendix.

3. Descent problem and Forms with maximal splitting

The main goal of this section is to prove Theorem 1.7. It should be noticed that in all cases except for dim q = 2^{n} − 7 this theorem was proved earlier:

• if dim q = 2^{n} or 2^{n} − 1, the theorem was proved by M. Knebusch and A. Wadsworth (independently);

• if dim q = 2^{n} − 2 or 2^{n} − 3, the theorem was proved by D. Hoffmann [4];

• if dim q = 2^{n} − 4 or 2^{n} − 5, the theorem was proved by B. Kahn [10, remerk after Th.4] (see also more elementary proofs in [5] or [7]);

• In the case dim q = 2^{n} − 6, the theorem follow easily from a result of A.

Laghribi [16]. To prove the theorem in the case dim q = 2^{n} − 7 we use the same method as in the paper of Bruno Kahn [10]. Namely, we reduce Theorem 1.7 to the study of a descent problem for quadratic forms (see Proposition 3.8 and Theorem 3.9). As in the paper of B. Kahn, we work modulo suitable power I^{n}(F) of the fundamental ideal I(F).

We start with the following notation.

Definition 3.1. Let ψ be a form over F and n ≥ 0 be an integer. We define dim_{n} ψ as follows:

dim_{n} ψ = min{dim ϕ | ϕ ≡ ψ (mod I^{n}(F))}

Lemma 3.2. Let ψ be a form over F and L/F be some field extension. Then dim_{n} ψ_{L} ≤ dim_{n} ψ. If L/F is unirational, then dim_{n} ψ_{L} = dim_{n} ψ.

Proof. The inequality dim_{n} ψ_{L} ≤ dim_{n} ψ is obvious. If L/F is unirational, the identity dim_{n} ψ_{L} = dim_{n} ψ follows easily from the standard specialization arguments.

Corollary 3.3. Let ψ and q_{0} ⊂ q be forms over F. Then dim_{n} ψ_{F (q_{0})} ≤ dim_{n} ψ_{F (q)}.

Proof. Since q_{0} ⊂ q, it follows that q_{F (q_{0})} is isotropic and hence the extension F (q, q_{0})/F (q_{0}) is purely transcendental. By Lemma 3.2, we have dim_{n} ψ_{F (q_{0})} = dim_{n} ψ_{F (q,q_{0})} ≤ dim_{n} ψ_{F (q)}.

Now, we recall an evident consequence of Merkurev’s index reduction formula: if A is a central simple algebra of index 2^{n} and q is a form of dimension > 2n + 2, then ind A_{F (q)} = ind A. The following lemma is an obvious generalization of this statement.

Lemma 3.4. Let A be a central simple F -algebra of index 2^{n} and q be a quadratic form over F. Let F_{0} = F, F_{1}, …, F_{h} be the generic splitting tower of q. Let i ≥ 1 be an integer such that dim((q_{F_{i−1}})_{an}) > 2n + 2. Then ind A_{F_{i}} = ind A.

Lemma 3.5. Let A be a central simple F -algebra of index 2^{n} and q be a quadratic form of dimension > 2n + 4. Then there exists a unirational extension E/F and a 3-dimensional form q_{0} ⊂ q_{E} such that ind(A_{E} ⊗ C_{0}(q_{0})) = 2^{n+1}

Proof. Let F̃ = F (X, Y, Z), q̃ = qF̃ ⊥ −X ⟨ ; ⟨Y, Z⟩ ; ⟩, and Ã = AF̃ ⊗ (Y, Z).

Clearly, ind Ã = 2 ind A = 2^{n+1}. Let F̃_{0} = F̃, F̃_{1}, …, F̃_{h} be the generic splitting tower for q̃. Let q̃_{i} = (q̃F̃_{i})_{an} for i = 0, …, h. Let s be the minimal integer such that dim q̃_{s} ≤ dim q − 2. We have dim q̃_{s−1} ≥ dim q > 2(n + 1) + 2. By Lemma 3.4, we have ind AF̃_{s} = ind Ã = 2^{n+1}.

We set E = F̃_{s}. Since q̃_{E} = q_{E} ⊥ −X ⟨ ; ⟨Y, Z⟩ ; ⟩, the forms q_{E} and X ⟨ ; ⟨Y, Z⟩ ; ⟩

contain a common subform of dimension

\frac{1}{2}(dim q + dim(X ⟨ ; ⟨Y, Z⟩ ; ⟩) − dim(q̃_{E})_{an}) = \frac{1}{2}(dim q + 4 − dim q̃_{s})

≥ \frac{1}{2}(dim q + 4 − (dim q − 2)) = 3.

Hence, there exists a 3-dimensional E-form q_{0} such that q_{0} ⊂ q_{E} and q_{0} ⊂X⟨ ; ⟨Y, Z⟩ ; ⟩_{E}. Clearly, C_{0}(q_{0}) = (Y, Z). Hence, ind(A_{E} ⊗_{E} C_{0}(q_{0})) = ind Ã_{E} = 2^{n+1}.

To complete the proof, it suffices to show that E/F is unirational. To prove this, let us write q in the form q = x ⟨1, −y, −z⟩ ⊥ q_{0} with x, y, z ∈ F^{∗}. Let us consider the field

K = F̃(\sqrt{X/x}, \sqrt{Y/y}, \sqrt{Z/z}) = F (X, Y, Z)(\sqrt{X/x}, \sqrt{Y/y}, \sqrt{Z/z}).

Clearly, K/F is purely transcendental. In the Witt ring W (K), we have q̃_{K} =

q_{K} − X ⟨ ; ⟨Y, Z⟩ ; ⟩_{K} = x ⟨1, −y, −z⟩_{K} + q_{0} − X ⟨1, −Y, −Z, −Y Z⟩ = q_{0} + ⟨XY Z⟩.

Hence dim(q̃_{K})_{an} ≤ dim q_{0} + 1 = dim q − 3 + 1 = dim q − 2. Since s is the minimal integer such that dim q̃_{s} ≤ dim q − 2, it follows that the extension (K · F̃_{s})/K is purely transcendental (see, e.g., [14, Cor. 3.9 and Prop. 5.13]), where K · F̃_{s} is the free composite of K and F̃_{s} over F̃. Since K/F is purely transcendental, it follows that (K · F̃_{s})/F is also purely transcendental. Hence F̃_{s}/F is unirational. Since E = F̃_{s}, we are done.

Lemma 3.6. Let ρ be a Pfister neighbor of ⟨ ; ⟨a, b⟩ ; ⟩ and n be a positive integer. Let ψ be a form such that dim_{n} ψ_{F (ρ)} < 2^{n−1}. Then there exist an F -forms μ such that dim μ = dim_{n} ψ_{F (ρ)} and ψ_{F (ρ)} ≡ μ_{F (ρ)} (mod I^{n}(F)).

Proof. Let ξ be a F (ρ)-form such that dim ξ = dim_{n} ψ_{F (ρ)} and ψ_{F (ρ)} ≡ ξ (mod I^{n}(F (ρ))). By [16, Lemme 3.1], we have ξ ∈ W_{nr}(F (ρ)/F). By the excellent property of F (ρ)/F (see [2, Lemma 3.1]), there exists an F -form μ such that ξ = μ_{F (ρ)}.

Corollary 3.7. Let ρ be a Pfister neighbor of ⟨ ; ⟨a, b⟩ ; ⟩ and n be a positive integer such that n ≤ 5. Let ψ be a form such that dim_{n} ψ_{F (ρ)} < 2^{n−1}. Then there exist an F -forms μ and γ such that dim μ = dim_{n} ψ_{F (ρ)} and ψ ≡ μ + ⟨ ; ⟨a, b⟩ ; ⟩ γ (mod I^{n}(F)).

Proof. Let μ be a form as in Lemma 3.6. We have (ψ − μ)_{F (ρ)} ∈ I^{n}(F (ρ)). By Corollary 2.4, we have ψ −μ ∈ ⟨ ; ⟨a, b⟩ ; ⟩ W (F)+I^{n}(F). Hence, there exists γ such

that ψ − μ ∈ ⟨ ; ⟨a, b⟩ ; ⟩ γ + I^{n}(F).

Proposition 3.8. Let q be an F -form of dimension > 16 and ψ be a form over F such that dim_{5} ψ_{F (q)} ≤ 7. Then dim_{5} ψ = dim_{5} ψ_{F (q)}. In particular, dim_{5} ψ ≤ 7.

