INTEGRAL MOTIVES OF QUADRICS

A.VISHIK ^{1}

Contents

0. Introduction 1 1. Some examples 3 1.1. On the kernels in Milnor’s K-theory under function field extensions 3 1.2. After the J-filtration conjecture 4 1.3. Direct sum decomposition of the Chow motive of a quadric 5 1.4. Chow motive of a quadric is determined by the Witt indexes of the

corresponding form over all field extensions 6

2. Basic definitions and facts 6 2.1. Motivic category of V.Voevodsky 6 2.2. Category of Chow motives 11 2.3. Standard simplicial schemes 12 2.4. Some facts about quadratic forms 15 2.5. Triangulated categories 24 2.6. Proof of the Statement 1.1.1 29 3. General theorems 33 3.1. Structure of the motive of a quadric 33 3.2. Ring of endomorphisms of M(Q) 43 3.3. Some unpleasant computations 50 4. Higher forms - generalization of Rost motives 62 5. Back from X’s to quadratic forms 72 6. Few remarks about the Witt numbers 75 References 80

0. Introduction

In this paper we will construct the motivic decomposition of arbitrary quadric in the triangulated category of motives DM^{eff} (with ℤ-coefficients) over a field k (see Definition 2.1.8 ), and then will give some applications of this decomposition to various questions from quadratic form theory.

1Supported by Graduate P.A.Sloan Dissertation Fellowship, and by Max-Planck Institut für Mathematik

We will work in the motivic category DM^{eff}(k), constructed by V.Voevodsky in [17]. The usual category of Chow motives (see Definition 2.2.3 ) has natural full embedding into DM^{eff}(k), and the image is closed under the taking of direct summands. In particular, for smooth projective variety Q, the corresponding motive M(Q) has the same direct sum decomposition in DM^{eff}(k), as in Chow(k).

The motive of a quadric with rational coefficients is very simple: for odd-dimensional quadric it is the same as the motive of a projective space of the same dimension. With integral coefficients situation is much more subtle.

The investigation of the structure of a motive of a quadric (with integral coefficients) (in the category of Chow motives) was initiated by M.Rost in [14], who showed that the motive of a Pfister quadric (see Definition 2.4.3 ), corresponding to the pure symbol α = {a_{1}, …, a_{n}} ∈ K_{n}^{M}(k)/2 (see Definition 2.4.4 ), can be decomposed in the category of Chow motives into a direct sum of 2^{n−1} simpler motives (so-called, Rost motives) M_{α}(i)[2i], each of which, shifted by appropriate Tate-motive ℤ(−i)[−2i] (see Definition 2.1.15 ), becomes, after extending k to k̄, equal to the sum of just two Tate-motives: ℤ and ℤ(2^{n−1} − 1)[2^{n} − 2]. As was shown by V.Voevodsky (see Theorem 4.5 from [18]), M_{α} itself, as an object of DM^{eff}(k), can be constructed as an extension of two simpler things (each of which becomes a Tate-motive over k̄); namely, M_{α} = Cone[−1] (X_{Q} → X_{Q}(2^{n−1} − 1)[2^{n} − 1]). Here X_{Q} is a motive of a standard simplicial scheme, corresponding to the morphism Q → Spec(k) (X_{Q}^{i} = Q ×_{k} ⋯ ×_{k} Q - i + 1-times, with natural maps of faces and degenerations (see Definition 2.3.1 )), in particular, we have a natural morphism from X_{Q} to M(Spec(k)) =: ℤ, which is an isomorphism iff Q has a rational k-point). Combining this result with the one of M.Rost about excellent quadrics (see [14], Proposition 4), we get that the motive M(Q) is an extension (see Definition 2.5.6 ) of 2[dim(Q)/2] + 2 “simple” parts for arbitrary excellent quadric (see [14], discussion after Proposition 4 for the definition).

The first attempts to try the non-excellent case were undertaken by D.Orlov in the case of 3-dimensional non-Pfister quadric, who shared his ideas with me.

In the Theorem 3.1 of Section 3 below we will show, how to generalize the motivic decomposition above (i.e.: the presentation of M(Q) as an extension of 2[dim(Q)/2]+ 2 “simple” parts) to the case of arbitrary quadric. Then in Theorem 3.7 we will prove that this decomposition is completely canonical. Motive of Q appears in our construction equipped with the canonical Postnikov tower (Postnikov tower in the sense of triangulated category - just the system of distinguished triangles) with graded parts - forms of Tate-motives (i.e. motives, which become equal to ℤ(i)[2i] over k̄). We can rearrange terms of this Postnikov tower (using octahedron axiom, see Definition 2.5.3 , axiom TR4) and get different presentations for M(Q), which are sometimes useful. This is done in Proposition 3.4 and Proposition 3.6 . Theorem 3.7 on it’s part give us more, than just canonicity, it shows, that any endomorphism of M(Q) can be prolonged to our Postnikov tower in a unique way. This gives us that any nontrivial direct summand in M(Q) should consist of the same “simple” parts as M(Q), in particular, it should be non-zero over k̄ (Lemma 3.23 ), and that idempotents of End(M(Q)) are mapped surjectively under natural morphism α : End(M(Q)) → End(M(Q|k̄)) onto idempotents of image(α) ∩ End(M(Q|k̄)) (Lemma 3.12 ). Using this, and the fact that the category of Chow motives Chow^{eff}(k) is a full subcategory of DM^{eff}(k) closed under direct summands, we give in Corollary 3.14 a criteria of decomposability of the motive of a quadric in the category of Chow motives Chow(k) (see Definition 2.2.3 and Theorem 2.2.4 ). The same considerations also permit to bound the order of any element of Aut(M(Q)), and bound “nonabelianness” of this group (Corollary 3.17 and Corollary 3.19 ).

In Section 4 I describe some operations F_{α}, α ∈ K_{n}^{M}(k)/2-pure, which act on the motives of quadrics (and, presumably, on all direct summands of them as well), producing higher forms of them (see Theorem 4.1 and discussion after it). This gives a generalization of Rost-motives, since F_{\{a_{2},...,a_{n}\}}(M(k(\sqrt{a_{1}}))) = M_{\{a_{1},...,a_{n}\}}.

In Section 5 I show that “elementary pieces” from Theorem 3.1 define motive of a quadric uniquely. Equivalently, if two quadrics have equivalent universal splitting towers (see Definition 2.4.19 ), then their motives are isomorphic. Taking into account the fact that we believe the motive of a quadric should define quadric itself up to isomorphism, this can provide us (after the proving of the last fact) with some method to check: are two quadratic forms proportional, or not.

In Section 6 I show some applications of our methods to the computation of the Witt numbers (see Definition 2.4.19 ), and formulate some open questions.

The main results of the paper are contained in sections 3,4,5 and 6.

Acknoledgements: This work uses the techniques, developed by V.Voevodsky in [18], [19], and by D.Orlov-V.Voevodsky in [13]. During my work I had constant discussions with Volodia Voevodsky, which cleared many things for me and helped me a lot. I’m very grateful to him, and to Dima Orlov (discussions with whom pushed me to consider the case of general quadric). Also, I would like to thank Lyonia Positselsky, Yuri Tschinkel and Vadim Vologodsky for very helpfull discussions.

Finally, I would like to thank Barry Mazur and Yuri Ivanovich Manin for many suggestions and remarks, which improved this text dramatically.

During the writing of this work I was supported by Graduate P.A.Sloan Dissertation Fellowship and by Max-Planck Institut für Mathematik. I’m very grateful for this support.

1. Some examples

Here we give few examples of to what in the classical world our methods can be applied. Everywhere we will assume that characteristic of our field k is 0.

1.1. On the kernels in Milnor’s K-theory under function field extensions. The first application settles some very particular case of the following question:

Question.

Let Q be a quadric, defined over the field k. Is it true, that the ideal Ker_{Q} := Ker (K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2) is generated by pure symbols?

The positive answer to the above question would imply the one to the following two things:

1) Conjecture by B.Kahn, M.Rost and R.Sujatha, see the original version of [8], Conjecture on p.6 (the new version does not contain it). Here the authors basically conjectured the same for the Ker_{Q} in degrees < log_{2}(dim(Q)+2) (among other things, which were settled in [13]).

2) Question: If any conservative quadric (see Definition 2.4.10 ) is embedable, i.e. can be represented as a subquadric in a big Pfister quadric (see Definition 2.4.3 ) ?

We have the positive answer to our question in the case of a Pfister neighbor (see Definition 2.4.6 ) in ([13], exact sequence (5) on p.19). Here we get it for the case of 2-dimensional non-Pfister quadric.

Statement 1.1.1 .

Let Q be a 2-dimensional quadric corresponding to the form ⟨1, −a, −b, −c⟩, −abc ∉ (k^{∗})^{2}. Then Ker_{Q} is generated by pure 3-symbols {a, b, x}, where x ∈ Nrm_{k\sqrt{−abc}/k} ((k\sqrt{−abc})^{∗}).

We will give a proof of Statement 1.1.1 in Section 2.6 , after we introduce some definitions.

1.2. After the J-filtration conjecture. Another application is related to the Jfiltration conjecture.

On the Witt ring of quadratic forms W (k) (see Definition 2.4.2 ) we have two natural filtrations: I and J - filtration. I filtration corresponds to the powers of the ideal I of even-dimensional forms (see the discussion after Definition 2.4.2 ). And J - filtration can be constructed in the following way: let q be even-dimensional anisotropic quadratic form, and Q - corresponding projective quadric, then over k(Q) Q will have a rational point, and so, q will represent 0, we have q|_{k(Q)} = h_{1} · ℍ ⊥ q_{1}, where h_{1} > 0, ℍ is the elementary hyperbolic form ⟨1, −1⟩, and q_{1} is some anisotropic form, defined over k(Q) (such form is unique). Certainly, dim(q_{1}) < dim(q). Over k(Q)(Q_{1}) we have: q_{1}|_{k(Q)(Q_{1})} = h_{2} · ℍ ⊥ q_{2}, where h_{2} > 0, and q_{2} is some anisotropic form defined over k(Q)(Q_{1}), again, dim(q_{2}) < dim(q_{1}). After few such steps we will have: q_{s} = 0. That means that q_{s−1} is hyperbolic over generic point of it’s projective quadric. The only quadric which has such a property is so-called Pfister quadric (see Definition 2.4.3 , and [6], Theorem 5.8), i.e. quadric corresponding to the 2^{n} - dimensional quadratic form q_{{a_{1},...,a_{n}}} = ⟨1, −a_{1}⟩ ⊗ ⟨1, −a_{2}⟩ ⊗ ⋯ ⊗ ⟨1, −a_{n}⟩, where a_{1}, …, a_{n} is some set of nonzero field elements.

So, to each quadratic form q we can assign it’s degree - the nonnegative integer - the number n, appearing in the Pfister form above. Now we can define J_{n}(W (k)) as the set of all quadratic forms of degree ⩾ n.

We have: I^{n} ⊂ J_{n} ([6], Corollary 6.6), and it was conjectured (see [6], Question 6.7), that I^{n} = J_{n}.

This conjecture was settled in [13], Statement 2 of Section 3.3. The numbers: h_{1}, h_{2}, …, h_{s} above are called Witt numbers (see Definition 2.4.19 ). The J-filtration conjecture can be reformulated as: if q ∈ I^{n}(W (k))\I^{n+1}(W (k)), then h_{s} = 2^{n−1}.

Now we can say something about h_{s−1}:

Statement 1.2.1 (see Statement 6.2 ).

Suppose our field k contains \sqrt{−1}, and q ∈ I^{n}\I^{n+1}. Then h_{s−1} ∈ {2^{r} − 2^{n}, for r > n; 2^{m}, for 0 ⩽ m ⩽ n − 1}.

We have some understanding of: what different values of h_{s−1} should mean - see Question 6.3 and Statement 6.2 (j_{i} := h_{s} + h_{s−1} + ⋯ + h_{s−i+1}, see the very end of Section 3.1 ). I would say that the Question 6.3 is an analog of the J-filtration conjecture for h_{s−1}.

Also we can give another bound on the possible behavior of the Witt numbers.

Statement 1.2.2 (see Statement 6.1 ).

Suppose our field k contains \sqrt{−1}. Let for some quadratic form q we have: h_{1} > h_{i}, for all i > 1. Then dim(Q) − h_{1} = 2^{m} − 2 for some m.

For the proof of Statement 1.2.1 and Statement 1.2.2 - see Statement 6.2 and Statement 6.1 .

1.3. Direct sum decomposition of the Chow motive of a quadric. Using our methods we can give a criteria of decomposability of a Chow motive of a quadric.

Let P be a cycle of dimension n = dim(Q) on Q × Q. Then it gives us a morphism p : M(Q) → M(Q) in the Chow(k)-category of Chow motives over k (see Definition 2.2.3 ). For all 0 ⩽ i < n/2 we can define p_{i} ∈ ℤ as an intersection number of P |\overline{k} and h^{i} × l_{i} ⊂ (Q × Q)\overline{k}, where h^{i} is a plane section of codimension i, and l_{i} is a projective subspace of dimension i on Q|\overline{k} (over \overline{k} Q is hyperbolic); in the same way, we can define p′_{i} ∈ ℤ as an intersection number of P |\overline{k} and l_{i} × h^{i}.

5

We get a map: α : CH^{n}(Q × Q) → (×_{i=0}^{[ \frac{n−1}{2} ]}ℤ) × (×_{j=0}^{[ \frac{n−1}{2} ]}ℤ) , and α/2 : CH^{n}(Q × Q) → ×_{l=1,...,2[n+1/2]}ℤ/2.

Statement 1.3.1 (Corollary 3.14 ).

M(Q) is not decomposable into a direct sum in Chow(k) if and only if: image(α/2) = diagonal ℤ/2 (generated by the image of Δ_{Q}).

The same techniques gives us some generalization of the result of M.Rost. I remind, that in [14], Proposition 4 M.Rost proved that the Chow motive of 2^{n} − 2dimensional Pfister quadric Q_{{a_{1},...,a_{n}}} with quadratic form: ⟨⟨a_{1}, …, a_{n}⟩⟩ := ⟨1, −a_{1}⟩× ⋯ × ⟨1, −a_{n}⟩ (see Definition 2.4.3 ) is a direct sum: ⊕_{i=0}^{2^{n−1}−1} M_{{a_{1},...,a_{n}}}(i)[2i], where M_{{a_{1},...,a_{n}}}(i)[2i] is a Rost motive M_{{a_{1},...,a_{n}}} (this is a definition of it) shifted by the Tate motive ℤ(i)[2i] (see Definition 2.1.15 ), and M_{\{a_{1},...,a_{n}\}}|\overline{k} = ℤ⊕ℤ(2^{n−1}−1)[2^{n}−2].