Proof. By Lemma 3.2, we have dim_{5} ψ ≥ dim_{5} ψ_{F (q)}. Hence, it suffices to verify that dim_{5} ψ ≤ dim_{5} ψ_{F (q)}. We start with the following case:

Case 1. either dim_{5} ψ_{F (q)} ≤ 6 or dim_{5} ψ_{F (q)} = 7 and ind C_{0}(ψ) ≠ 8.

By the definition of dim_{5} ψ_{F (q)}, there exists an F (q)-form ϕ such that dim ϕ = dim_{5} ψ_{F (q)} ≤ 7 and ϕ ≡ ψ_{F (q)} (mod I^{5}(F (q))). In particular, we have c(ϕ) = c(ψ_{F (q)}). Since dim ϕ ≤ 7 and dim q > 16, index reduction formula shows that ind c(ϕ) = ind c(ψ). By the assumption of Case 1, we see that

• either dim ϕ ≤ 6,

• or dim ϕ = 7 and ind C_{0}(ϕ) ≠ 8. Since ϕ ≡ ψ_{F (q)} (mod I^{5}(F (q))), it follows that ϕ ∈ im(W (F) → W (F (q))) + I^{5}(F (q)). The principal theorem of [16] shows that ϕ is defined over F. In other words, there exists an F -form μ such that ϕ = μ_{F (q)}. Therefore, ψ_{F (q)} ≡ ϕ ≡ μ_{F (q)} (mod I^{5}(F (q)). By Corollary 2.4, we see that ψ ≡ μ (mod I^{5}(F)). Hence, dim_{5}(ψ) ≤ dim μ = dim ϕ = dim_{5}(ψ_{F (q)}). This completes the proof in Case 1.

Case 2. dim_{5} ψ_{F (q)} = 7 and ind C_{0}(ψ) = 8.

Lemma 3.2 shows that we can change the ground field by an arbitrary unirational extension. After this, Lemma 3.5 (applying to A = C_{0}(ψ), n = 3 and q) shows, that we can assume that there exists a 3-dimensional subform q_{0} ⊂ q such that ind(C_{0}(ψ) ⊗ C_{0}(q_{0})) = 16.

Let a, b ∈ F^{∗} be such that q_{0} is a Pfister neighbor of ⟨ ; ⟨a, b⟩ ; ⟩.

By Corollary 3.3, we have dim_{5} ψ_{F (q_{0})} ≤ 7. By Corollary 3.7, there exists a form μ of dimension ≤ 7 and a form λ such that ψ ≡ μ + ⟨ ; ⟨a, b⟩ ; ⟩ λ (mod I^{5}(F)).

First, consider the case where dim λ is odd. Then c(ψ) = c(μ)+(a, b). Therefore ind C_{0}(μ) = ind(C_{0}(ψ) ⊗ (a, b)) = ind(C_{0}(ψ) ⊗ C_{0}(q_{0})) = 16. On the other hand, dim μ ≤ 7 and hence ind C_{0}(μ) ≤ 8. We get a contradiction.

Now, we can assume that dim λ is even. Then λ ≡ ⟨ ; ⟨c⟩ ; ⟩ (mod I^{2}(F)), where

c = d_{±}λ. Hence, ⟨ ; ⟨a, b⟩ ; ⟩ λ ≡ ⟨ ; ⟨a, b, c⟩ ; ⟩ (mod I^{4}(F)). Hence, ψ − μ ≡ ⟨ ; ⟨a, b⟩ ; ⟩ λ ≡

⟨ ; ⟨a, b, c⟩ ; ⟩ (mod I^{4}(F)). Let π = ⟨ ; ⟨a, b, c⟩ ; ⟩. We have ψ ≡ μ + π (mod I^{4}(F)).

Since π_{F (π)} is hyperbolic, it follows that ψ_{F (q,π)} ≡ μ_{F (q,π)} (mod I^{4}(F (q, π)). Since dim_{5} ψ_{F (q)} = 7, there exists a 7-dimensional F (q)-form ξ such that ψ_{F (q)} ≡ ξ (mod I^{5}(F (q))). This implies that μ_{F (q,π)} ≡ ψ_{F (q,π)} ≡ ξ_{F (q,π)} (mod I^{4}(F (q, π))). Since dim μ + dim ξ ≤ 7 + 7 = 14 < 2^{4}, APH shows that μ_{F (q,π)} = ξ_{F (q,π)}. Hence, ψ_{F (q,π)} ≡ ξ_{F (q,π)} ≡ μ_{F (q,π)} (mod I^{5}(F (q, π))). Since dim q > 16, Corollary 2.4 shows that ψ_{F (π)} ≡ μ_{F (π)} (mod I^{5}(F (π))). Hence (ψ − μ)_{F (π)} ∈ I^{5}(F (π)).

By Corollary 2.4, there exists an F -form γ such that ψ − μ ≡ πγ (mod I^{5}(F)). Since ψ − μ ≡ π (mod I^{4}(F)), it follows that either π is hyperbolic or dim γ is odd. In any case, we can assume that dim γ is odd. Then γ ≡ ⟨k⟩ (mod I^{2}(F)), where k = d_{±}γ. Hence, ψ − μ ≡ πγ ≡ kπ (mod I^{5}(F)). Therefore, ξ ≡ ψ_{F (q)} ≡ (μ + kπ)_{F (q)} (mod I^{5}(F (q))). Since dim ξ + dim μ + dim π = 7 + 7 + 8 < 2^{5}, APH shows that ξ = (ζ_{F (q)})_{an}, where ζ = (μ ⊥ kπ)_{an}. Since dim ζ ≤ dim(μ ⊥ kπ) ≤ 7 + 8 < 2^{4} < dim q, Hoffmann’s theorem shows that ζ_{F (q)} is anisotropic. Hence, ξ = ζ_{F (q)}. In particular, dim ζ = 7. We have ψ_{F (q)} ≡ ξ ≡ ζ_{F (q)} (mod I^{5}(F (q))). Since dim q > 16, Corollary 2.4 shows that ψ ≡ ζ (mod I^{5}(F)). Hence, dim_{5} ψ ≤ dim ζ = 7. On the other hand, dim_{5} ψ ≥ dim_{5} ψ_{F (q)} = 7. The proof is complete.

The essential part of the following theorem was proved by Ahmed Laghribi in [16].

Theorem 3.9. (cf. [16, Théorème principal]). Let q be a form of dimension > 16. Let ϕ be a form of dimension ≤ 7 over the field F (q). Then the following conditions are equivalent.

(1) ϕ is defined over F,

(2) ϕ ∈ im(W (F) → W (F (q))),

(3) ϕ ∈ im(W (F) → W (F (q))) + I^{5}(F (q)),

(4) ϕ ∈ W_{nr}(F (q)/F).

Proof. This theorem is proved in [16] except for the case where dim ϕ = 7 and ind C_{0}(ϕ) = 8. Implications (1)⇒(2)⇒(3) ⟺ (4) are also proved in [16]. It suffices to prove implication (3)⇒(1).

Condition (3) shows that there exists a form ψ over F such that ψ_{F (q)} ≡ ϕ (mod I^{5}(F (q))). Therefore dim_{5} ψ_{F (q)} ≤ dim ϕ ≤ 7. By Proposition 3.8, we have dim_{5} ψ ≤ 7. Hence there exists an anisotropic F -form μ of dimension ≤ 7 such that ψ ≡ μ (mod I^{5}(F)). Thus ϕ ≡ ψ_{F (q)} ≡ μ_{F (q)} (mod I^{5}(F (q))). Since dim ϕ + dim μ = 7 + 7 < 2^{5}, APH shows that ϕ_{an} = (μ_{F (q)})_{an}. Since dim μ < 8 < dim q, Hoffmann’s theorem shows that μ_{F (q)} is anisotropic. Hence ϕ_{an} = μ_{F (q)}. Therefore ϕ_{an} is defined over F. Hence, ϕ is defined over F.

Proof of theorem 1.7. (i)⇒(ii). Let ϕ = (q_{F (q)})_{an}. By [15, Th. 7.13], it suffices to prove that ϕ is defined over F. Since n ≥ 5, we have dim q ≥ 2^{n} −7 > 16. Clearly, ϕ ∈ im(W (F) → W (F (q))). Since q has maximal splitting, it follows that dim ϕ = 2^{n} − dim q ≤ 7. By Theorem 3.9, we see that ϕ is defined over F.

(ii)⇒(i). Obvious.

4. Elementary properties of AMS-forms

In this section we start studying the forms with absolutely maximal splitting (AMS-forms) defined in the introduction (Definition 1.10).

Lemma 4.1. Let ϕ be an anisotropic form and n be an integer such that 2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}. Suppose that ϕ has maximal splitting. Then ϕ has absolutely maximal splitting.

Proof. Let m = dim ϕ − 2^{n−1}. Clearly, dim ϕ = 2^{n−1} + m and 2^{n−3} < m ≤ 2^{n−1}. Since ϕ has maximal splitting, we have i_{1}(ϕ) = m. Let F = F_{0}, F_{1}, …, F_{h} be the generic splitting tower of ϕ. Let ϕ_{i} = (ϕ_{F_{i}})_{an} for i = 0, …, h. Let us fix r > 1. To prove that ϕ has absolutely maximal splitting, we need to verify that i_{r}(ϕ) < m. Clearly, i_{r}(ϕ) = i_{1}(ϕ_{r}). Thus, we need to verify that i_{1}(ϕ_{r}) < m. In the case where dim ϕ_{r} ≤ 2^{n−2}, we have i_{1}(ϕ_{r}) ≤ \frac{1}{2} dim ϕ_{r} ≤ \frac{1}{2}2^{n−2} = 2^{n−3} < m.

Thus, we can suppose that dim ϕ_{r} > 2^{n−2}. Since r ≥ 1, we have dim ϕ_{r} ≤ dim ϕ_{1} = dim ϕ − 2i_{1}(ϕ) = 2^{n−1} + m − 2m = 2^{n−1} − m. Hence, 2^{n−2} < dim ϕ_{r} ≤ 2^{n−2} + (2^{n−2} − m). By Theorem 1.2, we have i_{1}(ϕ_{r}) ≤ 2^{n−2} − m. Since m > 2^{n−3}, we have 2^{n−2} − m < m. Hence i_{1}(ϕ_{r}) ≤ 2^{n−2} − m < m.

From the results proven in the next sections (see Theorem 7.1) it follows that in dimension range we are interested in (2^{n−1} + 2^{n−3} < dim ϕ ≤ 2^{n}) the form has maximal splitting if and only if it has absolutely maximal splitting.

Remark 4.2. We cannot change the strict inequality 2^{n−1} + 2^{n−3} < dim ϕ by 2^{n−1} + 2^{n−3} ≤ dim ϕ in the formulation of the lemma. Indeed, for any n ≥ 3 there exists an example of (2^{n−1}+2^{n−3})-dimensional form ϕ with maximal splitting which is not an AMS-form. The simplest example is the following:

ϕ = ⟨ ; ⟨x_{1}, x_{2}, …, x_{n−3}⟩ ; ⟩ ⊗ ⟨1, 1, 1, 1, 1⟩ over the field ℝ(x_{1}, …, x_{n−3}).

In this case i_{1}(ϕ) = i_{2}(ϕ) = 2^{n−3}.

5. Motivic decomposition of AMS-Quadrics

In this section we will produce some “binary” motive related to AMS-quadric. Let X, Y and Z be smooth projective varieties over k of dimensions l, m and

n, respectively. Then we have natural (associative) pairing:

∘ : CH^{n+b}(Y × Z) ⊗ CH^{m+a}(X × Y) → CH^{n+a+b}(X × Z),

where v ∘ u := (π_{X,Z})_{∗}(π_{X,Y}^{∗} (u) ∩ π_{Y,Z}^{∗}(v)), and π_{X,Y} : X × Y × Z → X × Y, π_{Y,Z} : X × Y × Z → Y × Z, π_{X,Z} : X × Y × Z → X × Z are natural projections. In particular, taking X = Spec(k), we get a pairing:

CH^{n+b}(Y × Z) ⊗ CH_{r}(Y) → CH_{r−b}(Z).

In this case, we will denote v ∘ u as v(u).

Theorem 5.1. (cf. [25, Proof of Statement 6.1]) Let Q be an AMS-quadric. Let P ⊂ Q be subquadric of codimension = i_{1}(q) − 1. Then P possesses a Rost projector.

Proof. We say that “we are in the situation (∗)”, if we have the following data:

Q - some quadric; P ⊂ Q - some subquadric of codimension d; Φ ∈ CH^{m}(Q × Q), where m := dim(P).

In this case we have Ψ ∈ CH^{m+d}(P × Q) - the class of the graph of the natural embedding P ⊂ Q, and Ψ^{∨} ∈ CH^{m+d}(Q × P) - the dual cycle, and can define: ε := Ψ^{∨} ∘ Φ ∘ Ψ ∈ CH^{m}(P × P).

The action on CH_{∗}(P\overline{k}) identifies: CH^{m}(P\overline{k} × P\overline{k}) = ∏_{r} End(CH_{r}(P\overline{k})) (see [22, Lemma 7]), and we will denote as ε_{(r)} ∈ End(CH_{r}(P\overline{k})) the corresponding coordinate of ε\overline{k}.

• If 0 ≤ s < m/2, then CH_{s}(P\overline{k}) = Z with the generator l_{s} - the class of projective subspace of dimension s on P\overline{k} ;

• if m/2 < s ≤ m, then CH_{s}(P\overline{k}) = Z with the generator h^{m−s} - the class of plane section of codimension m − s on P\overline{k} ;

• if s = m/2, then CH_{s}(P\overline{k}) = Z ⊕ Z with the generators l_{m/2}^{1} and l_{m/2}^{2} - the classes of m/2-dimensional projective subspaces from two different families.

This permits to identify End(CH_{s}(P\overline{k})) with Z if 0 ≤ s ≤ m, s ≠ m/2, and with Mat_{2×2}(Z), if s = m/2. We should mention, that since for arbitrary field extension E/k, the natural map CH_{s}(P |\overline{k}) → CH_{s}(P |\overline{E}) is an isomorphism (preserving the generators above), we have an equality: (ε_{E})_{(s)} = ε_{(s)} (in Z, resp. Mat_{2×2}(Z)).

We will need the following easy corollary of Springer’s theorem. Under the degree of the cycle A ∈ CH_{s}(Q) we will understand the degree of the 0-cycle A∩h^{s}.

Lemma 5.2. Let 0 ≤ s ≤ dim(Q)/2. Then the following conditions are equivalent:

(1) q = (s + 1) · ℍ ⊥ q′, for some form q′;

(2) Q contains (projective subspace) ℙ^{s} as a subvariety;

(3) CH_{s}(Q) contains cycle of odd degree.

Proof. (1) ⇒ (2) and (2) ⇒ (3) - evident. (3) ⇒ (1) Use induction on s. For s = 0 the statement is equivalent to the Theorem of Springer. If now s > 0, then CH_{0}(Q) also contains a cycle of odd degree (obtained via intersection with h^{s}). So, q = ℍ ⊥ q′′. And we have the natural degree preserving isomorphism: CH_{s}(Q) = CH_{s−1}(Q′′). By induction, q′′ = s · ℍ ⊥ q′.

Lemma 5.3. In the situation of (∗), suppose, for some 0 ≤ s < m/2, ε_{(s)} ∈ Z is odd. Then for arbitrary field extension E/k, if Q_{E} contains a projective space of dimension s, then it contains a projective space of dimension s + d.

Proof. Let E/k be such an extension that l_{s} ∈ image(CH_{s}(Q_{E}) → CH_{s}(Q\overline{E})), and ε_{(s)} - odd. We have: Ψ ∘ Ψ^{∨} ∘ Φ(l_{s}) = λ · l_{s} ⊂ CH_{s}(Q\overline{E}), where λ ∈ Z is odd (since Ψ : CH_{s}(P\overline{E}) → CH_{s}(Q\overline{E}) is an isomorphism). On the other hand, the composition Ψ ∘ Ψ^{∨} : CH_{s+d}(Q\overline{E}) → CH_{s}(Q\overline{E}) is given by the intersection with the plane section of codimension d, so it preserves the degree of the cycle. This implies that Φ(l_{s}) ∈ image(CH_{s+d}(Q_{E}) → CH_{s+d}(Q\overline{E})) has odd degree. By Lemma 5.2, Q_{E} contains projective space of dimension s + d.