It appears that it is a particular case of the following:

Statement 1.3.2 (Theorem 4.1 ).

Let Q corresponds to the quadratic form q, and R corresponds to the quadratic form q × ⟨⟨a_{1}, …, a_{n}⟩⟩.

Then M(R) = ⊕_{i=0}^{2^{n}−1}F_{{a_{1},...,a_{n}}}(Q)(i)[2i], if dim(Q)-even, and = ⊕_{i=0}^{2^{n}−1}F_{{a_{1},...,a_{n}}}(Q)(i)[2i] ⊕ M(Q_{{a_{1},...,a_{n}}})(dim(R)/2 − 2^{n−1} + 1)[dim(R) − 2^{n} + 2], if dim(Q)-odd.

And: F_{\{a_{1},...,a_{n}\}}(Q)|\overline{k} = (⊕_{0⩽k⩽[dim(Q)/2]}ℤ(k · 2^{n})[k · 2^{n+1}]) ⊕ (⊕_{0⩽k⩽[dim(Q)/2]}ℤ((dim(Q) + 1 − k) · 2^{n} − 1)[(dim(Q) + 1 − k) · 2^{n+1} − 2]) .

Notice, that the number of Tate motives in F_{\{a_{1},...,a_{n}\}}(Q)|\overline{k} is the same as in M(Q)|\overline{k}. We get an action of the semigroup of pure symbols from K_{∗}^{M}(k)/2 (see Definition

2.4.4 ) on the Chow motives of quadrics. I would call F_{{a_{1},...,a_{n}}}(Q) - the higher form of Q.

In this language: the Rost motive M_{{a_{1},...,a_{n}}} is a higher form of a 0-dimensional quadric k\sqrt{a_{1}}: M_{\{a_{1},...,a_{n}\}} = F_{\{a_{2},...,a_{n}\}}(k\sqrt{a_{1}}).

1.4. Chow motive of a quadric is determined by the Witt indexes of the corresponding form over all field extensions. The following result gives a criteria for the Chow motives of two quadrics to be isomorphic.

Statement 1.4.1 (Proposition 5.1 ).

Let Q_{1},Q_{2} be projective quadrics. Then M(Q_{1}) = M(Q_{2}) if and only if for every field extension E/k, and i ⩾ 0, the quadric Q_{1}|_{E} contains a (rational) projective subspace of dimension i simultaneously with Q_{2}|_{E} (equivalently, q_{1}|_{E} = (i+1)·ℍ ⊥ q′_{1} iff q_{2}|_{E} = (i + 1) · ℍ ⊥ q′_{2}).

In some interesting cases we can check that two quadrics have equal Witt indexes (see the text after Definition 2.4.2 for the definition), which gives an isomorphism of their Chow motives (see Corollary 5.2 ).

2. Basic definitions and facts

2.1. Motivic category of V.Voevodsky. In [17] V.Voevodsky introduced the triangulated category of motives over a field.

Let us briefly give the definition of this category (all the definitions are from [17], and [18]).

Definition 2.1.1 ([17], p.3). Let X, Y are smooth schemes over k. Then denote as c(X, Y) the free abelian

group, generated by integral closed subschemes on X × Y, which are finite over X and surjective over connected component of X.

As usually, as Sm(k) we will denote the category of smooth schemes over k. Then we can introduce the additive category SmCor(k):

Definition 2.1.2 ([17], p.3). The objects of SmCor(k) are smooth schemes over k, and Hom_{SmCor(k)}(X, Y) :=

c(X, Y).

The composition of morphisms in SmCor(k) is defined as follows: if φ ∈ c(X, Y), and ψ ∈ c(Y, Z), then ψ ∘ φ := p_{13}_{∗}(p_{12}^{∗}(φ) ∩ p_{23}^{∗}(ψ)), where p_{ij} - is a projection from X × Y × Z onto corresponding double product.

The category SmCor(k) has a natural tensor structure given by direct product of schemes on objects and by external multiplication of correspondences on morphisms.

We, certainly, have natural functor Sm(k) → SmCor(k).

Definition 2.1.3 ([17], Definition 3.1.1). A presheaf with transfers is an additive contravariant functor from SmCor(k) to

the category of abelian groups.

A presheaf with transfers is called a Zariski sheaf with transfers, Nisnevich sheaf with transfers (see [4], Definition 3.2), Etale sheaf with transfers, if, restricted to Sm(k) it is a sheaf in the corresponding topology.

Category of such sheaves will be denote as Shv_{Zar}(SmCor(k)), Shv_{Nis}(SmCor(k)), and Shv_{et}(SmCor(k)) respectively.

Theorem 2.1.4 ([17], Theorem 3.1.4). The category of Nisnevich sheaves with transfers is abelian (see [5],II,§5,10).

Definition 2.1.5 . Denote as D^{−}(Shv_{Nis}(SmCor(k))) the derived category of complexes in

Shv_{Nis}(SmCor(k)) bounded from above (see [5],III,§2,1).

Definition 2.1.6 ([17], Definition 3.1.9). A presheaf with transfers F is called homotopy invariant if for any smooth scheme

X over k the morphism X × 𝔸^{1} → X induces isomorphism F (X) → F (X × 𝔸^{1}).

A Nisnevich sheaf with transfers is called homotopy invariant if it is homotopy invariant as a presheaf with transfers.

Theorem 2.1.7 ([17], Proposition 3.1.12). For any perfect field k the full subcategory HI(k) of the category Shv_{Nis}(SmCor(k)) which consists of homotopy invariant sheaves is abelian and the inclusion functor HI(k) → Shv_{Nis}(SmCor(k)) is exact.

Definition 2.1.8 ([17], p.20). Denote as DM_{−}^{eff}(k) the full subcategory of D^{−}(Shv_{Nis}(SmCor(k))) which consists

of complexes with homotopy invariant cohomology sheaves.

Theorem 2.1.7 implies that DM_{−}^{eff}(k) is a triangulated category (see Definition 2.5.3 ).

To see where the motives of schemes live in DM_{−}^{eff}(k), we need to introduce another category, which will be a full subcategory of DM_{−}^{eff}(k).

Definition 2.1.9 ([17], Definition 2.1.1). Let H^{b}(SmCor(k)) be the homotopy category of bounded complexes over SmCor(k). Let T be the thick subcategory in it generated by complexes of the form X ×𝔸^{1} → X, and U ∩ V → U ⊕ V → X, for all smooth schemes X over k and all Zariski open coverings X = U ∪ V.

The triangulated category of mixed motives DM_{gm}^{eff}(k) is the pseudo-abelian envelope (see Definition 2.5.2 ) of the localization (see [5],III,§2,2) of H^{b}(SmCor(k)) with respect to the morphisms with cones in T.

It is a tensor triangulated category.

The full embedding DM_{gm}^{eff}(k) → DM_{−}^{eff}(k) can be obtained in the following way: Denote as Δ^{·} the standard cosimplicial object in Sm(k); i.e. Δ^{n} = 𝔸^{n} given as a

subscheme in 𝔸^{n+1}: Δ^{n} = Spec(k[x_{1}, …, x_{n}]/(∑ x_{i} − 1)) with natural morphisms of faces and degenerations.

Definition 2.1.10 ([18], p.7). For any presheaf on Sm(k) denote by \underline{C}_{∗}(F) the bounded from above complex of

presheaves on Sm(k), s.t. for any smooth scheme U \underline{C}_{∗}(F)(U) is the normalization of the simplicial abelian group F (U × Δ^{·}). \underline{C}_{∗}(F) will be a complex with homotopy invariant cohomology sheaves.

Definition 2.1.11 ([17], p.13). For any variety X denote as L(X) the presheaf with transfers, representable by X

on SmCor(k).

Theorem 2.1.12 ([17], Lemma 3.1.2). For any smooth scheme X/k L(X) is a sheaf in Nisnevich topology. The rule: X ↦ \underline{C}_{∗}(L(X)) gives us an additive functor i′ : SmCor(k) → DM_{−}^{eff}(k).

Theorem 2.1.13 ([17], Theorem 3.2.6). i′ defines the triangulated functor i : DM_{gm}^{eff}(k) → DM_{−}^{eff}(k), which is a full

embedding.

Definition 2.1.14 . Any smooth projective variety X has it’s natural image M(X) in DM_{gm}^{eff}(k) (X is

an object of the additive category SmCor(k)), and, via i, also in DM_{−}^{eff}(k). Now we can define Tate-motives. Definition 2.1.15 . Denote M(Spec(k)) := ℤ. This is a unit object with respect to the tensor structure

on DM (any of them).

Denote ℤ(1) to be Cone(M(ℙ^{1}) → M(k))[−3]. Denote by ℤ(m)[n] the motive ℤ(1)^{⊗m}[n], where [n] is a cohomological shift in

triangulated category.

Also for arbitrary object X, denote X(n) to be X ⊗ ℤ(n). Notice, that the map M(ℙ^{1}) → M(k) has a section given by any rational point on

ℙ^{1}. Hence, M(ℙ^{1}) = ℤ ⊕ ℤ(1)[2].

Definition 2.1.16 . Motivic cohomology of a motive Y are groups H_{M}^{j,i}(Y) := Hom_{DM_{−}^{eff} (k)}(Y, ℤ(i)[j]).

If Y is an object of DM_{gm}^{eff}(k) then H_{M}^{j,i}(Y) = Hom_{DM_{gm}^{eff} (k)}(Y, ℤ(i)[j]), since DM_{gm}^{eff}(k) is a full subcategory of DM_{−}^{eff}(k).

For smooth projective varieties we have the following relation between motivic cohomology and Higher Chow groups of S.Bloch (see [2], p.1):

Theorem 2.1.17 ([17], Theorem 4.2.9). Let X be smooth projective variety. Then H_{M}^{j,i}(X) = CH^{i}(X, 2i − j). In particular, Hom_{DM_{gm}^{eff} (k)}(X, ℤ(i)[2i]) = CH^{i}(X).

From the previous theorem and the definition of Higher Chow groups (see [2], p.1) it follows that:

Theorem 2.1.18 . For smooth projective variety X Hom_{DM_{gm}^{eff} (k)}(X, ℤ(i)[j]) = 0 in the following

cases:

1) i < 0;

2) j − i > dim(X);

3) j > 2i;

Also, we have: Theorem 2.1.19 ([15], Proposition 2.2).

For smooth connected scheme X

Hom_{DM_{gm}^{eff} (k)}(X, ℤ(0)[j]) = \begin{cases}ℤ, for j = 0; \\ 0, for j ≠ 0. \end{cases}

and

Hom_{DM_{gm}^{eff} (k)}(X, ℤ(1)[j]) = \begin{cases}O^{∗}, for j = 1; \\ P ic(X), for j = 2; \\ 0, for j ≠ 1, 2. \end{cases}

Some part of motivic cohomology of ℤ is given by the following:

Theorem 2.1.20 ([18], Proposition 2.7).

Hom_{DM}(ℤ, ℤ(n)[n]) = K_{n}^{M}(k).

To introduce duality we need to enlarge the category DM_{gm}^{eff}(k).

Definition 2.1.21 ([17], the end of p.5).

DM_{gm}(k) is the following category: Objects of DM_{gm}(k) are pairs (A, n), where A is an object of DM_{gm}^{eff}(k), and n is integer number. The morphisms are defined by the following formula: Hom_{DM_{gm}(k)}((A, n), (B, m)) = lim_{k⩾−m,−n} Hom_{DM_{gm}^{eff} (k)}(A(n + k), B(m + k))

Theorem 2.1.22 ([17], p.6, Corollary 2.1.5, and Theorem 4.3.1).

Suppose k admits resolution of singularities. Then DM_{gm}(k) is a tensor triangulated category and the functor A ↦ (A, 0) is a full embedding.

Theorem 2.1.23 ([17], p.10, Theorem 4.3.7).

For any field k which admits resolution of singularities the category DM_{gm}(k) is a “rigid tensor triangulated category”. More precisely, one has:

1) For any pair of objects A, B in DM_{gm}(k) there exists the internal Hom-object \underline{Hom}_{DM_{gm}(k)}(A, B). We set A^{∗} to be \underline{Hom}_{DM_{gm}(k)}(A, ℤ).

2) For any object A in DM_{gm}(k) the canonical morphism A → (A^{∗})^{∗} is an isomorphism.

3) For any pair of objects A, B in DM_{gm}(k) there are canonical isomorphisms: \underline{Hom}_{DM_{gm}(k)}(A, B) = A^{∗} ⊗ B; (A ⊗ B)^{∗} = A^{∗} ⊗ B^{∗}.

4) For smooth projective variety X/k of dimension n we have natural identification: M(X) → \underline{Hom}_{DM_{gm}(k)}(M(X), ℤ(n)[2n]), given by the generic cycle of diagonal X → X ×X, considered as an element of Hom(M(X)⊗M(X), ℤ(n)[2n]) via Theorem 2.1.17 .

For each n > 0 we have natural element n ∈ Hom(ℤ, ℤ) = ℤ.

Definition 2.1.24 .

ℤ/n := Cone(ℤ →^{n} ℤ).

H_{M}^{j,i}(X, ℤ/n) := Hom_{DM_{gm}}(X, ℤ/n)(i)[j]).

We have action of motivic cohomological operations in motivic cohomology of motives of simplicial smooth schemes with finite coefficients.

From Theorem 2.1.20 , Theorem 2.1.19 it follows that Hom_{DM}(ℤ/2, ℤ/2(1)) = (Hom_{DM}(ℤ, ℤ(1)[1]))_{2}, and the later is ℤ/2 with the only nontrivial element represented by {−1} via identification Hom_{DM}(ℤ, ℤ(1)[1]) = K_{1}^{M}(k) = k^{∗}

Definition 2.1.25 .

Denote as τ the element of the group Hom_{DM}(ℤ/2, ℤ/2(1)), which correspond to {−1} via identification above.

Denote ρ := β(τ) ∈ Hom_{DM}(ℤ/2, ℤ/2(1)[1]) (it will be also the image of {−1} under natural map: Hom_{DM}(ℤ, ℤ(1)[1]) → Hom_{DM}(ℤ/2, ℤ/2(1)[1]).

We can consider cohomological operations - compositions with the above elements. We will denote them also as τ and ρ.

Theorem 2.1.26 ([20], Theorem 3.16 and Theorem 3.17).

Let k has characteristic 0.

There exist cohomological operations P^{i}, i ⩾ 0, where P^{i} : Hom(Y, ℤ/2(a)[b]) → Hom(Y, ℤ/2(a + i)[b + 2i]), s.t.:

1) P^{0} = id;

2) For any smooth simplicial scheme Y and any u ∈ H_{M}^{n,i}(X, ℤ/2), P^{i}(u) = 0 for n < 2i, and P^{i}(u) = u^{i} for n = 2i.