Lemma 5.4. In the situation of (∗), suppose, for some m/2 < s ≤ m, ε_{(s)} ∈ Z is odd. Then for arbitrary field extension E/k, if Q_{E} contains a projective space of dimension (m − s), then it contains a projective space of dimension (m − s + d).

Proof. Consider the cycle ε^{∨} ∈ CH^{m}(P × P) dual to ε. Since (A ∘ B)^{∨} = B^{∨} ∘ A^{∨}, we have: ε^{∨} = Ψ^{∨} ∘ Φ^{∨} ∘ Ψ. On the other hand, (ε^{∨})_{(s)} = ε_{(m−s)}. Now, the statement follows from Lemma 5.3.

Lemma 5.5. In the situation of (∗), if d > 0, then ε\overline{k}(l_{m/2}^{1}) = ε\overline{k}(l_{m/2}^{2}) = c · h^{m/2}, where c ∈ Z.

Proof. Clearly, Ψ(l_{m/2}^{i}) = l_{m/2} ∈ CH_{m/2}(Q\overline{k}). On the other hand, Φ ∘ Ψ(l_{m/2}^{i}) ∈ CH_{m/2+d}(Q\overline{k}), the later group is generated by h^{m/2} (since (m/2) + d> (m + d)/2), and Ψ^{∨}(h^{m/2}) = h^{m/2}.

Let now Q be AMS-quadric, and P ⊂ Q be subquadric of codimension i_{1}(q)−1. By the definition of AMS-quadrics, either dim(Q) = 0, or i_{1}(q) > 1 and P is a proper subform of Q. Clearly, it is enough to consider the second possibility.

By the definition of i_{1}(q), we have: q_{k(Q)} = i_{1}(q) · ℍ ⊥ q_{1}. So, the quadric Q_{k(Q)} contains (i_{1}(q) − 1)-dimensional projective subspace l_{(i_{1}(q)−1)}. Denote: d:= i_{1}(q) − 1, and m := dim(P). Let Φ ∈ CH^{m}(Q × Q) be the class of the closure of l_{d} ⊂ Spec(k(Q)) × Q ⊂ Q × Q. Let us denote this particular case of (∗) as (∗∗).

Lemma 5.6. In the situation of (∗∗), ε\overline{k} = P\overline{k} × l_{0} + ∑_{0<i<m} b_{i} · (h^{m−i} × h^{i}) + a · l_{0} × P\overline{k}, where b_{1}, …, b_{m−1}, a ∈ Z.

Proof. If for some 0 ≤ i < m/2, the coordinate ε_{(i)} is odd, then by Lemma 5.3, in the generalized splitting tower k = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{h} for the quadrics Q, there exists 0 ≤ t < h such that i_{W} (q_{F_{t}}) ≤ i < i + i_{1}(q) − 1 < i_{W} (q_{F_{t+1}}). Since q is AMS-form, this can happen only if i = 0. In the same way, using Lemma 5.4, we get that for all m/2 < i < m, the coordinates ε_{(i)} are even.

This implies that on the group CH_{i}(P\overline{k}), where 0 < i < m, i ≠ m/2, the map ε\overline{k} acts as some (integral) multiple of h^{m−i} × h^{i} (notice also that h^{m−i} × h^{i} acts trivially on all CH_{j}(P\overline{k}), j ≠ i). The same holds for i = m/2 by Lemma 5.5.

Clearly, P\overline{k} ×l_{0} (resp. l_{0} ×P\overline{k}) acts on CH_{0}(P\overline{k}) (resp. CH_{m}(P\overline{k})) as a generator of End(CH_{0}(P\overline{k})) = Z (resp. End(CH_{m}(P\overline{k})) = Z), and acts trivially on CH_{j}(P\overline{k}), j ≠ 0 (resp. j ≠ m). So, we need only to observe that ε_{(0)} = 1 (since Ψ^{∨}∘Φ∘Ψ(l_{0}) = Ψ^{∨}∘Φ(l_{0}) = Ψ^{∨}(l_{d}) = l_{0})(this is the only place where we use the specifics of Φ).

Now we can use ε to construct the desired projector in End(M(P)), where M(P) is a motive of the quadric P, considered as an object of the classical Chowmotivic category of Grothendieck Chow^{eff}(k) (see [3],[18],[24],[22]). We remind that End(M(P)) is naturally identified with CH^{m}(P × P) with the composition given by the pairing ∘.

Take ω := ε − ∑_{0<i<m} b_{i} · (h^{i} × h^{m−i}) − [a/2] · (h^{m} × P) ∈ End(M(P)). Then ω\overline{k} is a projector equal to either (P\overline{k} × l_{0} + l_{0} × P\overline{k}), or to P\overline{k} × l_{0} (depending on the oddness of a).

We have the following easy consequence of the Rost Nilpotence Theorem ([22, Corollary 10]):

Lemma 5.7. ([25, Lemma 3.12]) If for some ω ∈ End(M(P)), ω\overline{k} is an idempotent, then for some r, ω^{2^{r}} is an idempotent.

Mutually orthogonal idempotents P\overline{k} ×l_{0} and l_{0} ×P\overline{k} give the direct summands ℤ and ℤ(m)[2m] in M(P\overline{k}). By lemma 5.7, we get a direct summand L in M(P) such that either L\overline{k} = ℤ ⊕ ℤ(m)[2m], or L\overline{k} = ℤ. The later possibility is excluded by the following lemma.

Lemma 5.8. Let L be a direct summand of M(P), s.t. L\overline{k} ≃ ℤ. Then P is isotropic.

Proof. Let w ∈ CH^{dim(P)}(P × P) be the projector, corresponding to L. We have: End(M(P\overline{k})) = ∏_{r} End(CH_{r}(P\overline{k})). So, if dim(P) > 0, then the restriction w\overline{k} of our projector to \overline{k} has no choice but to be P\overline{k} × l_{0} ∈ CH^{dim(P)}(P\overline{k} × P\overline{k}). Then, evidently, degree(w ∩ Δ_{P}) = 1, and on P × P, and, consequently, on P we get a point of odd degree. By Springer’s Theorem, P is isotropic. If dim(P) = 0, then End(M(P)) has nontrivial projector if and only if det_{±}(p) = 1 (⇔ p is isotropic).

Theorem 5.1 is proven.

6. Binary direct summands in the motives of quadrics

The following result was proven (but not formulated) by the second author in his thesis (see the proof of Statement 6.1 in [25]). We will reproduce it’s proof here for the reader’s convenience.

Theorem 6.1. ([25]) Let k be a field of characteristic 0, and P be smooth anisotropic projective quadric of dimension n whose Chow-motive M(P) contains a direct summand N such that N|\overline{k} = ℤ ⊕ ℤ(n)[2n]. Then n = 2^{s} − 1 for some s.

Proof of Theorem 6.1. The construction we use here is very close to that used by V.Voevodsky in [28].

The category of Chow-motives Chow^{eff}(k) which we used in the previous section is a full additive subcategory (closed under taking of direct summands) in the triangulated category DM_{−}^{eff}(k) - see [26]. The category DM_{−}^{eff}(k) contains the “motives” of all smooth simplicial schemes over k. If P is a smooth projective variety over k, we denote as Č(P)^{·} the standard simplicial scheme corresponding to the pair P → Spec(k) - see Definition A.8. We will denote it’s motive as X_{P}.

From the natural projection: Č(P)^{·} →^{pr} Spec(k), we get a map: X_{P} ⟶^{M(pr)} ℤ. By Theorem A.9, M(pr)\overline{k} is an isomorphism. From this point, we will denote M(pr) simply as pr (since we will not use simplicial schemes themselves any more).

By Theorem A.11, we get that in DM_{−}^{eff}(k),

N := Cone[−1](X_{P} ⟶^{μ′} X_{P} (n)[2n + 1]),

where μ′ is some (actually, the only) nontrivial ^{3} element from

Hom(X_{P} , X_{P} (n)[2n + 1]).

By Theorem A.14, pr : X_{P} → ℤ induces the natural isomorphism for all a, b:

pr_{∗} : Hom(X_{P} , X_{P} (a)[b]) → Hom(X_{P} , ℤ(a)[b]).