3) Δ(P^{i}) = ∑_{a+b=i} P^{a} ⊗ P^{b} + τ ∑_{a+b=i−2} βP^{a} ⊗ βP^{b}

Define the following operations Q_{i} of bidegree (2^{i} − 1)[2^{i+1} − 1] inductively as follows: Q_{0} = β-bokstein, and Q_{i+1} = [Q_{i}, P^{2^{i}}]. These operations have the following properties:

4) Q_{i}Q_{j} = Q_{j}Q_{i}, Q_{i}^{2} = 0.

5) Δ(Q_{i}) = 1 ⊗ Q_{i} + Q_{i} ⊗ 1 + ∑ ρ^{n_{j}}φ_{j} ⊗ ψ_{j}, where n_{j} > 0, and φ_{j}, ψ_{j} are some operations of bidegree (q)[p] with p > 2q.

6) for any i > 0 there exists operation q_{i}, s.t. Q_{i} = [β, q_{i}].

2.2. Category of Chow motives. Let us now define the category of Chow motives. Definition 2.2.1 ([17], p.6). Consider a category C whose objects are smooth projective varieties over k and

morphisms are given by the formula Hom_{C}(X, Y) = ⊕_{X_{i}}A_{dim(X_{i})}(X_{i} × Y), where X_{i} are connected components of X, and A_{j}(−) is the group of cycles of dimension j modulo rational equivalence.

The Pseudo Abelian envelope of C (see the Definition 2.5.2 ) is called the category of effective Chow motives over k: Chow^{eff}(k).

We have natural functor Chow : SmP roj/k → Chow^{eff}(k) from the category of smooth projective varieties to the category of effective Chow motives.

Theorem 2.2.2 ([17], Proposition 2.1.4).

There exists a functor Chow^{eff}(k) → DM_{gm}^{eff}(k), s.t. the following diagram is

commutative: \begin{matrix}SmP roj/k & ⟶ & Sm/k \\ ^{Chow}↓ & & ↓ \\ Chow^{eff}(k) & ⟶ & DM_{gm}^{eff}(k) \end{matrix}

As usually we will denote: Chow motive of Spec(k) := ℤ. In Chow^{eff}(k) we have a decomposition: Chow(ℙ^{1}) = ℤ ⊕ Y, where direct summand ℤ is given by the idempotent p ∈ Hom_{Chow^{eff} (k)}(ℙ^{1}, ℙ^{1}) = CH^{1}(ℙ^{1} × ℙ^{1}) represented by the cycle Spec(k) × ℙ^{1} ⊂ ℙ^{1} × ℙ^{1}. Define: ℤ(1)[2] := Y. It is easy to see that corresponding idempotent is represented by the cycle ℙ^{1} × Spec(k) ⊂ ℙ^{1} × ℙ^{1}.

In the same way as we obtained DM_{gm}(k) from DM_{gm}^{eff}(k), we can obtain category of Chow motives Chow(k) from the category of effective Chow motives Chow^{eff}(k):

Definition 2.2.3 .

Objects of Chow(k) are pairs (X, n), where X is an object of Chow^{eff}(k), and n ∈ ℤ.

Morphisms are given by the formula: Hom_{Chow(k)}((X, n), (Y, m)) = lim_{l⩾−m,−n} Hom_{Chow^{eff} (k)}(X(n + l), Y (m + l))

Theorem 2.2.4 ([17], end of p.11, Corollary 4.2.6).

Let our field k admits resolution of singularities. Then the natural functor Chow^{eff}(k) → Chow(k) is a full embedding and the functor Chow^{eff}(k) → DM_{gm}^{eff}(k) from Theorem 2.2.2 can be prolonged to a functor Chow(k) → DM_{gm}(k) which is again a full embedding.

The last functor performs an equivalence between Chow(k) and the full tensor additive subcategory of DM_{gm}(k) which is closed under direct summands and is generated by the objects of the form M(X)(n)[2n], where X/k is a smooth projective variety, and n ∈ ℤ.

Any object of the above category is called pure motive.

2.3. Standard simplicial schemes. Again, I remind that everywhere we assume that char(k) = 0.

Here we will describe some objects of DM^{eff}(k) with which we will work.

Let X^{·} be a smooth simplicial scheme. Then it has it’s image M(X^{·}) in DM^{eff}(k) in the following way: Let X^{n} be n-th simplicial part of X^{·}.

M(X^{·}) is equal to \underline{C}_{∗}(L(X^{·})), where L(X^{·}) is a complex of sheaves with r-th term L(X^{−r}), and morphisms given by the alternating sum of face maps.

Definition 2.3.1 .

Let P/k be some smooth connected projective scheme. Then we will call standard simplicial scheme corresponding to the pair P → Spec(k) to be the simplicial scheme X^{·}_{P}, s.t. X_{P}^{n} = P ×_{k} P ×_{k} ⋯ ×_{k} P - n + 1-times, and the maps of faces and degenerations are given by partial projections, and partial diagonals.

We will often denote the motive M(X^{·}_{P}) by the same symbol X^{·}_{P}, or by X_{P}. We have natural projection M(X_{P}) → ℤ given by the natural morphism of simpli-

cial schemes X^{·}_{P} → Spec(k).

The following theorem describes how this map acts on the corresponding representable and corepresentable functors.

Theorem 2.3.2 ([18], Lemma 4.9).

Let P be some smooth projective scheme over k.

If A is an object of the thick subcategory of DM^{eff}(k), generated by the objects of the form P ⊗ L for all L (i.e.: the minimal subcategory, containing specified set of objects and closed under taking direct summands), then M(X_{P}) → ℤ induces an isomorphism: Hom(ℤ, A(i)[j]) → Hom(M(X_{P}), A(i)[j]).

If B is an object of localizing subcategory of DM^{eff}(k), generated by the objects of the form P ⊗ L for all L (i.e.: the minimal subcategory, containing specified set of objects and closed under taking direct summands and arbitrary direct sums), and C some object of DM^{eff}(k), then Hom(B(i)[j], M(X_{P}) ⊗ C) → Hom(B(i)[j], ℤ ⊗ C).

M(X_{P}) is an object of localizing subcategory of DM^{eff}(k), generated by M(P). Remark In Lemma 4.9 of [18] everything is formulated in the context of P = Q_{α} -

the Pfister quadric (and there is no C there), but the proof does not use any specific, and is general.

Theorem 2.3.3 .

Let P be smooth projective variety. Then:

1) Hom_{DM}(M(X_{P}), ℤ(i)[j]) = 0 for all i < 0, or i = 0 and j < 0 and Hom_{DM}(M(X_{P}), ℤ) = Hom_{DM}(ℤ, ℤ) = ℤ (the first map is induced by the natural projection M(X_{P}) → ℤ, and the second equality via Theorem 2.1.20 ).

2) The image of Hom_{DM}(ℤ, M(X_{P})) in ℤ = Hom_{DM}(ℤ, ℤ) (see Theorem 2.1.20 ) (via the map induced by the projection X^{·}_{P} → Spec(k)) is a subgroup which is generated by the greatest common divisor of the degrees of finite points on P, and it coincides with the image(Hom_{DM}(ℤ, M(P)) → Hom_{DM}(ℤ, ℤ)).

Proof

1) The first part is just [20], Corollary 2.2, and the fact that Hom_{DM}(M(X_{P}), ℤ) = ℤ follows from [15], Proposition 2.2.

2) Consider Y := Cone(M(P) → M(X^{·}_{P})) (the map here is induced by natural isomorphism: X_{P}^{0} = P). Then by [17], Proposition 3.1.8, and since Y as a complex of Nisnevich sheaves with transfers is concentrated in negative degrees, we have that Hom(ℤ, Y) = H_{Nis}(Spec(k), Y) = 0 (the second group here is a hypercohomology of Spec(k) with coefficients in the specified complex of sheaves in Nisnevich topology) (see the proof of Proposition 3.1.8 from [17]).

So, we get a surjection: Hom_{DM}(ℤ, M(P)) → Hom(ℤ, M(X^{·}_{P})), and image(Hom(ℤ, M(X^{·}_{P}))→Hom_{DM}(ℤ, ℤ))=image(Hom(ℤ, M(P))→Hom_{DM}(ℤ, ℤ)).

Any element of Hom(ℤ, M(P)) is represented by a cycle of dimension 0 on P (see Theorem 2.1.17 ), and it’s image in ℤ = Hom_{DM}(ℤ, ℤ) is equal to the degree of a cycle.

It appears that the motive M(X^{·}_{P}) contains much less information, than P itself. Namely, we have the following:

Theorem 2.3.4 . Let P have a zero-cycle of degree 1. Then M(X^{·}_{P}) = ℤ. Moreover, let P and Q are smooth (connected) projective varieties over k, s.t. Q

has a zero cycle of degree 1 over k(P), and P has a zero cycle of degree 1 over k(Q). Then there is unique up to change of sign isomorphism M(X^{·}_{P}) → M(X^{·}_{Q}). Moreover, the condition above is equivalent to: M(X^{·}_{P}) = M(X^{·}_{Q}).

Proof Suppose P has a zero-cycle of degree 1, i.e. we have a section j : ℤ → M(P) (see

Theorem 2.1.17 , Theorem 2.1.23 ) of the standard projection, then j defines a direct sum decomposition M(X^{·}_{P}) = ℤ ⊕ Y, and a homotopy of id_{Y} to 0.

The second part follows from the following general result

Theorem 2.3.5 . Suppose we have some morphism φ : A → B in DM_{−}^{eff}(k), and smooth projective

variety P/k. Then if φ|_{k(P)} is an isomorphism (as a morphism in DM_{−}^{eff}(k(P)), then φ × id : A ⊗ M(P) → B ⊗ M(P) is also an isomorphism.

Proof A × M(P) → B × M(P) is an isomorphism ⇔ Cone[−1](φ) × M(P) → 0 is an

isomorphism.

Now Cone[−1](φ) as an object of DM_{−}^{eff}(k) is a bounded from above complex of Nisnevich sheaves with transfers with homotopy invariant cohomology sheaves. So, to prove that it is 0, it is equivalent to prove that all cohomology sheaves are.

We have a homotopy invariant Nisnevich sheaf with transfers, which is zero in the generic point. Since any Nisnevich covering contains some Zariski open subset (see [4], Definition 3.2), we have that our sheaf is zero on some Zariski open subscheme.

Then everything follows from [16], Corollary 4.18 (see also Definition 3.1 and Definition 3.6 there).

Now we have natural map: M(X^{·}_{P}) ⊗ M(Q) → M(Q) (induced by the projection M(X^{·}_{P}) → ℤ). But if we change Q by k(Q), then the corresponding map is an isomorphism (as we just saw), and so, our map is an isomorphism as well.

Since M(X^{·}_{Q}) belongs to the localizing category generated by the object M(Q), we have that the natural map M(X^{·}_{P}) ⊗ M(X^{·}_{Q}) → M(X^{·}_{Q}) is an isomorphism as well. Similarly we can prove symmetric isomorphism.

So, M(X^{·}_{P}) = M(X^{·}_{Q}). But Hom_{DM}(M(X^{·}_{P}), M(X^{·}_{P})) = Hom_{DM}(M(X^{·}_{P}), ℤ) by Theorem 2.3.2 , and the later group is identified with Hom_{DM}(ℤ, ℤ) = ℤ (equality by Theorem 2.1.20 ) via the projection M(X^{·}_{P}) → ℤ by Theorem 2.3.3 .

So, there are only two invertible elements there: 1 and −1.

Conversely, if M(X^{·}_{P}) = M(X^{·}_{Q}), then M(P) × M(X^{·}_{P}) = M(P) × M(X^{·}_{Q}), in particular, over k(P): M(X_{Q} × k(P)) = M(X^{·}_{P} × k(P)) = ℤ|_{k(P)} (as was proven above (P has a rational point over k(P))). By Theorem 2.3.3 (2) it follows that over k(P), Q has a zero-cycle of degree 1.

Remark From Theorem 2.3.4 , and the fact that over algebraically closed field any variety of finite type has a rational point, it follows that M(X^{·}_{P}) is a form of Tate-motive, i.e. M(X^{·}_{P})|\overline{k} = ℤ.

From the proof of Theorem 2.3.4 we by the way get the following fact:

Theorem 2.3.6 .

If P, Q are smooth projective varieties and P has a zero cycle of degree 1 over k(Q), then M(X^{·}_{P}) ⊗ M(Q) = M(Q), and M(X^{·}_{P}) ⊗ M(X^{·}_{Q}) = M(X^{·}_{Q}).

The following result is very useful in computation of ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2) for smooth projective variety Q/k.

Theorem 2.3.7 ([18], Lemma 6.4, [20], Theorem 4.1).

The map β : Hom_{DM}(M(X_{Q}), ℤ/2(n−1)[n]) → Hom_{DM}(M(X_{Q}), ℤ_{(2)}(n−1)[n+1]) (see Definition 2.1.25 ) identifies the first group with the 2-torsion subgroup of the second.

τ : Hom_{DM}(M(X_{Q}), ℤ/2(n−1)[n]) → Hom_{DM}(M(X_{Q}), ℤ/2(n)[n]) = K_{n}^{M}(k)/2 (see Definition 2.1.25 ,Theorem 2.1.18 and Theorem 2.1.20 ) identifies the first group with ker(K_{n}^{M}(k)/2 → K_{n}^{M}(k(Q))/2).

So, we get isomorphism: τ ∘β^{−1} : (Hom_{DM}(M(X_{Q}), ℤ_{(2)}(n−1)[n+1]))_{2} → Ker_{n}^{Q} := Ker(K_{n}^{M}(k)/2 → K_{n}^{M}(k(Q))/2).

Remark In Lemma 6.4 of [18] everything is formulated in the context of Q = Q_{α} - the Pfister quadric, but the proof does not use any specific, and is general.

Definition 2.3.8 .

Denote M(X̃_{P}) := Cone[−1](M(X_{P}) → ℤ).

Since M(X_{P}) is a form of ℤ (i.e.: they are equal in étale topology), then, considering the just defined object, we get:

Theorem 2.3.9 ([20], Corollary 2.12 + the main result of the cited paper (Hilbert90 for ℤ/2-coeff.).

For any smooth projective variety P/k, and any n ∈ ℤ we have:

Hom_{DM}(M(X_{P}), ℤ_{(2)}(n)[m]) = Hom_{DM}(ℤ, ℤ_{(2)}(n)[m]) for m ⩽ n + 1.

In particular, by Theorem 2.1.18 we get:

Theorem 2.3.10 .