Denote: μ := pr_{∗}(μ′) ∈ Hom(X_{P} , ℤ(n)[2n + 1]).

Sublemma 6.2. The map

(μ′)^{∗} : Hom(X_{P} , ℤ(c)[d]) → Hom(X_{P} , ℤ(c + n)[d + 2n + 1])

coincides with the multiplication by μ ∈ Hom(X_{P} , ℤ(n)[2n + 1]).

Proof. The maps Δ_{X_{P}} : X_{P} → X_{P} ⊗ X_{P}, and π_{i} : X_{P} ⊗ X_{P} → X_{P} are mutually inverse isomorphisms (by Theorem A.13). Clearly, μ · u = Δ_{X_{P}} (μ ⊗ u).

The map μ ⊗ u : X_{P} ⊗ X_{P} → ℤ(n)[2n + 1] ⊗ ℤ(c)[d] coincides with the composition:

X_{P} ⟶^{μ′} X_{P} (n)[2n + 1] ⟶^{pr(n)[2n+1]} ℤ(n)[2n + 1]
⊗  ⊗  ⊗
X_{P} ⟶^{id}  X_{P}  ⟶^{u}  ℤ(c)[d]

which can be identified with the composition:

X_{P} →^{μ′} X_{P} (n)[2n + 1] →^{u} ℤ(n + c)[2n + 1 + d],

which is equal to (μ′)^{∗}(u).

Sublemma 6.3. Multiplication by μ performs an isomorphism:

Hom(X_{P} , ℤ(c)[d]) → Hom(X_{P} , ℤ(c + n)[d + 2n + 1])

for any d − c > 0, and is surjective for d = c. The same is true about cohomology with ℤ/2-coefficients.

3Since, otherwise, X_{P} would be a direct summand in M(P), and by Lemma 5.8, p would be isotropic

Proof. Since N is a direct summand in M(P), Hom(N, ℤ(a)[b]) = 0, for b − a > n = dim(P), by Theorem A.2(1). And the same is true about cohomology with ℤ/2-coefficients.

Consider Hom’s from the exact triangle N → X_{P} →^{μ′} X_{P} (n)[2n + 1] → N[1] to ℤ(n + c)[2n + d + 1]. We have: Hom(N, ℤ(n + c)[2n + d + 1]) = 0, if d − c ≥ 0, and Hom(N, ℤ(n + c)[2n + d]) = 0, if d − c > 0. This, combined with Sublemma 6.2, implies the statement for ℤ-coefficients. Considering Hom’s to ℤ/2(n+c)[2n+d+1], we get the ℤ/2-case.

We can also consider X̃_{P} := Cone[−1](X_{P} →^{pr} ℤ).

Sublemma 6.4. Let a and b be integers such that b > a. Then

• Hom(X_{P} , ℤ(a)[b]) is a 2-torsion group,

• Hom(X_{P} , ℤ(a)[b]) embeds into Hom(X_{P} , ℤ/2(a)[b]),

• the natural map X̃_{P} →^{δ} X_{P} induces an isomorphism

Hom(X_{P} , ℤ/2(a)[b]) →^{=} Hom(X̃_{P} , ℤ/2(a)[b]).

Proof. For the finite field extension E/k we have the action of transfers on motivic cohomology:

Tr : Hom(X|_{E}, ℤ(a)[b]) → Hom(X, ℤ(a)[b]),

which is induced by the natural map ℤ → M(Spec(E)) (given by the generic cycle on Spec(k) × Spec(E) = Spec(E)). The main property of the transfer is: Tr ∘j = ·[E : k], where

j : Hom(X, ℤ(a)[b]) → Hom(X|_{E}, ℤ(a)[b])

is the natural restriction.

Quadric P has a point E of degree 2, and over E, X_{P} becomes ℤ (by Theorem A.9), so we have that Hom(X_{P} |_{E}, ℤ(a)[b]) = 0 for b − a > 0 (by Theorem A.2(1)).

Considering the composition Tr ∘j = ·[E : k] we get that Hom(X_{P} , ℤ(a)[b]) is a 2-torsion group for b > a. In particular, the natural map Hom(X_{P} , ℤ(a)[b]) → Hom(X_{P} , ℤ/2(a)[b]) is injective for b > a.

Since Hom(ℤ, ℤ/2(a)[b]) = 0 for any b > a (see Theorem A.2(1)), we also have that for b > a, δ^{∗} : Hom(X_{P} , ℤ/2(a)[b]) → Hom(X̃_{P} , ℤ/2(a)[b]) is an isomorphism.

We have the action of motivic cohomological operations Q_{i} on Hom(X_{P} , ℤ(∗)[∗′]) and Hom(X̃_{P} , ℤ(∗)[∗′]) - see Theorems A.5 and A.6. The differential Q_{i} acts without cohomology on Hom(X̃_{P} , ℤ/2(∗)[∗′]) for any i ≤ [log_{2}(n+1)]

- see Theorem A.15.

Denote η := μ(mod2), i.e. the image of μ in the cohomology with ℤ/2 coefficients. From Sublemma 6.4 it follows that η ≠ 0.

Denote r = [log_{2}(n)].

Sublemma 6.5. Q_{i}(η) = 0, for all i ≤ r.

Proof. Really, Q_{i}(η) ∈ Hom(X_{P} , ℤ/2(n + 2^{i} − 1)[2n + 2^{i+1}]), and the later group is an extension of 2-cotorsion in Hom(X_{P} , ℤ(n + 2^{i} − 1)[2n + 2^{i+1}]), and 2-torsion in Hom(X_{P} , ℤ(n + 2^{i} − 1)[2n + 2^{i+1} + 1]).

But, by Sublemma 6.3, the multiplication by μ performs a surjections Hom(X_{P} , ℤ(2^{i} − 1)[2^{i+1} − 1]) → Hom(X_{P} , ℤ(n + 2^{i} − 1)[2n + 2^{i+1}]), and Hom(X_{P} , ℤ(2^{i} −1)[2^{i+1}]) → Hom(X_{P} , ℤ(n+2^{i} −1)[2n+2^{i+1} +1]). And the groups: Hom(X_{P} , ℤ(2^{i} − 1)[2^{i+1} − 1]), Hom(X_{P} , ℤ(2^{i} − 1)[2^{i+1}]) are zero.

Really, from the exact triangle N → X_{P} → X_{P} (n)[2n + 1] → N[1], we have exact sequence: Hom(N, ℤ(2^{i} − 1)[2^{i+1} − 1]) ← Hom(X_{P} , ℤ(2^{i} − 1)[2^{i+1} − 1]) ← Hom(X_{P} (n)[2n + 1], ℤ(2^{i} − 1)[2^{i+1} − 1]) The first group is zero since N is a direct summand in the motive of a smooth projective variety, and (consequently) Hom(N, ℤ(a)[b]) = 0 for b > 2a (see Theorem A.2(2)). The third group is zero, since n > 2^{i} − 1 (see Theorem A.1). So, the second is zero as well. Analogously, in the case of Hom(X_{P} , ℤ(2^{i} − 1)[2^{i+1}]).

So, Q_{i}(η) = 0.

Sublemma 6.6. Let 0 ≤ j ≤ r. Then Q_{j} is injective on Hom(X̃_{P} , ℤ/2(c)[d]), if d − c = n + 1 + 2^{j}.

Proof. Let ṽ ∈ Hom(X̃_{P} , ℤ/2(c)[d]), where d − c = n + 1 + 2^{j}. If Q_{j}(ṽ) = 0, then ṽ = Q_{j}(w̃), for some w̃ ∈ Hom(X̃_{P} , ℤ/2(c − 2^{j} + 1)[d − 2^{j+1} + 1]) (since Q_{j} acts without cohomology on Hom(X̃_{P} , ℤ/2(∗)[∗′]), by Theorem A.15). Since (d − 2^{j+1} + 1) − (c − 2^{j} + 1) = n + 1 > 0, we have that δ^{∗} : Hom(X_{P} , ℤ/2(c − 2^{j} + 1)[d − 2^{j+1} + 1]) → Hom(X̃_{P} , ℤ/2(c − 2^{j} + 1)[d − 2^{j+1} + 1]) is an isomorphism, and there exists w ∈ Hom(X_{P} , ℤ/2(c − 2^{j} + 1)[d − 2^{j+1} + 1]), s.t. w̃ = δ^{∗}(w).