For any smooth projective variety P/k, and any n ∈ ℤ we have:

Hom_{DM}(M(X_{P}), ℤ_{(2)}(n)[n + 1]) = 0

Theorem 2.3.11 ([20], Theorem 3.25, Lemma 4.11). Let P be a quadric of dimension ⩾ 2^{i}−1. Then Q_{i} (see Theorem 2.1.26 ) (considered as a differential (it has square =0)) acts without cohomology on H_{M}^{∗,∗′}(M(X̃_{P}), ℤ/2).

From the existence of transfers on H_{M}^{∗,∗′}(M(X̃_{P}), ℤ_{(2)}), and the fact that P has quadratic point, we have by Theorem 2.3.4 , that H_{M}^{∗,∗′}(M(X̃_{P}), ℤ_{(2)}) ⊂ H_{M}^{∗,∗′}(M(X̃_{P}), ℤ/2), and H_{M}^{∗,∗′}(M(X̃_{P}), ℤ_{(2)}) is a group of exponent 2 (in particular, we can change ℤ_{(2)} by ℤ).

2.4. Some facts about quadratic forms. Let k be the field of characteristic different from 2. Quadratic form over k is pair (V, q), where V/k-finite dimensional vector space and q : V → k is a quadratic function (i.e., diagonal part of some symmetric bilinear function V × V → k). If (e_{1}, …, e_{n}) is a basis of V, then q is defined by it’s matrix (q_{ij}). We, certainly, can always diagonalize the matrix of q. So, q is isomorphic to some form ⟨a_{1}, …, a_{n}⟩, where the latest denotes quadratic form corresponding to the diagonal matrix with eigenvalues - a_{i}’s . Of course, for two different sets a_{1}, …, a_{n} and b_{1}, …, b_{n} we can have: ⟨a_{1}, …, a_{n}⟩ isomorphic to ⟨b_{1}, …, b_{n}⟩ (for example, ⟨1, −1⟩ = ⟨a, −a⟩); so, such presentation is not unique (not to say that each eigenvalue can be multiplied by a square). Due to Witt Chain equivalence theorem: we can transform one such presentation into arbitrary other by changing only two of our entries at a time.

The form ⟨1, −1⟩ := ℍ is called elementary hyperbolic form. Notice, that this form represents 0 (i.e., we can find nonzero vector v ∈ V, s.t. q(v) = 0; in our case v = (1, 1)). Direct sum of elementary hyperbolic forms is called hyperbolic. Notice: q is hyperbolic iff V_{q} contains isotropic subspace of dimension dim(V)/2. The following observation is trivial: quadratic form q represents 0 iff it has ℍ as a direct summand. It follows from the fact that every subform is a direct summand (q = q′ ⊥ q′^{⊥}). The form is called anisotropic, if it does not represent 0.

Also we should take into account the Witt cancellation theorem :

Theorem 2.4.1 ([9], I, Theorem 4.2).

If for some quadratic forms q, s, s′ we have q ⊥ s = q ⊥ s′, then s = s′.

Each form q can be represented as q = d·ℍ ⊥ q′, where q′ is the unique anisotropic form with such a property, and d is called the Witt (isotropity) index.

Definition 2.4.2 .

The Witt ring of quadratic forms over k is a Grothendieck group of quadratic forms over k modulo hyperbolic forms; addition is as in Groth. group, and multiplication via tensor product of quadratic spaces (in terms of diagonal presentations: ⟨a_{1}, …, a_{n}⟩ · ⟨b_{1}, … b_{m}⟩ =⊥_{i=1,...,n;j=1,...,m} ⟨a_{i} · b_{j}⟩).

Each element of W (k) can be represented by a unique anisotropic quadratic form, i.e.: as a set, W (k) is the same as the set of anisotropic forms.

W (k) contains one natural ideal: the ideal of even-dimensional forms I (notice, that hyperbolic forms are even-dimensional). Powers of the ideal I provide us with the multiplicative decreasing filtration on W (k): W (k) ⊃ I ⊃ I^{2} ⊃ . . .. Denote as gr^{∗}(W (k)) the associated graded ring. It is easy to see that k^{∗}/(k^{∗})^{2} = I/I^{2} via map: a ↦ ⟨1, −a⟩.

Some quadratic forms are better than others. And the best possible forms are Pfister forms.

Definition 2.4.3 .

Let a_{1}, a_{2}, …, a_{n} are elements of k^{∗}. Then the (big) Pfister form, associated with this set is the form ⟨1, −a_{1}⟩ × ⟨1, −a_{2}⟩ × ⋯ × ⟨1, −a_{n}⟩. We will denote such form as q_{{a_{1},...,a_{n}}} = ⟨⟨a_{1}, …, a_{n}⟩⟩.

Definition 2.4.4 .

Milnor’s K-theory of the field k is a homogeneous quadratic algebra (over ℤ) with generators k^{∗} and relations a ⊗ (1 − a), a ∈ k^{∗}\1 (K_{∗}^{M}(k) = T (k^{∗})/R, where R is double-sided ideal generated by relations as above).

We will denote element of K_{1}^{M}(k) as {a}. Element of K_{n}^{M}(k) is called pure if it is multiplicative, i.e. the product of some elements of degree 1 : {a_{1}, …, a_{n}}.

It is not difficult to see that K_{∗}^{M}(k) is a skew-symmetric ring (hence, K_{∗}^{M}(k)/2 is symmetric).

We have natural map ρ : K_{∗}^{M}(k)/2 → gr^{∗}(W (k)), given on the generators by the rule: {a} ↦ ⟨1, −a⟩. Since gr^{∗}(W (k)) is generated by the first degree component, it will be a surjection. The well-known Milnor conjecture states that ρ is an isomorphism. Finally, it was proven by V.Voevodsky (see [13], p.14).

Notice, that ρ({a_{1}, …, a_{n}}) = ⟨⟨a_{1}, …, a_{n}⟩⟩. The following result is due to R.Elman and T.Y.Lam:

Theorem 2.4.5 ([3], Main Theorem 3.2).

The following conditions are equivalent:

1) {a_{1}, …, a_{n}} = {b_{1}, …, b_{n}} ∈ K_{n}^{M}(k)/2

2) ⟨⟨a_{1}, …, a_{n}⟩⟩ − ⟨⟨b_{1}, …, b_{n}⟩⟩ ∈ I^{n+1}(W (k))

3) The forms ⟨⟨a_{1}, …, a_{n}⟩⟩ and ⟨⟨b_{1}, …, b_{n}⟩⟩ are isomorphic.

Moreover, if ⟨⟨a_{1}, …, a_{n}⟩⟩ is isotropic, then it is hyperbolic.

Definition 2.4.6 . A Pfister relative of the big Pfister form q_{{a_{1},...,a_{n}}} = ⟨⟨a_{1}, …, a_{n}⟩⟩ is a subform of it

which has dimension > 1/2 dim(⟨⟨a_{1}, …, a_{n}⟩⟩) = 2^{n−1}.

A small Pfister form is a subform q_{{a_{1},...,a_{n}}} of dimension 2^{n−1} + 1 (in particular, it is a relative).

Two forms q_{1} and q_{2} are called Pfister half-neighbors if dim(q_{1}) = dim(q_{2}), and q_{1} ⊥ q_{2} = q_{{a_{1},...,a_{n}}}.

Let some element h ∈ K_{n}^{M}(k)/2 be represented as a sum of two pure symbols: h = {a_{1}, …, a_{n}} + {b_{1}, …, b_{n}}.

Definition 2.4.7 . An Albert form associated with h is a form A_{h} = ⟨⟨a_{1}, …, a_{n}⟩⟩ ⊥ −⟨⟨b_{1}, …, b_{n}⟩⟩. (Notice that it is defined non-uniquely, since h can have different presentations as

a sum of two pure symbols.)

Theorem 2.4.8 ([3], Theorem 4.5). The only possible values for the dim((A_{h})|_{anis.}) are: 2^{n+1} − 2^{i+1}, where 0 ⩽ i ⩽ n. Moreover, the following conditions are equivalent:

1) dim((A_{h})|_{anis.}) = 2^{n+1} − 2^{i+1};

2) {a_{1}, …, a_{n}} and {b_{1}, …, b_{n}} have common pure divisor {c_{1}, …, c_{i}} of degree i, and do not have one of degree > i.

3) h has pure divisor {c_{1}, …, c_{i}} of degree i, and does not have one of degree > i.

Theorem 2.4.9 . Let A_{h} be an Albert form corresponding to the symbol h = {a, b} + {c, d} in

K_{2}^{M}(k)/2. Then the following conditions are equivalent:

1) h is pure (see Definition 2.4.4 );

2) (A_{h})_{anis.} has dimension ⩽ 4;

3) A_{h} is hyperbolic over some field E, where [E : k] = 2;

4) A_{h} is hyperbolic over some field E, where [E : k] = 2 · d, with d-odd.

Proof (1 ⇔ 2) By Theorem 2.4.8 . (1 ⇒ 3) Again by Theorem 2.4.8 , we have that there exist f_{1}, f_{2}, g ∈ k^{∗}, that

{a, b} = {f_{1}, g}, {c, d} = {f_{2}, g} (in K_{2}^{M}(k)/2). Then A_{h} = ⟨1, −f_{1}, −g, f_{1} · g⟩ − ⟨1, −f_{2}, −g, f_{2} · g⟩, and so it is hyperbolic over k\sqrt{g}.

(3 ⇒ 1) Let A_{h} becomes hyperbolic over some quadratic extension E = k\sqrt{g}. This means that ⟨1, −a, −b, a · b⟩|_{E} = ⟨1, −c, −d, c · d⟩|_{E}, and by Theorem 2.4.5 {a, b}|_{E} =

{c, d}|_{E}, i.e.: h ∈ Ker(K_{2}^{M}(k)/2 →^{j} K_{2}^{M}(E)/2).

By the theorem of A.Merkurjev (see [11], Theorem and Proposition 2), we have the

following commutative diagram: \begin{matrix}K_{1}^{M}(k)/2 & ⟶^{·\{g\}} & K_{2}^{M}(k)/2 & ⟶^{j} & K_{2}^{M}(k\sqrt{g})/2 \\ ∥ & & ∥ & & ∥ \\ H_{et}^{1}(k, ℤ/2) & ⟶^{·\{g\}} & H_{et}^{2}(k, ℤ/2) & ⟶^{j} & H_{et}^{2}(k, π_{∗}(ℤ/2)), \end{matrix}

where π_{∗}(ℤ/2) is a Gal(k)-module, induced from the subgroup Gal(k\sqrt{g}) ⊂ Gal(k). The lower row here is an exact sequence (coming from the short exact sequence 0 → ℤ/2 → π_{∗}(ℤ/2) → ℤ/2 → 0). So, if h ∈ Ker(K_{2}^{M}(k)/2 → K_{2}^{M}(E)/2), then h is divisible by {g} and is pure.

(3 ⇒ 4) Trivial. (4 ⇒ 2) dim((A_{h})_{anis.}) ⩽ 4 ⇔ ⟨−a, −b, ab, c, d, −cd⟩ is isotropic. Suppose A_{h} is

hyperbolic over some field E of degree 2d with d-odd. We have a tower of fields k ⊂ F ⊂ E, where [F : k] = d (F corresponds to some 2-Sylow subgroup of Gal(E/k)). Over F the form ⟨−a, −b, ab, c, d, −cd⟩ is isotropic (by (3 ⇒ 2)). But then by Springer’s theorem (see Theorem 2.4.14 ) this form is isotropic over k.

Definition 2.4.10 . Projective quadric is called conservative, if Ker_{∗}^{Q} := Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2)

is nontrivial.

Theorem 2.4.11 ([13], Statement 1 of Section 3.2). For any quadric Q of dimension ⩾ 2^{n} − 1 we have: Ker_{n}^{Q} = 0. Remark B.Kahn, M.Rost and R.Sujatha obtained the same result for n ⩽ 4 (see

[8], Theorem 1).

We can introduce the notion of Chern classes of a quadratic form: Definition 2.4.12 . For n-dimensional quadratic form q = ⟨a_{1}, …, a_{n}⟩ the Chern character

Ch(q) ∈ K_{0}^{M}(k)/2 × K_{1}^{M}(k)/2 × K_{2}^{M}(k)/2 × … can be defined as ∏_{1⩽j⩽n}(1+{a_{i}})·(∏_{1⩽i⩽[n/2]}(1+{−1}))^{−1}. (In our definition Ch is multiplicative only on even-dimensional forms but it has the advantage of being equal to 1 on hyperbolic forms (even and odd-dimensional), in particular, Ch(q) = Ch(q|_{anis.}).)

c_{m}(q) is a K_{m}^{M}(k)/2-component of Ch(q).

Definition 2.4.13 . For an even-dimensional projective quadric Q define det(Q) as c_{1}(q) for arbitrary

form q representing Q.

For an odd-dimensional projective quadric Q define c_{2}(Q) as c_{2}(q), where q-unique form representing Q with c_{1}(q) = 0 (or, in other words: det(q) = (−1)^{dim(q)−1/2}).

The following theorem belongs to T.A.Springer:

Theorem 2.4.14 ([9], VII, Theorem 2.3). Let quadric Q/k contains a point of odd degree. Then it contains rational point. Let q be a quadratic form over k. We can associate with q the following varieties:

Projective quadric Q, defined by this form; Q^{1} - variety of lines on Q; Q^{2} - variety of 2-dimensional planes on Q;....;Q^{i} - variety of i-dimensional planes on Q, for 0 ⩽ i ⩽ [n/2].

Also we can define the variety \underline{Q}^{i} as a variety of flags: (l_{0} ⊂ l_{1} ⊂ l_{2} ⊂ ⋯ ⊂ l_{i}), where l_{j} is a projective subspace of dimension j living on Q.

We have natural projection \underline{Q}^{i} → Q^{i}, in particular, Q^{i} has a point over k(\underline{Q}^{i}). On the other hand, from the existence of rational i-dimensional plane on Q follows the existence of i-dimensional rational flag on it.

So, using Theorem 2.3.4 we get: Theorem 2.4.15 . M(X\underline{Q}^{i}) = M(X_{Q^{i}}).

Since, evidently, from the existence of rational j-dimensional plane on Q follows the existence of rational i-dimensional plane on it, for j ⩾ i, we have by Theorem 2.3.6 :

Theorem 2.4.16 . M(X^{·}_{Q^{j}}) ⊗ M(X^{·}_{Q^{i}}) = M(X^{·}_{Q^{j}}) for i ⩽ j. Since big Pfister form q_{{a_{1},...,a_{n}}} is hyperbolic as soon as it is isotropic (by Theorem

2.4.5 ), we have that Q_{{a_{1},...,a_{n}}}^{dim(Q)/2} has a rational point over k(Q_{{a_{1},...,a_{n}}}). So, by Theorem 2.3.4 we have that M(X^{·}_{Q_{{a_{1},...,an}}^{dim(Q)/2}}) = M(X^{·}_{Q_{{a_{1},...,an}}}). Now, from Theorem 2.4.16 we

have the following result.

Theorem 2.4.17 . For big Pfister quadric Q = Q_{{a_{1},...,a_{n}}} we have: M(X^{·}_{Q}) = M(X^{·}_{Q^{1}}) = ⋯ =

M(X^{·}_{Q^{dim(Q)/2}}). (We remind that the dimension of the projective quadric is 2 less than of the corresponding form.)

Remark Actually, big Pfister quadric and a hyperplane section in a big Pfister quadric are the only examples of quadrics having such a property. We will not use this fact.

Theorem 2.4.18 . For projective quadric Q: the variety Q^{i} has a rational point if and only if it has a

zero-cycle of degree 1.

In particular, M(X_{Q^{i}}) = M(X_{R^{j}}) for two quadrics Q and R, if and only if Q^{i} has a rational point over R^{j} and R^{j} has a rational point over Q^{i}.

Proof

If the variety \underline{Q}^{i} has a zero-cycle of degree 1, then it has point of odd degree. So, projecting from \underline{Q}^{i} to Q, we get a point of odd degree on Q. By Springer’s theorem (see Theorem 2.4.14 ), q is isotropic. Let q = ℍ ⊥ q′. Then M(X_{Q′}^{i−1}) = M(X_{Q^{i}}) = ℤ (by Theorem 2.3.4 ). So, in the same way, \underline{Q′}^{i−1} has a point of odd degree, and, again, that means that q′ is isotropic by Springer theorem (Theorem 2.4.14 ). After i such steps we get: Q^{i} has a rational point. (See also Theorem 2.4.15 and discussion before it.)

The second statement follows from the first by Theorem 2.3.4 .

Definition 2.4.19 .

Let q be an anisotropic quadratic form, and Q-corresponding projective quadric. Then q|_{k(Q)} = h_{1} · ℍ ⊥ q_{1}, where q_{1} is an anisotropic form defined over k(Q) (such form is unique). Then we have: q_{1}|_{k(Q)(Q_{1})} = h_{2} · ℍ ⊥ q_{2}, where q_{2} is an anisotropic form defined over k(Q)(Q_{1}), and we can continue this until we get q_{s} = 0.

The numbers h_{1}, h_{2}, …, h_{s} are called Witt numbers of a quadric Q.

The tower of fields: k ⊂ k(Q) ⊂ k(Q)(Q_{1}) ⊂ ⋯ ⊂ k(Q) … (Q_{s−1}) is called the Universal splitting tower of M.Knebusch.

We will say that for two quadrics Q, R such towers are equivalent, if their Witt numbers are equal (pairwise) (in particular, dim(Q) = dim(R)) and for each i ⩾ 0, Q^{h_{1}+···+h_{i}} has a rational point over R^{h_{1}+···+h_{i}} and for each j, R^{h_{1}+···+h_{j}} has a rational point over Q^{h_{1}+···+h_{j}} (see also discussion below). From the definition of Witt numbers we see that: over k(Q) all varieties: Q, ..., Q^{h_{1}−1} have a rational point, i.e.: M(X_{Q}) = ⋯ = M(X_{Q^{h_{1}−1}}) (by Theorem 2.4.16 and Theorem 2.3.6 ). Moreover, Q^{h_{1}} does not have a rational point over k(Q) (again, by definition). Repeating this arguments with k(Q^{h_{1}}), k(Q^{h_{1}+h_{2}}), etc. ..., we get:

Theorem 2.4.20 .

M(X_{Q}) = ⋯ = M(X_{Q^{h_{1}−1}}) ≠ M(X_{Q^{h_{1}}}) = ⋯ = M(X_{Q^{h_{1}+h_{2}−1}}) ≠ … ≠ M(X_{Q^{h_{1}+···+h_{s−1}}}) = ⋯ = M(X_{Q^{[dim(Q)/2]}})

For two smooth projective varieties P and R we will say that M(X_{P}) ⩽ M(X_{R}), if M(X_{P}) × M(X_{R}) = M(X_{R}), and that M(X_{P}) < M(X_{R}), if moreover, M(X_{P}) × M(X_{R}) ≠ M(X_{P}).

In this light we can put statement (using Theorem 2.4.16 and Theorem 2.4.18 ) in the following form: M(X_{Q})=…=M(X_{Q^{h_{1}−1}})<M(X_{Q^{h_{1}}})=…=M(X_{Q^{h_{1}+h_{2}−1}})<…< M(X_{Q^{h_{1}+···+h_{s−1}}})=…=M(X_{Q^{[dim(Q)/2]}}).

Notice that if P ⊂ Q is a subquadric of codimension m, then from the existence of rational point on Q^{m} follows the existence of rational point on P (just intersect corresponding rational m-dimensional projective subspace with P), hence, by Theorem 2.3.6 (since over k(Q^{m}) P has a rational point) we get: M(X_{P})×M(X_{Q^{m}}) = M(X_{Q^{m}}).

In the notations as above: M(X_{P}) ⩽ M(X_{Q^{m}}). On the other hand, since Q has a rational point over k(P), we get (again by Theorem 2.3.6 ) that M(X_{Q}) ⩽ M(X_{P}). So, M(X_{Q}) ⩽ M(X_{P}) ⩽ M(X_{Q^{m}}).

Also, notice, that if for some smooth projective varieties P and R we have M(X_{P}) ⩽ M(X_{R}) and M(X_{R}) ⩽ M(X_{P}), then M(X_{P}) = M(X_{R}) by the very definition.

Applying this result to the big Pfister quadric Q_{{a_{1},...,a_{n}}} (see Definition 2.4.3 ), and using the fact that Q_{{a_{1},...,a_{n}}} is hyperbolic as soon as it is isotropic (see Theorem 2.4.5 ), i.e. Q_{{a_{1},...,a_{n}}}^{dim(Q)/2} has a rational point over k(Q), and so, M(X_{Q_{{a_{1},...,an}}}) = M(X_{Q_{{a_{1},...,an}}^{dim(Q)/2}}), we get:

Theorem 2.4.21 .

Let Q_{{a_{1},...,a_{n}}} be a big Pfister quadric, and P ⊂ Q_{{a_{1},...,a_{n}}} - it’s relative (see Definition 2.4.6 ). Then M(X_{Q_{{a_{1},...,an}}}) = M(X_{P}).

Remark Actually, these two conditions are equivalent.

In conclusion, we will say few words about Chow motives of isotropic and hyperbolic quadrics. Using some abuse of notations we will call odd dimensional quadratic form r · ℍ ⊥ ⟨1⟩ also hyperbolic.

Theorem 2.4.22 .

1) Let q = r·ℍ be a hyperbolic quadratic form of dimension 2r, and Q -corresponding projective quadric.

Then

M(Q) = ∑_{\begin{matrix}i=0,...,r−2 \\ r,...,2r−2 \end{matrix}} ℤ(i)[2i] ⊕ (ℤ(r − 1)[2r − 2] ⊕ ℤ(r − 1)[2r − 2]).

2) Let q′ = r · ℍ ⊥ ⟨1⟩ be a hyperbolic quadratic form of dimension 2r + 1, and Q′ -corresponding projective quadric.

Then

M(Q′) = ∑_{i=0,...,2r−1} ℤ(i)[2i].

Proof

1) We have the following set of mutually orthogonal idempotents in Hom_{Chow}(Q, Q) = CH^{dim(Q)}(Q × Q): h^{i} × l_{i}; l_{j} × h^{j} for 0 ⩽ i, j ⩽ r − 2, and l′_{r−1} × l′′_{r−1}, l′′_{r−1} × l′_{r−1} , or l′_{r−1} × l′_{r−1}, l′′_{r−1} × l′′_{r−1} (depending on: is r - even, or odd), where h^{i} is a plane section of codimension i and l_{j} is a projective subspace of dimension j on Q (exists since Q is hyperbolic), l′_{r−1} and l′′_{r−1} denote projective subspaces of middle dimension from different families.

These idempotents define our decomposition.

2) The same proof.

The same result can be obtained from the iterated application of the following: Suppose now that Q is just isotropic, i.e.: it contains a rational point, and corre-

sponding quadratic form contains a elementary hyperbolic summand: q = ⟨1, −1⟩ ⊥ q′. Notice, that Q′ will be precisely quadric of lines on Q passing some rational point p. Q is a union of three strata: p, (T_{p}(Q) ∩ Q)\p, and Q\(T_{p}(Q) ∩ Q) = 𝔸^{dim(Q)}. From this it is not difficult to deduce the following result (see Theorem 2.4.25 for the global result).

Theorem 2.4.23 ([14], Proposition 1).

M(Q) = ℤ ⊕ M(Q′)(1)[2] ⊕ ℤ(dim(Q))[2 dim(Q)]

Remark 1 By Theorem 2.2.4 Chow(k) is a full subcategory of DM_{gm}(k), closed under direct summands, so everything is valid in DM_{gm}(k) as well.

Remark 2 Notice that the map M(Q) → ℤ is given by the generic cycle (see Definition 2.2.1 ), and ℤ(dim(Q))[2 dim(Q)] → M(Q) is dual to it via duality: \underline{Hom}(M(Q), ℤ(dim(Q))[2 dim(Q)]) = M(Q) (see Theorem 2.1.23 ).

Definition 2.4.24 .

Let P - is a smooth projective variety.

We will say that n-dimensional quadric Q is globally i + 1-times isotropic over P, if the natural projection P × Q^{i} → P has a section, or, which is the same, there is a map φ : P → Q^{i}.

We will say that n-dimensional quadric Q is globally i + 1-times strongly isotropic over P, if the natural projection P × \underline{Q^{i}} → P has a section.

Let P be a smooth projective variety, over which quadric Q is globally i + 1-times isotropic.

Let R will be a variety of pairs: (p, π), where p is a point of P, and π is a i + 1dimensional projective plane on Q, containing i-dimensional plane φ(p). R has natural projection to P with fibers - n−2(i+1)-dimensional projective quadrics. In particular, R is a smooth projective variety.

The same construction works in a strongly isotropic case.

Theorem 2.4.25 .

Let P be a smooth projective variety, over which quadric Q is globally i + 1-times strongly isotropic.

Then M(P ×Q) = (⊕_{j=0,...,i}M(P)(j)[2j])⊕M(R)(i+1)[2i+1]⊕(⊕_{j=0,...,i}M(P)(n− j)[2n − 2j]).

Moreover the maps: M(P × Q) → M(P)(j)[2j] (resp. M(P)(n − j)[2n − 2j] → M(P ×Q)) are just id_{M(P)} ×α_{j} (resp. id_{M(P)} ×β_{j}) (see the definition before Theorem 3.1 ).

Proof

We have a map ψ : P → \underline{Q^{i}}, and then a map φ : P → Q^{i} (by the natural projection \underline{Q^{i}} → Q^{i} (see the discussion before Theorem 2.4.15 ).

We have the following subvariety F of P × Q: F consists of pairs (p, q), where q belongs to i-dimensional projective subspace φ(p). So, F is a i-dimensional projective bundle over P (in Zariski topology).

Consider the variety G, which consists of pairs: (p, q), where p - point on P, and the i + 1-dimensional projective space generated by q and φ(p) lives on Q (in particular, q ∉ φ(p)).

We have natural projection G → R with fibers - ℙ^{i+1}\ℙ^{i} = 𝔸^{i+1}.

Consider the complement to G ∪ F in P × Q - W. It will be a fibration over P and the fiber is the complement to the T_{l_{i},Q}, where T_{l_{i},Q} is a set of points q, s.t. the subspace generated by l_{i} and q lives on Q; such T_{l_{i},Q} has dimension n − i − 1.

Since over our P, Q is strongly i + 1-times isotropic, we have flag of projective bundles F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{i} = F inside the bundle F. Complements to their T_{−,Q}’s will provide us with the filtration W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{i} = W on W. The complements: W_{j+1}\W_{j} will be a fibration over P with fibers 𝔸^{n−j}.

By Gysin distinguished triangle (see [17], Proposition 3.5.4, and also: the Definition 2.1.1 there (for the definition of a motive of nonprojective smooth variety)) we have that M(P × Q) is an extension (see Definition 2.5.6 ) of: M(F)(n − i)[2n − 2i], M(G)(i + 1)[2i + 2] and M(W).

By [17], Proposition 3.5.1 we have: M(F) = ⊕_{j=0,...,i}M(P)(j)[2j]. By homotopy invariance of a motive: M(G) = M(R). And by homotopy invariance of a motive and by Gysin distinguished triangle ([17], Proposition 3.5.4) we have: M(W) = ⊕_{j=0,...,i}M(P)(j)[2j].

Since all the motives are pure (i.e.: the direct summands of the motives of smooth projective varieties), we have by duality (see Theorem 2.1.23 ) and Theorem 2.1.18 that this extension is trivial (for two pure motives A and B we have Hom_{DM}(A, B(i)[2i+ j]) = 0 for all j > 0 (can assume that A, B are motives of smooth projective varieties, let B has dimension n, then \underline{Hom}(B, ℤ(n)[2n]) = B, and Hom(A, B(i)[2i+j]) = Hom(A × B, ℤ(n + i)[2n + 2i + j]) = 0 (by Theorem 2.1.18 )), and an elementary extension (see Definition 2.5.6 ) is defined by some element u ∈ Hom(A, B[1])).

So, M(P × Q) = (⊕_{j=0,...,i}M(P)(j)[2j]) ⊕ M(R)(i + 1)[2i + 1] ⊕ (⊕_{j=0,...,i}M(P)(n − j)[2n − 2j]).

2.5. Triangulated categories.

Definition 2.5.1 ([5],II,§5,10).

An additive category is a category C, s.t. the following properties are satisfied: A1. For every X, Y ∈ Ob(C), the set Hom_{C}(X, Y) has a structure of an abelian

group, and the composition map: Hom_{C}(X, Y) × Hom_{C}(Y, Z) → Hom_{C}(X, Z) is bilinear.

A2. There exist “zero object”, i.e.: such object 0, that Hom_{C}(0, 0) = 0 A3. For every pair of objects X_{1}, X_{2} ∈ C there exists object Y (a direct sum), and

morphisms p_{1,2}, i_{1,2}:

X_{1}←^{p_{1}} ; i_{1}^{→}Y →^{p_{2}} ; i_{2}^{←}X_{2}

that p_{1}i_{1} = id_{X_{1}}, p_{2}i_{2} = id_{X_{2}}, i_{1}p_{1} + i_{2}p_{2} = id_{Y}, p_{2}i_{1} = p_{1}i_{2} = 0.

An additive functor is a functor F : C_{1} → C_{2}, which preserves group structure on Hom’s, “zero object”, and direct sums.