By Sublemma 6.3, w = η·u, for some u ∈ Hom(X_{P} , ℤ/2(c−2^{j} +1−n)[d−2^{j+1}− 2n]). By Theorem A.6(2), Q_{j}(η · u) = Q_{j}(η) · u + η · Q_{j}(u) + ∑{−1}^{x_{i}}ϕ_{i}(η) · ψ_{i}(u), where x_{i} > 0, and ϕ_{i}, ψ_{i} are cohomological operations of some bidegree (∗)[∗′], where ∗′ > 2∗ ≥ 0.

Notice, that c − 2^{j} + 1 − n = d − 2^{j+1} − 2n =: s. But by Theorem A.17, we have that Hom(ℤ, ℤ/2(a)[b]) →^{pr^{∗}} Hom(X_{P} , ℤ/2(a)[b]) is an isomorphism for a ≥ b. So, u = pr^{∗}(u_{0}), where u_{0} ∈ Hom(ℤ, ℤ/2(s)[s]) = K_{s}^{M}(k)/2. We have: Q_{j}(u_{0}) = 0 and ψ_{i}(u_{0}) = 0 (since Hom(ℤ, ℤ/2(a)[b]) = 0, for b > a).

But pr^{∗} commutes with Q_{j} and ψ_{i}. So, Q_{j}(u) = 0 and ψ_{i}(u) = 0. That means: Q_{j}(w) = Q_{j}(η · u) = Q_{j}(η) · u = 0, by Sublemma 6.5.

We get: ṽ = Q_{j}(w̃) = Q_{j} ∘ δ^{∗}(w) = δ^{∗} ∘ Q_{j}(w) = 0. I.e., Q_{j} is injective on Hom(X̃_{P} , ℤ/2(c)[d]).

Denote: η̃ := δ^{∗}(η) ∈ Hom(X̃_{P} , ℤ/2(n)[2n + 1]). Since η ≠ 0, we have: η̃ ≠ 0 (by Sublemma 6.4).

Sublemma 6.7. Let 0 ≤ m < r, and η̃ = Q_{m} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}(η̃_{m}), for some η̃_{m} ∈ Hom(X̃_{P} , ℤ/2(n−2^{m+1} +m+2)[2n−2^{m+2} +m+4]), Then there exists η̃_{m+1}, s.t. η̃_{m} = Q_{m+1}(η̃_{m+1}).

Proof. Since Q_{m+1} acts without cohomology on Hom(X̃_{P} , ℤ/2(∗)[∗′]), it is enough to show that Q_{m+1}(η̃_{m}) = 0.

Denote: ṽ := Q_{m+1}(η̃_{m}). We have: ṽ ∈ Hom(X̃_{P} , ℤ/2(n + m + 1)[2n + m + 3]). Since Q_{i} commutes with Q_{j} (by Theorem A.6(1)), we have: Q_{m}∘Q_{m−1}∘⋯∘Q_{0}(ṽ) = Q_{m}∘Q_{m−1}∘⋯∘Q_{0}∘Q_{m+1}(η̃_{m}) = Q_{m+1}∘Q_{m}∘Q_{m−1}∘⋯∘Q_{0}(η̃_{m}) = Q_{m+1}(η̃) = 0, by Sublemma 6.5.

But, for any 0 ≤ t ≤ m, Q_{t−1} ∘ ⋯ ∘ Q_{0}(ṽ) ∈ Hom(X̃_{P} , ℤ(c)[d]), where d − c = n + 1 + 2^{t}, and Q_{t} is injective on Hom(X̃_{P} , ℤ(c)[d]), by Sublemma 6.6. So, from the equality: Q_{m} ∘ Q_{m−1} ∘ ⋯ ∘ Q_{0}(ṽ) = 0, we get: ṽ = 0.

From Sublemma 6.7 it follows that η̃ = Q_{r} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}(η̃_{r}). Denote: γ̃ := η̃_{r}. We have: γ̃ ∈ Hom(X̃_{P} , ℤ/2(n − 2^{r+1} + 2 + r)[2n − 2^{r+2} + 4 + r]).

But, (2n − 2^{r+2} + 4 + r) − (n − 2^{r+1} + 2 + r) = n − 2^{r+1} + 2, and r = [log_{2}(n)], so: 2^{r} ≤ n < 2^{r+1}. Since we know, that Hom(X̃_{P} , ℤ/2(a)[b]) = 0, for a ≥ b (by Theorem A.17), and η̃ ≠ 0, the only possible choice for n is n = 2^{r+1} − 1.

Theorem 6.1 is proven.

Lemma 6.8. Let 0 ≤ j ≤ r. Then Q_{j} is injective on Hom(X̃_{P} , ℤ/2(c)[d]), if d − c = 2^{j}.

Proof. Let ṽ ∈ Hom(X̃_{P} , ℤ/2(c)[d]), where d − c = 2^{j}. If Q_{j}(ṽ) = 0, then ṽ = Q_{j}(w̃), for some w̃ ∈ Hom(X̃_{P} , ℤ/2(c − 2^{j} + 1)[d − 2^{j+1} + 1]) (since Q_{j} acts without cohomology on Hom(X̃_{P} , ℤ/2(∗)[∗′]), by Theorem A.15). But (c−2^{j} +1) = (d−2^{j+1}+1), and Hom(X̃_{P} , ℤ/2(c−2^{j} +1)[d−2^{j+1}+1]) = 0, by Theorem A.17.

Theorem 6.9. (compare with [8, Theorem 3.1]) Let k be a field of characteristic 0, P be smooth n-dimensional anisotropic projective quadric over k, and N be a direct summand in M(P), s.t. N|\overline{k} = ℤ⊕ℤ(n)[2n]. Then n = 2^{s−1} −1, and there exists α ∈ K_{s}^{M}(k)/2, s.t. for any field extension E/k, the following conditions are equivalent:

1) α|_{E} = 0; 2) P |_{E} is isotropic.

In particular, α ∈ Ker(K_{s}^{M}(k)/2 → K_{s}^{M}(k(P))/2) ≠ 0.

Proof. It follows from Theorem 6.1 that n = 2^{r+1} − 1, for some r, and there exists γ̃ ∈ Hom(X̃_{P} , ℤ/2(r + 1)[r + 2]), s.t. η̃ = Q_{r} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}(γ̃). By Sublemma 6.4, the map δ^{∗} : Hom(X_{P} , ℤ/2(r + 1)[r + 2]) → Hom(X̃_{P} , ℤ/2(r + 1)[r + 2]) is an isomorphism, and γ̃ = δ^{∗}(γ) for some γ ∈ Hom(X_{P} , ℤ/2(r + 1)[r + 2]). Let τ be the only nontrivial element of Hom(ℤ/2, ℤ/2(1)) = Z/2. Denote as α the element corresponding to τ ∘γ via identification (by Theorem A.17) Hom(X_{P} , ℤ/2(r+2)[r+ 2]) = Hom(ℤ, ℤ/2(r + 2)[r + 2]) = K_{r+2}^{M}(k)/2. Then, by Theorem A.18, for any field extension E/k, α|_{E} = 0 if and only if γ|_{E} = 0. But γ|_{E} = 0 ⇔ γ̃|_{E} = 0. By Lemma 6.8, γ̃|_{E} = 0 ⇔ η̃|_{E} = 0. By Sublemma 6.4, η̃|_{E} = 0 ⇔ η|_{E} = 0 ⇔ μ|_{E} = 0. Finally, μ|_{E} = 0 ⇔ X_{P} |_{E} is a direct summand in N and, consequently, in M(P), which by Lemma 5.8, is equivalent to: P |_{E} is isotropic.

Remark 6.10. 1) Theorem 6.9 basically says that under the mentioned

conditions, the quadric P is a norm-variety for α ∈ K_{s}^{M}(k)/2.

2) Taking into account the Milnor’s conjecture ([28]) and the definition of the Rost projector, we see that Theorem 6.9 implies Theorem 1.12.

3) It should be mentioned that in small-dimensional cases it is possible to prove the result (in arbitrary characteristic ≠ 2) without the use of Voevodsky’s technique. For example, the case n = 7 was considered in [13].

7. Properties of forms with absolutely maximal splitting

In this section we work with fields satisfying the conditions char F = 0. We begin with the following modification of Theorem 1.11.