Definition 2.5.2 . Let C be an additive category. The pseudo-abelian envelope of C is the category P A(C), whose objects are pairs (X, p_{X}), where X is an object of C, and p_{X} ∈ End(X) is a projector: p^{2} = p. The morphism from (X, p_{X}) to (Y, p_{Y}) is a morphism φ : X → Y, s.t. p_{Y} ∘ φ = φ, and φ = φ ∘ p_{X}. We have a natural functor F : C → P A(C) (F (X) = (X, id)). P A(C) is naturally an additive category, and F is a full embedding (by definition). For any idempotent p ∈ End(X) we have a direct sum decomposition in P A(C): F (X) = (X, p) ⊕ (X, (1 − p)).

Definition 2.5.3 ([5],IV,§1,1).

The triangulated category is an additive category D together with:

a) An additive automorphism T : D → D, called: the shift functor. (We will denote T^{k}(X) as X[k].)

b) Class of distinguished (exact) triangles, (triangle is a diagram of the type:

X →^{u} Y →^{v} Z →^{w} X[1];

the morphism of two triangles is a commutative diagram:

\begin{matrix}X & ⟶^{u} & Y & ⟶^{v} & Z & ⟶^{w} & X[1] \\ ^{f}↓ & & ^{g}↓ & & ^{h}↓ & & ^{f[1]}↓ \\ X′ & ⟶^{u′} & Y′ & ⟶^{v′} & Z′ & ⟶^{w′} & X′[1] \end{matrix}).

This data should satisfy the following axioms:

T R1. a. The triangle: X →^{id} X → 0 → X[1] is distinguished.

b. If triangle is distinguished, then any triangle isomorphic to it is distinguished

as well. c. Any morphism X →^{u} Y can be completed to a distinguished triangle:

X →^{u} Y →^{v} Z →^{w} X[1].

T R2. The triangle: X →^{u} Y →^{v} Z →^{w} X[1] is distinguished if and only if

Y →^{v} Z →^{w} X[1] →^{−u[1]} Y [1] is.

T R3. Let we have two distinguished triangles: X →^{u} Y →^{v} Z →^{w} X[1], and X′ →^{u′} Y′ →^{v′}

Z′ →^{w′} X′[1], and a commutative diagram: \begin{matrix}X & ⟶^{u} & Y \\ ^{f}↓ & & ^{g}↓ \\ X′ & ⟶^{u′} & Y′ \end{matrix}. Then there is morphism

h (not necessarily unique), which together with f and g gives us the morphism of triangles:

\begin{matrix}X & ⟶^{u} & Y & ⟶^{v} & Z & ⟶^{w} & X[1] \\ ^{f}↓ & & ^{g}↓ & & ^{h}↓ & & ^{f[1]}↓ \\ X′ & ⟶^{u′} & Y′ & ⟶^{v′} & Z′ & ⟶^{w′} & X′[1] \end{matrix}.

T R4. (so-called: octahedron axiom) Let us denote as U →^{[1]} V the map from U to V [1].

Then the triangle can be denoted as: \begin{aligned}& Z \\ ↗ & ↓ [1] \\ Y & ⟵ & X \end{aligned} . We will denote distinguished

triangle as: \begin{aligned}& Z \\ ↗ ⋆ & ↓ [1] \\ Y & ⟵ & X \end{aligned} .

Let we have a diagram of the type: \begin{matrix}X′ \\ [1] ↙ & [1]↓ & ⋆ & ↖ \\ Z′ & ⟵ & Y & ⟶ & Z \\ [1] ↘⋆ & ↑ & ↗ \\ X \end{matrix} , where the “triangles”

without ⋆’s are commutative. Then this diagram can be completed to a whole “octahedron”, i.e.: there exists a diagram of the type:

\begin{matrix}X′ \\ [1] ↙⋆ & ↑ & ↖ \\ Z′ & ⟶ & Y′ & ⟵ & Z \\ [1] ↘ & [1]↓ & ⋆ & ↗ \\ X \end{matrix} , (with the same morphisms: X′ →^{[1]} Z′, Z′ →^{[1]} X, X → Z,

Z → X′).

From the axioms T R1 − T R4 it is easy to deduce the following:

Theorem 2.5.4 ([5], IV,§1, Proposition 3). If X →^{f} Y →^{g} Z →^{h} X[1] is a distinguished triangle, then for any object U we have

exact sequences:

… → Hom(U, X[i]) →^{f_{∗}[i]} Hom(U, Y [i]) →^{g_{∗}[i]} Hom(U, Z[i]) →^{h_{∗}[i]} Hom(U, X[i + 1]) → …;

… → Hom(X[i + 1], U) →^{h^{∗}[i]} Hom(Z[i], U) →^{g^{∗}[i]} Hom(Y [i], U) →^{f^{∗}[i]} Hom(X[i], U) → . . ..

From the above result, TR1 c) and TR3 follows that the following definition is correct.

Definition 2.5.5 . Let we have some morphism f : X → Y in D. Then: Cone(f) is a unique up to

noncanonical isomorphism object Z, s.t. there exists a distinguished triangle of the

type: X →^{f} Y → Z → X[1].

If f = 0, then Cone(f) = Y ⊕ X[1]; and f is an isomorphism iff Cone(f) = 0.

Definition 2.5.6 . Let A, B, C ∈ Ob(C). We say that C is an elementary extension of A and B, if

there exist either an exact triangle of the type: A → C → B → A[1], or of the type: B → C → A → B[1].

Let A_{1}, …, A_{n}, C ∈ Ob(C). We say that C is an extension of A_{1}, …, A_{n}, if there exist 1 ⩽ j ⩽ n, and B ∈ Ob(C), s.t. C is an elementary extension of A_{j} and B, and B is an extension of (A_{i})_{1⩽i⩽n;i≠j}.

From Theorem 2.5.4 it follows that if Hom(U, A_{i}) = 0 for all 1 ⩽ i ⩽ n, and some U ∈ Ob(C), and C is an extension of A_{1}, …, A_{n}, then Hom(U, C) = 0 (the same about Hom(C, U)).

From Theorem 2.5.4 we also have the following:

Theorem 2.5.7 . Let we have pair of endomorphisms φ, ψ of some distinguished triangle A → C →

B → A[1]. If φ_{B} = 0, ψ_{A} = 0, then (ψ ∘ φ)_{C} = 0.

From TR4 it is easy to get:

Theorem 2.5.8 . Let we have a diagram:

\begin{matrix}A_{1} & A_{2} \\ ↗ ⋆ & ↓[1] & ↘[1] & ↗ ⋆ & ↓[1] & ↘ [1] \\ X & ⟵ & C_{1} & ⟶ & D & ⟵ & C_{2} & ⟶ & Y \\ ↖ & ↑ & ⋆ ↙[1] & ↖ & ↑ & ⋆ ↙ [1] \\ B_{1} & B_{2} \end{matrix} ,

where all triangles without ⋆ are commutative. Then there exists a diagram of the type:

\begin{matrix}A \\ ↗ ⋆ & ↓[1] & ↘[1] \\ X & ⟵ & D′ & ⟶ & Y \\ ↖ & ↑ & ⋆ ↙[1] \\ B \end{matrix} ,

where A = Cone[−1](A_{1} → A_{2}[1]), and B = Cone[−1](B_{2} → B_{1}[1]), and A_{1} → A_{2}[1], B_{2} → B_{1}[1] are maps from the diagram.

Moreover, if we had maps: α_{1} : U → C_{1}, α_{2} : U → C_{2}, s.t. the diagram:

\begin{matrix}U & ⟶^{α_{1}} & C_{1} \\ ^{α_{2}}↓ & & ↓ \\ C_{2} & ⟶ & D \end{matrix} is commutative, then there exists β : U → D′, s.t. the diagram:

\begin{matrix}U & ⟶^{β} & D′ \\ ^{α_{2}}↓ & & ↓ \\ C_{2} & ⟶ & Y \end{matrix} is commutative.

Applying few times the observation that the Cone of a zero map is a direct sum, we get the following:

Theorem 2.5.9 .

Suppose we have a diagram:

\begin{aligned}& & A_{1} & & A_{2} & & & A_{m} \\ ↗ ⋆ & ↓[1] & ↗ ⋆ & ↓[1] & … & ↗ ⋆ & ↓[1] \\ A & ⟵ & Y_{1} & ⟵ & Y_{2} & Y_{m−1} & ⟵ & Y_{m} \end{aligned} ,

s.t. for each 1 ⩽ i ⩽ m − 1 the map Y_{i} → A_{i+1} can be lifted to a map from A to A_{i+1}, and Y_{m} = 0.

Then A = ⊕_{i=1,...m}A_{i}.

In the same way, if in the diagram: \begin{aligned}& & B_{1} & & B_{2} & & & B_{m} \\ ↙ ⋆ & ↑[1] & ↙ ⋆ & ↑[1] & … & ↙ ⋆ & ↑[1] \\ B & ⟶ & Z_{1} & ⟶ & Z_{2} & Y_{m−1} & ⟶ & Z_{m} \end{aligned} , with

Z_{m} = 0, all the maps: B_{i+1} → Z_{i} can be lifted to maps from B_{i+1} to B, then we have: B = ⊕_{i=1,...,m}B_{i}.

From the previous two theorems we get:

Theorem 2.5.10 .

Let we have a diagram:

\begin{matrix}A_{1} & A_{2} & A_{3} & A_{m} \\ ↗ ⋆ & ↓[1] & ↘[1] & ↗ ⋆ & ↓[1] & ↘ [1] & ↗ ⋆ & ↓[1] & ↘ [1] & ↗ ⋆ & ↓[1] & ↘ [1] \\ X & ⟵ & C_{1} & ⟶ & X_{1} & ⟵ & C_{2} & ⟶ & X_{2} & ⟵ & C_{3} & ⟶ & X_{3} … X_{m−1} & ⟵ & C_{m} & ⟶ & X_{m} \\ ↖ & ↑ & ⋆ ↙[1] & ↖ & ↑ & ⋆ ↙ [1] & ↖ & ↑ & ⋆ ↙ [1] & ↖ & ↑ & ⋆ ↙ [1] \\ B_{1} & B_{2} & B_{3} & B_{m} \end{matrix} ,

then there exists a diagram of the type:

\begin{matrix}A \\ ↗ ⋆ & ↓[1] & ↘[1] \\ X & ⟵ & Y & ⟶ & X_{m} \\ ↖ & ↑ & ⋆ ↙[1] \\ B \end{matrix} ,

where A and B have “decompositions”: \begin{aligned}& & A_{1} & & A_{2} & & & A_{m} \\ ↗ ⋆ & ↓[1] & ↗ ⋆ & ↓[1] & … & ↗ ⋆ & ↓[1] \\ A & ⟵ & Y_{1} & ⟵ & Y_{2} & Y_{m−1} & ⟵ & Y_{m} \end{aligned} ,

and \begin{aligned}& & B_{1} & & B_{2} & & & B_{m} \\ ↙ ⋆ & ↑[1] & ↙ ⋆ & ↑[1] & … & ↙ ⋆ & ↑[1] \\ B & ⟶ & Z_{1} & ⟶ & Z_{2} & Y_{m−1} & ⟶ & Z_{m} \end{aligned} , where Z_{m} = Y_{m} = 0.

Moreover, suppose in the original diagram for all i = 2, …, m the maps B_{i} → C_{i} “can be lifted to” the maps B_{i} → C_{1}, i.e. for all 1 ⩽ j < i ⩽ m there exist maps

γ_{i,j} : B_{i} → C_{j}, that all the diagrams: \begin{matrix}B_{i} & ⟶^{γ_{i,j}} & C_{j} \\ ↓^{γ_{i,j+1}} & ↓ \\ C_{j+1} & ⟶ & X_{j} \end{matrix} are commutative.

Then all the maps: B_{i} → Z_{i−1} can be lifted to the maps: B_{i} → B, and, consequently, B = ⊕_{i=0,...,m}B_{i}.

2.6. Proof of the Statement 1.1.1. Now we can give a proof of Statement 1.1.1 . Proof First of all we will construct some decomposition of the motive M(Q) in DM_{−}^{eff}(k)

(see Definition 2.1.8 ).

Below we will omit symbols M(−), and will denote the motives of varieties and simplicial schemes in the same way as the objects themselves. Also we will write just Hom(−, −) instead of Hom_{DM}(−, −).

So, let Q be a projective quadric over k given by the quadratic form ⟨1, −a, −b, −c⟩. Denote α_{0} : Q → ℤ the element corresponding to the generic cycle in CH^{0}(Q) via

identification: CH^{0}(Q) = Hom(Q, ℤ) (see Theorem 2.1.17 ) (this map is also induced by projection Q → Spec(k)).

Denote β_{0} : ℤ(2)[4] → Q the dual map to α_{0} via duality: Hom(ℤ(2)[4], Q) = Hom(ℤ(2)[4], \underline{Hom}(Q, ℤ(2)[4])) = Hom(Q(2)[4], ℤ(2)[4]) = Hom(Q, ℤ) (see Theorem 2.1.23 ).

By Theorem 2.3.2 : Hom(Q, ℤ) = Hom(Q, X_{Q}), where X_{Q} - is the motive of the standard simplicial scheme corresponding to the pair Q → Spec(k) (see Definition 2.3.1 ), and the map is induced by the natural projection X_{Q} → ℤ. In the same way (by Theorem 2.3.2 ) we have: Hom(ℤ(2)[4], Q) = Hom(X_{Q}(2)[4], Q), and the map is induced again by the projection X_{Q} → ℤ. We can consider α_{0} and β_{0} as maps Q → X_{Q} and X_{Q}(2)[4] → Q respectively.

Denote R := Cone(β_{0} : X_{Q}(2)[4] → Q). Consider the composition X_{Q}(2)[4] → Q → X_{Q}, it lives in the group Hom(X_{Q}(2)[4], X_{Q}) = Hom(X_{Q}(2)[4], ℤ) (by Theorem 2.3.2 ) = 0 (by Theorem 2.3.3 ). In the same way, Hom(X_{Q}(2)[4], X_{Q}[1]) = Hom(X_{Q}(2)[4], ℤ[1]) = 0. So, α_{0} : Q → X_{Q} can be uniquely lifted to a map α′_{0} : R → X_{Q}.

Denote Q⟨1⟩(1)[2] := Cone[−1](α′_{0} : R → X_{Q}). We have the following diagram:

X_{Q}

↗ ↑ ⋆ ↘[1]
Q ⟶ R ⟵ Q⟨1⟩(1)[2],
↖ ⋆ ↓[1] ↙[1]
X_{Q} (2) [4]

Let us multiply this diagram by Q. We have (since X_{Q}×Q=Q, by Theorem 2.3.6) the following diagram:

Q

↗ ↑ ⋆↘[1]
Q × Q ⟶R⟵ Q × Q⟨1⟩(1)[2],
↖ ⋆↓[1] ↙[1]
Q (2) [4]

But, now the map Q × Q →^{id×α_{0}} Q (which is induced by the projection on the first factor) has section - morphism Q → Q × Q induced by the diagonal map. By duality (see Theorem 2.1.23 ) the map Q(2)[4] →^{id×β_{0}} Q × Q also has a splitting.