Theorem 7.1. Let ϕ be an anisotropic quadratic form over a field F satisfying the conditions char F = 0. Suppose that ϕ is an AMS-form. Then

(1) ϕ has maximal splitting, (2) the group H^{s}(F (ϕ)/F) is nontrivial, where s is the integer such that 2^{s−1} <

dim ϕ ≤ 2^{s}.

Proof. (1) Let ψ be subform of ϕ of codimension i_{1}(q) − 1. Let X be the projective quadric corresponding to ψ. By Theorem 5.1, X possesses a Rost projector. Theorem 6.1 shows that dim(X) = 2^{s−1} −1 for suitable s. Hence dim ψ = 2^{s−1} +1. By the definition of ψ, we have dim ϕ − i_{1}(ϕ) = dim ψ − 1 = 2^{s−1}. Therefore dim ϕ = 2^{s−1} + m, where m = i_{1}(ϕ). To prove that ϕ has maximal splitting, it suffices to verify that m ≤ 2^{n}. It is obvious because 2^{s−1} + m = dim ϕ ≥ 2i_{1}(ϕ) = 2m.

(2) Obvious in view of Theorem 6.9 and the isomorphism k_{s}(F) ≃ H^{s}(F).

Theorem 7.1 and Conjecture 1.1 make natural the following

Conjecture 7.2. If an anisotropic quadratic form has absolutely maximal splitting, then it is a Pfister neighbor.

In the proof of Theorem 1.8 we will need some deep results related to Milnor conjecture.

Theorem 7.3. (see [28],[20]). Let F be a field of characteristic 0. Then for any n ≥ 0

(1) there exists an isomorphism e^{n} : I^{n}(F)/I^{n+1}(F) →^{≃} H^{n}(F) such that

e^{n}(⟨ ; ⟨a_{1}, …, a_{n}⟩ ; ⟩) = (a_{1}, …, a_{n}).

(2) If ϕ is a Pfister neighbor of π ∈ GP_{n}(F). Then H^{n}(F (ϕ)/F) is generated by e^{n}(π).

(3) If dim τ > 2^{n}, then H^{n}(F (τ)/F) = 0.

(4) The ideal I^{n}(F) coincides with the Knebusch’s ideal J_{n}(F). In other words, for any τ ∈ I^{n}(F)\I^{n+1}(F), we have deg τ = n.

We need also the following easy consequence of a Hoffmann’s result:

Lemma 7.4. Let ϕ be an anisotropic form such that dim ϕ ≤ 2^{n}. Let τ be an anisotropic quadratic form and F_{0} = F, F_{1}, …, F_{h} be the generic splitting tower of τ. Let j be such that dim(τ_{F_{j−1}})_{an} > 2^{n}. Suppose that ϕ_{F_{j}} has maximal splitting. Then ϕ has maximal splitting.

Proof. Obvious in view of [4, Lemma 5].

Proposition 7.5. Let F be a field of characteristic 0. Let ϕ be a quadratic form over F and n be such that 2^{n−1}< dim ϕ ≤ 2^{n}. Suppose that H^{n}(F (ϕ)/F) ≠ 0. Then H^{n}(F (ϕ)/F) ≃ Z/2Z and ϕ has maximal splitting.

Proof. Let u be an arbitrary nonzero element of the group H^{n}(F (ϕ)/F). Since the homomorphism e^{n} : I^{n}(F)/I^{n+1}(F) → H^{n}(F) is an isomorphism, there exists anisotropic τ ∈ I^{n}(F) such that τ ∉ I^{n+1}(F) and e^{n}(τ) = u ∈ H^{n}(F (ϕ)/F). Let F_{0} = F, F_{1}, …, F_{h} be the generic splitting tower of τ. Let τ_{i} = (τ_{F_{i}})_{an}. Since τ ∈ I^{n}(F)\I^{n+1}(F), Item (4) of Theorem 7.3 shows that deg τ = n. Therefore, τ_{h−1} is a nonhyperbolic form from GP_{n}(F_{h−1}). Since e^{n}(τ) ∈ H^{n}(F (ϕ)/F), we have e^{n}((τ_{h−1})_{F_{h−1}(ϕ)}) = 0. Hence, τ_{h−1} is hyperbolic over the function field of ϕ_{F_{h−1}}. Since τ_{h−1} is anisotropic form from GP_{n}(F_{h−1}), Cassels–Pfister subform theorem shows that ϕ_{F_{h−1}} is a Pfister neighbor of τ_{F_{h−1}}. Hence ϕ_{F_{h−1}} has maximal splitting. Lemma 7.4 shows that ϕ has maximal splitting.

Since ϕ_{F_{h−1}} is a Pfister neighbor of τ_{F_{h−1}}, Item (2) of Theorem 7.3 shows that |H^{n}(F_{h−1}(ϕ)/F_{h−1})| ≤ 2. By Item (3) of Theorem 7.3, we have H^{n}(F_{h−1}/F) = 0. Hence |H^{n}(F_{h−1}(ϕ)/F)| ≤ 2. Since H^{n}(F (ϕ)/F) ⊂ H^{n}(F_{h−1}(ϕ)/F), we get |H^{n}(F (ϕ)/F)| ≤ 2. Now, since H^{n}(F (ϕ)/F) ≠ 0, we have H^{n}(F (ϕ)/F) ≃ Z/2Z.

Corollary 7.6. Let n ≥ 5 and ϕ be an anisotropic form such that 2^{n} − 7 ≤ dim ϕ ≤ 2^{n} Then the following conditions are equivalent:

(a) ϕ has maximal splitting,

(b) ϕ is a Pfister neighbor,

(c) H^{n}(F (ϕ)/F) ≃ Z/2Z.

(d) H^{n}(F (ϕ)/F) ≠ 0.

Proof. (a)⇒(b) follows from Theorem 1.7. (b)⇒(c) follows from Theorem 7.3; (c)⇒(d) is obvious; (d)⇒(a) is proved in Proposition 7.5.

Proof of Theorem 1.8. Let ϕ and n be as in Theorem 1.8. If ϕ has maximal splitting, then Lemma 4.1 shows that ϕ has absolutely maximal splitting. Then Theorem 7.1 shows that H^{n}(F (ϕ)/F) ≠ 0. Conversely, if we suppose that H^{n}(F (ϕ)/F) ≠ 0, then Proposition 7.5 shows that ϕ has maximal splitting.

Appendix A.

In this section we will list some results of V.Voevodsky, which we use in the proof of Theorems 6.1 and 6.9.

We will assume everywhere that char(k) = 0.

First of all, we need some facts about triviality of motivic cohomology of smooth simplicial schemes. If not specified otherwise, under Hom(−, −) we will mean Hom_{DM_{−}^{eff} (k)}(−, −).

Theorem A.1. ([28, Corollary 2.2(1)]) Let X be smooth simplicial scheme over k. Then Hom(M(X)(a)[b], ℤ(c)[d]) = 0, for any a > c.

In the case of smooth variety we have further restrictions on motivic cohomology:

Theorem A.2. ([28, Corollary 2.3]) Let N be a direct summand in M(X), where X is a smooth scheme over k. Then Hom(N, ℤ(a)[b]) = 0 in the following cases:

1 If b − a > dim(X);
2 If b > 2a.

The same is true about cohomology with ℤ/2-coefficients.

In [28] the Stable homotopy category of schemes over Spec(k), SH(k) was defined (see also [19]). SH(k) is a triangulated category, and there is functor S : SmSimpl/k → SH(k), and triangulated functor G : SH(k) → DM_{−}^{eff}(k), s.t. the composition G ∘ S : SmSimpl/k → DM_{−}^{eff}(k) coincides with the usual motivic functor: X ↦ M(X) (here SmSimpl/k is the category of smooth simplicial schemes over Spec(k). In [28], Section 3.3, the Eilenberg-MacLane spectrum H_{ℤ/2} (as an object of SH(k)) is defined, together with it’s shifts H_{ℤ/2}(a)[b], for a, b ∈ Z.

Theorem A.3. ([28, Theorem 3.12]) If X is a smooth simplicial scheme, then there exist canonical isomorphisms

Hom_{SH(k)}(S(X), H_{ℤ/2}(a)[b]) = Hom_{DM_{−}^{eff} (k)}(M(X), ℤ/2(a)[b]).