Hence, Q × Q = Q ⊕ Q × Q⟨1⟩(1)[2] ⊕ Q(2)[4].

Consider now the variety \underline{Q^{1}} of flags :(l, p), where l is a line on Q, and p is a point on this line. We have natural projection \underline{Q}^{1} → Q, and the fiber is constant - zero-dimensional quadric k\sqrt{det(Q)}.

By Theorem 2.4.25 , where we take P = Q with φ : P → Q - identical map, we have: Q × Q = Q ⊕ R(1)[2] ⊕ Q(2)[4]. Notice that R here ( see Theorem 2.4.25 for the definition) is nothing else but a variety \underline{Q}^{1}, and the maps Q × Q → Q and Q(2)[4] → Q × Q are just id × α_{0} and id × β_{0}.

So, we get that Q × Q⟨1⟩ = \underline{Q}^{1}.

Over E = k\sqrt{det(Q)} we have \underline{Q}^{1}|_{E} = Q|_{E} ∪ Q|_{E} (I mean varieties - not motives). Consider cycle D ⊂ Q×_{k}Q×_{k}E∪Q×_{k}Q×_{E} = \underline{Q}^{1}×_{k}E×_{k}Q - the diagonal embedding (over E) of Q×_{k} E into the first product. This gives us an isomorphism Q×E → \underline{Q}^{1}.

Finally, we get: Q × Q⟨1⟩ = Q × k\sqrt{det(Q)}.

Let us show that Q⟨1⟩ = X_{Q} × k\sqrt{det(Q)} (see Definition 2.3.1 ).

First, we will construct a map Q⟨1⟩ → k\sqrt{det(Q)}.

Notice, that by Theorem 2.3.6 Q×X_{Q} = Q, and X_{Q}×X_{Q} = X_{Q} (since X_{Q} belongs to the localizing subcategory of DM_{−}^{eff}(k) generated by Q, see Theorem 2.3.2 ). Hence, the natural map Q⟨1⟩ × X_{Q} → Q⟨1⟩ (induced by the projection X_{Q} → ℤ) is an isomorphism as well (Q⟨1⟩ is a consecutive Cone of Q, X_{Q} and X_{Q}(2)[4]).

Really, denote Y := Cone(Q → X_{Q}) (Q = X_{Q}^{0}). Y is an extension of Q × Q[1], Q × Q × Q[2], etc. ... (this has sense, since as a complexes of sheaves (see the definition of DM_{−}^{eff}: Definition 2.1.8 ) for each k < 0 only finitely many of the above objects have the k-th cohomology), and we have natural map Q × Q[1] → Y, s.t. the composition Q × Q[1] → Y → Q[1] is the map induced by the difference of two projections (on different factors).

Consider the exact triangle: Q⟨1⟩ × (Q → X_{Q} → Y → Q[1]), which is the same as Q × Q⟨1⟩ → Q⟨1⟩ → Y × Q⟨1⟩ → Q × Q⟨1⟩[1]. Consider Hom’s from this triangle to E. Since Q × Q × ⋯ × Q × Q⟨1⟩ = Q × Q × ⋯ × Q × E (as was proven above), and Hom(S[j], E) = Hom(S × E, ℤ[−j]) = 0, for j > 0 for arbitrary smooth projective variety S by duality (Theorem 2.1.23 ) and Theorem 2.1.19 , we have that the map Hom(Y [−1], E) → Hom(Q × Q, E) is an embedding. Then the following sequence is exact: Hom(Q⟨1⟩, E) → Hom(Q × Q⟨1⟩, E) → Hom(Q × Q × Q⟨1⟩, E), where the last map is induced by the difference of two projections.

Since Q×Q⟨1⟩ = Q×E (as was proven above), we have the map μ : Q×Q⟨1⟩ → E (coming from the projection on E-factor). To show that it can be lifted to Q⟨1⟩, we need to prove that the restriction of μ to Q × Q × Q⟨1⟩ is trivial. But Q × Q × Q⟨1⟩ is a smooth projective variety (since Q × Q⟨1⟩ = \underline{Q^{1}} is), and for any smooth projective variety S, Hom(S, ℤ) = ⊕_{conn.comp.(S)}ℤ ↪ ⊕_{conn.comp.(S|\overline{k})}ℤ = Hom(S|\overline{k}, ℤ).

So, it is enough to check it over \overline{k}, where it is clear, since E|\overline{k} = Q⟨1⟩|\overline{k} = ℤ ⊕ ℤ. So, we get a map π : Q⟨1⟩ → E. Let us prove that id_{X_{Q}} ×π induces an isomorphism

Q⟨1⟩ → X_{Q} × Q⟨1⟩ → X_{Q} × E.

Really, since X_{Q} belongs to the localizing subcategory of DM^{eff}(k) generated by Q (see Theorem 2.3.2 ), it is enough to check that Q×Q⟨1⟩ → Q×E is an isomorphism, but this follows from the way our morphism π was constructed (it was constructed as a lifting of the natural projection Q × E → E).

So, we proved that Q⟨1⟩ = X_{Q} × k\sqrt{det(Q)}. And we have the following decomposition for the motive of Q:

X_{Q}
↗ ↑ ⋆ ↘[1]
Q ⟶ R ⟵ k\sqrt{det(Q)} × X_{Q}(1)[2],
↖ ⋆ ↓[1] ↙[1]
X_{Q} (2) [4]

where morphisms u ∈ Hom(X_{Q}, k\sqrt{det(Q)} × X_{Q}(1)[3]), and v ∈ Hom(k\sqrt{det(Q)} × X_{Q}, X_{Q}(1)[3]) can be described in the following way: Hom(X_{Q}, k\sqrt{det(Q)}×X_{Q}(1)[3])= Hom(k\sqrt{det(Q)} × X_{Q}, X_{Q}(1)[3]) by duality, and the later group is equal to Hom(k\sqrt{det(Q)} × X_{Q}, ℤ(1)[3]) (by Theorem 2.3.2 ). But over E = k\sqrt{det(Q)} = k\sqrt{−abc} Q becomes 2-fold Pfister quadric, corresponding to the symbol \{a, b\}|_{E} ({a, b}|_{E} = {b, c}|_{E} = {a, c}|_{E}) (see Definition 2.4.3 ).

Hom(X_{Q} ×E, ℤ(1)[3]) is the same as Hom(X_{Q}|_{E}, ℤ(1)[3]) in the category of motives over E. This group is ℤ/2 with generator β ∘ τ^{−1}({a, b}|_{E}), where β : Hom_{E}(X_{Q}|_{E}, ℤ/2(1)[2]) → Hom_{E}(X_{Q}|_{E}, ℤ(1)[3]) is a bokstein, and τ : Hom_{E}(X_{Q}|_{E}, ℤ/2(1)[2]) → Hom_{E}(X_{Q}|_{E}, ℤ/2(2)[2]) = Hom_{E}(ℤ_{E}, ℤ/2(2)[2]) = K_{2}^{M}(E)/2 is a composition with the unique nonzero element τ ∈ Hom(ℤ/2, ℤ/2(1)) which corresponds to the choice of −1 as a square root of 1 (see Definition 2.1.25 )(by Theorem 2.3.7 ). Really, {a, b} is the only nonzero element from the Ker(K_{2}^{M}(E)/2 → K_{2}^{M}(E(Q))/2) (see, for example, [13], exact sequence (1) on p.4).

By Theorem 2.3.7 we have an isomorphisms: τ (see Definition 2.1.25 ) from Hom(X_{Q}, ℤ/2(n−1)[n]) to the subgroup of Hom(X_{Q}, ℤ/2(n)[n]) = Hom(ℤ, ℤ/2(n)[n]) (by Theorem 2.3.9 ) = K_{n}^{M}(k)/2 (by Theorem 2.1.20 and Theorem 2.1.18 ) equal to the Ker(K_{n}^{M}(k)/2 → K_{n}^{M}(k(Q))/2).

And the bokstein β performs an isomorphism: Hom(X_{Q}, ℤ/2(n − 1)[n]) → Hom(X_{Q}, ℤ(n − 1)[n + 1]) (by Theorem 2.3.7 , and since the last group is 2-torsion (by transfer arguments, since Q has a point of degree 2, see discussion after Theorem 2.3.11 )).

So, to find a kernel Ker(K_{n}^{M}(k)/2 → K_{n}^{M}(k(Q))/2) it is enough to compute the group Hom(X_{Q}, ℤ(n − 1)[n + 1])

From the exact triangle X_{Q}(2)[4] → Q → R^{1} → X_{Q}(2)[5] we have an exact sequence: Hom(X_{Q}, ℤ(n−2)[n−1]) → Hom(R^{1}, ℤ(n)[n+4]) → Hom(Q, ℤ/2(n)[n+4]).

The first of these groups is zero by Theorem 2.3.10 , and by discussion after Theorem 2.3.11 . The last one is zero since dim(Q) = 2 < 4 (see Theorem 2.1.18 ). So, Hom(R^{1}, ℤ(n)[n + 4]) = 0. Now, from the exact triangle R^{1} → X_{Q} → X_{Q} × k\sqrt{det(Q)}(1)[3] → R^{1}[1], we get: Hom(X_{Q} × k\sqrt{det(Q)}, ℤ(n − 1)[n + 1]) → Hom(X_{Q}, ℤ(n)[n + 4]) → 0 = Hom(R^{1}, ℤ(n)[n + 4]). The surjection Hom(X_{Q} × k\sqrt{det(Q)}, ℤ(n − 1)[n + 1]) ↠ Hom(X_{Q}, ℤ(n)[n + 4]) here is given by multipli-

cation by u, and, so, it is a composition: Hom(X_{Q}|_{E}, ℤ(n − 1)[n + 1]) →^{β∘τ^{−1}({a,b})} Hom(X_{Q}|_{E}, ℤ(n)[n + 4]) ⟶ Hom(X_{Q}, ℤ(n)[n + 4]).

Again, since Q has a point of degree 2, by transfer arguments (and by Theorem 2.3.4 ) we have that Hom(X_{Q}, ℤ(a)[b]), for all b > a has exponent 2, and the map Hom(X_{Q}, ℤ(a)[b]) → Hom(X_{Q}, ℤ/2(a)[b]) is injective.

We have an action of motivic operation Q_{1} in Hom(X_{Q}, ℤ/2(a)[b]) (see Theorem 2.1.26 ). Let us denote it as d^{1} (we have too many Q’s without it).

As usually, let X̃_{Q} := Cone[−1](X_{Q} → ℤ) (see Definition 2.3.8 ). We have: d^{1} acts without cohomology on Hom(X̃_{Q}, ℤ/2(a)[b]) (see Theorem 2.3.11 ), and Hom(X̃_{Q}, ℤ/2(a)[b]) = Hom(X_{Q}, ℤ/2(a)[b]) for b > a (by Theorem 2.1.18 ). Since for b ⩽ a Hom(X̃_{Q}, ℤ(a)[b]) = 0 (by Theorem 2.3.9 ), we get that d^{1} performs an isomorphism from Hom(X_{Q}, ℤ/2(n − 1)[n + 1]) = Hom(X̃_{Q}, ℤ/2(n − 1)[n + 1]) to the group Ker(d^{1} : Hom(X_{Q}, ℤ/2(n)[n + 4]) → Hom(X_{Q}, ℤ/2(n + 1)[n + 7])).

By Theorem 2.1.26 (6), any element of the above subgroup of Hom(X_{Q}, ℤ/2(n)[n+ 4]) comes from the cohomology with integral coefficients: Hom(X_{Q}, ℤ(n)[n + 4]).

Since our map Hom(X_{Q} × E, ℤ(n − 1)[n + 1]) → Hom(X_{Q}, ℤ(n)[n + 4]) (induced by the multiplication by u) is surjective, the transfer map: Hom(X_{Q} × E, ℤ(n)[n + 4]) → Hom(X_{Q}, ℤ(n)[n + 4]) is surjective as well.

So, any element of the Ker : (d^{1} : Hom(X_{Q}, ℤ/2(n)[n + 4]) → Hom(X_{Q}, ℤ/2(n + 1)[n + 7])) can be lifted to the element of the group Hom(X_{Q} × E, ℤ(n)[n + 4]), and, consequently, to some element of Hom(X_{Q} × E, ℤ/2(n)[n + 4]).

Since Q|_{E} is a 2-dimensional Pfister quadric (corresponding to the pure symbol {a, b} (see Definition 2.4.3 )), all the elements of the later group are killed by d^{1} (see [13], Theorem 1’). Hence (since d^{1} commutes with transfers in motivic cohomology (V.Voevodsky, unpublished)), Ker(d^{1} : Hom(X_{Q}, ℤ/2(n)[n + 4]) → Hom(X_{Q}, ℤ/2(n + 1)[n+7])) coincides with the image of Tr_{E/k} : Hom(X_{Q}×E, ℤ(n)[n+4]) → Hom(X_{Q}× E, ℤ/2(n)[n + 4]) → Hom(X_{Q}, ℤ/2(n)[n + 4]).

On Hom(X_{Q} × E, ℤ/2(n − 1)[n + 1]) the map d^{1} coincides with the multiplication with β ∘ τ^{−1}({a, b}) (see [13], Theorem 1’).

Now from the commutative diagram:

Hom(X_{Q} × E, ℤ/2(n − 1)[n + 1]) ⟶ Hom(X_{Q}, ℤ/2(n − 1)[n + 1]) ↓^{·β∘τ^{−1}({a,b})}  ↓_{d^{1}} Hom(X_{Q} × E, ℤ/2(n)[n + 4]) ⟶ Hom(X_{Q}, ℤ/2(n)[n + 4]), and the fact

that d^{1} : Hom(X_{Q}, ℤ/2(n − 1)[n + 1]) → Hom(X_{Q}, ℤ/2(n)[n + 4]) is a monomorphism

(see above), we get that the map Hom(X_{Q} × E, ℤ(n − 1)[n + 1]) ⟶^{Tr_{E/k}} Hom(X_{Q}, ℤ(n − 1)[n + 1]) → Hom(X_{Q}, ℤ/2(n − 1)[n + 1]) is surjective.

Since transfers commute with τ and β, we get the surjection (see above): Ker(K_{n}^{M}(E)/2 → K_{n}^{M}(E(Q))/2) ↠ Ker(K_{n}^{M}(k)/2 → K_{n}^{M}(k(Q))/2).