Definition A.4. ([28, p.31]) The Motivic Steenrod algebra is the algebra of endomorphisms of H_{ℤ/2} in SH(k), i.e.:

A^{b,a}(k, ℤ/2) = Hom_{SH(k)}(H_{ℤ/2}, H_{ℤ/2}(a)[b]).

The composition gives pairing:

Hom_{SH(k)}(U, H_{ℤ/2}(c)[d]) ⊗ A^{b,a}(k, ℤ/2) → Hom_{SH(k)}(U, H_{ℤ/2}(c + a)[d + b]),

which is natural on U.

Let now f : X → Y be morphism in SmSimpl/k. In SH(k) we have an exact

triangle: cone(S(f))[−1] →^{δ′} S(X) →^{S(f)} S(Y) → cone(f).

Theorem A.3 imply:

Theorem A.5. We have an action of Motivic Steenrod algebra A^{∗,∗}(k, ℤ/2) on ⊕_{a,b} Hom_{DM_{−}^{eff} (k)}(M(X), ℤ/2(a)[b]), ⊕_{a,b} Hom_{DM_{−}^{eff} (k)}(M(Y), ℤ/2(a)[b]), and ⊕_{a,b} Hom(cone(M(f))[−1], ℤ/2(a)[b]), which is compatible with M(f)^{∗} and δ^{∗}.

We have some special elements Q_{i} ∈ A^{2^{i+1}−1,2^{i}−1}(k, ℤ/2) - see [28, p.32], which satisfy:

Theorem A.6. ([28, Theorems 3.17 and 3.14])

1) Q_{i}^{2} = 0, and Q_{i}Q_{j} + Q_{j}Q_{i} = 0.

2) Let u, v ∈ Hom_{DM_{−}^{eff} (k)}(M(X), ℤ(∗)[∗′]), for smooth simplicial scheme X. Then Q_{i}(u · v) = Q_{i}(u) · v + u · Q_{i}(v) + ∑{−1}^{n_{j}}ϕ_{j}(u) · ψ_{j}(v), where n_{j} > 0, and ϕ_{j}, ψ_{j} ∈ A(k, ℤ/2) are some (homogeneous) elements of bidegree (b, a), where b > 2a ≥ 0.

3) Q_{i} = [β, q_{i}], where β is Bockstein, and q_{i} ∈ A(k, ℤ/2).

Following [28], we define:

Definition A.7. ([28, p.32]) Morgulis motivic cohomology H̃M_{i}^{b,a}(U) are co-

homology groups of the complex: Hom_{SH(k)}(U, H_{ℤ/2}(a − 2^{i} + 1)[b − 2^{i+1} + 1]) →^{Q_{i}}

Hom_{SH(k)}(U, H_{ℤ/2}(a)[b]) →^{Q_{i}} Hom_{SH(k)}(U, H_{ℤ/2}(a + 2^{i} − 1)[b + 2^{i+1} − 1]), for any

U ∈ Ob(SH(k)).

If U is Cone[−1](S(f)), for some morphism f : X → Y of simplicial schemes, then by Theorems A.3 and A.5, H̃M_{i}^{b,a}(U) coincides with the cohomology of the complex

H_{M}^{b−2^{i+1}+1,a−2^{i}+1}(M(U), ℤ/2) →^{Q_{i}} H_{M}^{b,a}(M(U), ℤ/2) →^{Q_{i}} H_{M}^{b+2^{i+1}−1,a+2^{i}−1}(M(U), ℤ/2),

where H_{M}^{d,c}(∗, ℤ/2) := Hom_{DM_{−}^{eff} (k)}(∗, ℤ/2(c)[d]), and M(U) = Cone[−1](M(f)).

Since H̃M_{i}^{b,a}(Cone[−1](S(f))) depends only on M(f), we can denote it simply as H̃M_{i}^{b,a}(Cone[−1](M(f))).

Let P be some smooth projective variety over Spec(k).

Definition A.8. The standard simplicial scheme Č(P)^{·}, corresponding to the pair P → Spec(k) is the simplicial scheme, s.t. Č(P)^{n} = P × ⋯ × P (n + 1-times), with faces and degeneration maps given by partial projections and diagonals.

In SmSimpl/k we have natural projection: pr : Č(P)^{·} → Spec(k). Let us denote X_{P} := M(Č(P)^{·}). We get the natural map M(pr) : X_{P} → ℤ.

Theorem A.9. ([28, Lemma 3.8]) If P has a rational k-point, then M(pr) : X_{P} → ℤ is an isomorphism.

Remark A.10. Actually, M(pr) is an isomorphism if and only if P has a 0cycle of degree 1 (see [25, Theorem 2.3.4]).

Theorem A.9 shows that X_{P} |\overline{k} = ℤ, which means that X_{P} is a form of Tatemotive.

Theorem A.11. ([28, Theorem 4.4]) Let P be anisotropic projective quadrics of dimension n. Let N be a direct summand in M(P), s.t. N|\overline{k} = ℤ ⊕ ℤ(n)[2n]. Then in DM_{−}^{eff}(k) there exist distinguished triangle of the form:

X_{P} (n)[2n] → N → X_{P} →^{μ′} X_{P} (n)[2n + 1].

Remark A.12. Actually, Theorem 4.4 of [28] is formulated only for the case of Rost motive (as a direct summand in the motive of the small Pfister quadric). But the proof does not use any specific of the Pfister case, and works with any “binary” direct summand of dimension = dim(P). At the same time, Theorem A.11 is a very particular case of [25, Lemma 3.23].

Theorem A.13. ([28, Lemma 3.8]) The natural diagonal map Δ_{X_{P}} : X_{P} → X_{P} ⊗ X_{P} is an isomorphism.

Theorem A.14. ([28, Lemma 4.7]) M(pr)_{∗} : Hom(X_{P} , X_{P} (a)[b]) → Hom(X_{P} , ℤ(a)[b]) is an isomorphism, for any a, b.

Let us denote: X̃_{P} := Cone[−1](M(pr)). By above, it has sense to speak about Morgulis cohomology H̃M_{i}^{b,a}(X̃_{P}) of X̃_{P}.

Suppose now P be a smooth projective quadric of dimension ≥ 2^{i} − 1.

The following result of V.Voevodsky is the main tool in studying of motivic cohomology of quadrics:

Theorem A.15. ([28, Theorem 3.25 and Lemma 4.11]) Let P be a smooth projective quadric of dimension ≥ 2^{i} − 1, then H̃M_{i}^{b,a}(X̃_{P}) = 0, for any a, b.

Remark A.16. In [28, Lemma 4.11], the result is formulated only for the case of (2^{i} − 1)-dimensional Pfister quadric (corresponding to the form ⟨ ; ⟨a_{1}, …, a_{i}⟩ ; ⟩ ⊥ − ⟨a_{i+1}⟩). But the proof does not use any specific of the Pfister case (the only thing which is used is: for any j ≤ i, P has plane section of dimension 2^{j} − 1, which is again a quadric).

The following statement is a consequence of Beilinson-Lichtenbaum Conjecture for ℤ/2-coefficients.

Theorem A.17. ([28, Proposition 2.7, Corollary 2.13(2) and Theorem 4.1]) The map M(pr)^{∗} : Hom(ℤ, ℤ/2(a)[b]) → Hom(X_{P} , ℤ/2(a)[b]) is an isomorphism for any b ≤ a.

Motivic cohomology of X_{P} can be used to compute the kernel on Milnor’s K-theory ( mod 2):

Theorem A.18. ([27, Lemma 6.4], [28, Theorem 4.1] ; or [11, Theorem A.1]) Let τ ∈ Hom_{DM_{−}^{eff} (k)}(ℤ/2, ℤ/2(1)) = Z/2 be the only nontrivial element. Let P be smooth projective quadric over k. Then the composition

Hom(X_{P} , ℤ/2(m − 1)[m])→^{τ∘} Hom(X_{P} , ℤ/2(m)[m]) = Hom(ℤ, ℤ/2(m)[m])= K_{m}^{M}(k)/2

identifies the first group with the ker(K_{m}^{M}(k)/2 → K_{m}^{M}(k(P))/2).

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Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, GERMANY

    E-mail address: oleg@mathematik.uni-bielefeld.de

School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540, USA

    E-mail address: vishik@ias.edu