But, by [13], exact sequence (1), p.4, we know that for Pfister quadric Q|_{E} (see Definition 2.4.3 ), this kernel is generated by pure 2-symbol (see Definition 2.4.4 ) {a, b}:

Ker(K_{∗}^{M}(E)/2 → K_{∗}^{M}(E(Q))/2) = {a, b} · K_{∗−2}^{M}(E)/2.

By the lemma of H.Bass and J.Tate (see [1], Corollary 5.3) we have that as a module over K_{∗}^{M}(k) the Milnor’s K-theory of E, K_{∗}^{M}(E) is generated by the components of degree 0 and 1.

The transfer Tr_{E/k} : K_{∗}^{M}(E) → K_{∗}^{M}(k) is a homomorphism of K_{∗}^{M}(k) - modules. So, since the zero-degree component of K_{∗}^{M}(E)/2 is mapped to 0 via Tr_{E/k}, we

get that the ideal Ker(K_{∗}^{M}(k)/2 → K_{∗}^{M}(k(Q))/2) is generated by the elements of the form: {a, b} · Tr_{E/k}(E^{∗}).

Statement 1.1.1 is proven.

3. General theorems

3.1. Structure of the motive of a quadric. In this section we will describe motivic decomposition for quadrics in the motivic category DM^{eff} (see Definition 2.1.8 ). Everywhere below we will assume that our field k has characteristic 0.

Below we will usually omit M(−), and denote motives of schemes in the same way as schemes themselves.

Let Q be some quadric of dimension n . Denote as Q^{1}, Q^{2}, …, Q^{[n/2]} - varieties of lines, planes, etc. ... on a quadric Q . As usually, for a variety P we will denote as X_{P} the motive of a standard simplicial scheme, corresponding to the pair P → Spec (k) (see Definition 2.3.1 ). X_{P} is a form of Tate-motive (see remark after the proof of Theorem 2.3.4 ).

For 2-dimensional quadric Q we had decomposition, which can be drawn in the following way:

∗

↗ ↑ ⋆ ↘
∗ ⟶ ∗ ⟵ ∗
↖ ⋆ ↓ ↙

∗ or, considering the other side of the same octahedron:

∗
↗ ⋆ ↓ ↘
∗ ⟵ ∗ ⟶ ∗
↖ ↑ ⋆ ↙
∗

It appears that in the case of the quadric of arbitrary dimension we have similar decomposition:

∗  ∗  ∗
↗ ⋆ ↓ ↘ ↗ ⋆ ↓ ↘  ↗ ⋆ ↓ ↘
∗ ⟵ ∗ ⟶ ∗ ⟵ ∗ ⟶ ∗ … ∗ ⟵ ∗ ⟶ ∗
↖ ↑ ⋆ ↙ ↖ ↑ ⋆ ↙  ↖ ↑ ⋆ ↙
∗  ∗  ∗

consisting of n exact triangles for dim(Q) = n - even, and of the form:

∗  ∗  ∗  ∗
↗ ⋆ ↓ ↘ ↗ ⋆ ↓ ↘  ↗ ⋆ ↓ ↘ ↗ |
∗ ⟵ ∗ ⟶ ∗ ⟵ ∗ ⟶ ∗ … ∗ ⟵ ∗ ⟶ ∗ ⋆ |
↖ ↑ ⋆ ↙ ↖ ↑ ⋆ ↙  ↖ ↑ ⋆ ↙ ↖ ↓
∗  ∗  ∗  ∗

with n exact triangles for n-odd.

So, the motive of our quadric Q is an extension (see Definition 2.5.6 ) of n + 1 “elementary pieces”, living in the upper and lower row of the diagram (also + the last element from the middle row, if n-even).

These “elementary pieces” appears to be standard simplicial schemes (see Definition 2.3.1 ), associated with the pairs Q^{i} → Spec(k), for all i = 0, …, [n/2], where Q^{i} is a variety of i-dimensional planes on Q (see above).

More precisely, we have:

Theorem 3.1 . M (Q) is an extension (see Definition 2.5.6 ) of: X_{Q}, X_{Q^{1}} (1) [2], …, X_{Q^{(n−1)/2}} (n − 1/2) [n − 1], X_{Q^{(n−1)/2}} (n + 1/2) [n + 1], …, X_{Q^{1}} (n − 1) [2n − 2], X_{Q} (n) [2n], if n is odd, and an extension of X_{Q}, X_{Q^{1}} (1) [2], …, X_{Q^{n/2−1}} (n/2 − 1) [n − 2],

X_{Q^{n/2}}(n/2) [n]×k(\sqrt{det (Q)}) , X_{Q^{n/2−1}}(n/2+1)[n+2], ..., X_{Q^{1}}(n−1)[2n−2], X_{Q}(n)[2n], if n is even.

More precisely, there exist motives Q⟨1⟩, Q⟨2⟩, …, Q⟨[n/2]⟩ , which fit in the following Postnikov tower (under this I mean nothing more than just a system of exact triangles in a triangulated category) for Q :

X_{Q}  X_{Q^{[n/2]−1}} ([n/2] − 1) [2[n/2] − 2]
↗ ⋆ ↓[1] ↘[1] ↗ ⋆ ↓ [1] ↘ [1]
Q ⟵ R^{1} ⟶ Q′⟨1⟩ , …, Q′⟨[n/2] − 1⟩⟵ R^{[n/2]} ⟶ Q′⟨[n/2]⟩
↖ ↑ ⋆ ↙[1] ↖ ↑ ⋆ ↙ [1]
X_{Q} (n) [2n]  X_{Q^{[n/2]−1}} ([n/2] + 1) [2[n/2] + 2]

where Q′⟨i⟩ = Q⟨i⟩ (i) [2i] .

and in the case n is even, Q⟨n/2⟩ = X_{Q^{n/2}} × k (\sqrt{det (Q)}) ;

in the case n is odd there is an exact triangle:

X_{Q^{n−1/2}} (n + 1/2) [n + 1] → Q′⟨(n − 1) /2⟩ → X_{Q^{n−1/2}} (n − 1/2) [n − 1].

Proof of Theorem 3.1

The proof is similar to the one of Theorem 4.4 from [20].

Denote as α_{i} the standard morphism Q → ℤ (i) [2i], corresponding to the plane section of codimension i (see Theorem 2.1.17 ), and as β_{i} the dual map ℤ (n − i) [2n− 2i] → Q (see (4) from Theorem 2.1.23 ).

Denote as R^{1} the Cone[−1] (Q → X_{Q}) (map Q → X_{Q} comes from the fact that X_{Q}^{0} = Q, or from α_{0} via identification Hom(Q, X_{Q})=Hom(Q, ℤ) (see Theorem 2.3.2)).

Since Hom (X_{Q}, X_{Q} (a) [b]) = Hom (X_{Q}, ℤ (a) [b]) (by Theorem 2.3.2 ), by Theorem 2.3.3 (1) we see that the composition: X_{Q} (n) [2n] ⟶^{β_{0}} Q ⟶^{α_{0}} X_{Q} is zero,

and we get a map X_{Q} (n) [2n] → R^{1} (which is unique precisely by the same reason). Denote as Q⟨1⟩ the Cone (X_{Q} (n) [2n] → R^{1}) (−1) [−2]. We claim, that the natural map Q⟨1⟩ × X_{Q^{1}} → Q⟨1⟩ (coming from projection

X_{Q} → ℤ) is an isomorphism. Really, consider the following diagram :

Q⟨1⟩ × X_{Q^{1}} × X_{Q} ⟶ Q⟨1⟩ × X_{Q}
↓  ↓
Q⟨1⟩ × X_{Q^{1}} ⟶ Q⟨1⟩

Both vertical maps are isomorphisms , since Q has point over k (Q^{1}) and Q⟨1⟩ is constructed from Q (use Theorem 2.3.6 ) . So, it is enough to prove that upper horizontal map is an isomorphism, and for this it is enough to prove that Q⟨1⟩×X_{Q^{1}} × Q → Q⟨1⟩ × Q is an isomorphism (since X_{Q} belongs to the localizing subcategory, generated by Q (see Theorem 2.3.2 )) and the later is equivalent to the fact that Q⟨1⟩ × X_{Q^{1}} × k (Q) → Q⟨1⟩ × k (Q) is an isomorphism (by Theorem 2.3.5 ). But over k (Q) Q has a point, and, so, Q⟨1⟩ × k (Q) become just the quadric Q′ of lines passing through this point (see Theorem 2.4.23 and Remark 2 after it), and X_{Q^{1}} × k (Q) = X_{Q′/k(Q)}. Hence, the isomorphism in this case is evident.

Now, composition X_{Q} (n) [2n] ⟶^{β_{0}} R^{1} ⟶^{α_{1}|_{R^{1}}} ℤ (1) [2] is zero by Theorem 2.3.2 and Theorem 2.3.3 (1), since n > 1. So, we get morphism α′′_{1} : Q⟨1⟩ (1) [2] → ℤ (1) [2] and since Q⟨1⟩ × X_{Q^{1}} = Q⟨1⟩ also a morphism α′_{1} : Q⟨1⟩ (1) [2] → X_{Q^{1}} (1) [2] (by Theorem 2.3.2 ).

Since Q has a point over Q^{1} ,we see that Hom (X_{Q^{1}}, X_{Q} (a) [b])=Hom (X_{Q^{1}}, ℤ (a) [b]) by Theorem 2.3.2 and Theorem 2.3.6 , and β_{1} can be lifted to β′_{1} : X_{Q^{1}} (n − 1) [2n − 2] → Q⟨1⟩ (1) [2]. If n > 2, then the composition : X_{Q^{1}} (n − 1) [2n − 2] ⟶^{β′_{1}} Q⟨1⟩ (1) [2] ⟶^{α′_{1}} X_{Q^{1}} (1) [2] is zero, by Theorem 2.3.3 (1) and since Hom (X_{Q^{1}}, X_{Q^{1}} (a) [b]) = Hom (X_{Q^{1}}, ℤ (a) [b]) (by Theorem 2.3.2 ), and if we denote

R^{2} = Cone[−1] (Q⟨1⟩ (1) [2] → X_{Q^{1}} (1) [2]) , then β′_{1} can be lifted to a map from X_{Q^{1}} (n − 1) [2n − 2] to R^{2}

Denote as Q⟨2⟩ the Cone (X_{Q^{1}} (n − 1) [n − 2] → R^{2}) (−2) [−4] . We claim, that the natural map : Q⟨2⟩ × X_{Q^{2}} → Q⟨2⟩ is an isomorphism. Again, consider a diagram

\begin{matrix}Q⟨2⟩ × X_{Q^{2}} × X_{Q^{1}} & ⟶ & Q⟨2⟩ × X_{Q^{1}} \\ ↓ & & ↓ \\ Q⟨2⟩ × X_{Q^{2}} & ⟶ & Q⟨2⟩ \end{matrix} Both vertical maps are isomorphisms, since Q^{1}

has point over k (Q^{2}) and Q⟨2⟩ is made of Q⟨1⟩ (for which we have corresponding property) and Q^{1} (we use Theorem 2.3.6 here). So, it is again enough to prove, that the upper horizontal map is an isomorphism, and in the same way as before for this it is enough to prove (using Theorem 2.3.5 ) that Q⟨2⟩ × X_{Q^{2}} × k (Q^{1}) → Q⟨2⟩ × k (Q^{1}) is an isomorphism. But over k (Q^{1}) Q has a line and Q⟨2⟩ is isomorphic to the quadric Q′′ of planes on Q ,containing this line (apply Theorem 2.4.23 twice). Hence, X_{Q^{2}} × k (Q^{1}) = X_{Q′′/k(Q^{1})} , and we get what we need. Continuing the same way we get Q⟨i⟩ ,where i < n/2. Since Hom (X_{Q^{i}}, X_{Q^{j}} (a) [b]) = Hom (X_{Q^{i}}, ℤ (a) [b]) for all j < i by Theorem 2.3.2 and Theorem 2.3.6 (Q^{j} has a point over k (Q^{i}) ), and Q⟨i⟩ × X_{Q^{i}} = Q⟨i⟩ we can lift α_{i} and β_{i} to a maps : X_{Q^{i}} (n − i)[2n − 2i] ⟶^{α′_{i}} Q⟨i⟩(i)[2i] ⟶^{β′_{i}} X_{Q^{i}} (i)[2i] Since n/2 > i the composition will be zero (by Theorem 2.3.2 and Theorem 2.3.3 (1)) and we can construct Q⟨i + 1⟩ .

If i + 1 < n/2 ,then using precisely the same arguments as before we get that Q⟨i + 1⟩ × X_{Q^{i+1}} = Q⟨i + 1⟩ . Otherwise we have two possibilities :

1) n is odd and i = (n − 1) /2

We claim that Q⟨i+1⟩ = 0 . Really, Consider a diagram : \begin{matrix}Q⟨i + 1⟩ × X_{Q^{i}} & ⟶ & 0 \\ ↓ & & ↓ \\ Q⟨i + 1⟩ & ⟶ & 0 \end{matrix}

Vertical maps here are isomorphisms (since Q⟨i+1⟩ is constructed from Q⟨i⟩, for which we have this property, and X_{Q^{i}}’s,), and to check that Q⟨i+1⟩×X_{Q^{i}} = 0 , it is enough to prove that Q⟨i + 1⟩ × Q^{i} = 0 or that Q⟨i + 1⟩ × k (Q^{i}) = 0 (by Theorem 2.3.5 ), which follows from Theorem 2.4.22 , since over k(Q^{n−1/2}) Q is hyperbolic.

2) n is even and i = n/2 − 1

Since Q⟨n/2⟩ × X_{Q^{n/2}} = Q⟨n/2⟩, by Theorem 2.3.2 we have : Hom(Q′⟨n/2⟩, X_{Q^{n/2}}(n/2)[n]×k (\sqrt{det (Q)}))=Hom(Q′⟨n/2⟩, k(\sqrt{det (Q)})(n/2)[n])

Now we can take element of the Hom(Q′⟨n/2⟩, ℤ(n/2)[n]), corresponding to the plane section of Q of dimension n/2 (see Theorem 2.1.17 ). This map is actually the composition: Q′⟨n/2⟩→ k(\sqrt{det (Q)}) (n/2)[n] → ℤ(n/2)[n]. To see this we need the following:

Together with the varieties Q^{i} we can consider varieties \underline{Q^{i}} - the varieties of flags point ⊂ line ⊂ ⋯ ⊂ i-dim.plane on Q ^{2}. Evidently, \underline{Q^{i}} is a grassmanian bundle over Q^{i}, and the existence of rational point on both varieties is equivalent. So, X_{Q^{i}} = X\underline{Q^{i}} (by Theorem 2.3.4 , see also Theorem 2.4.15 ).

Motive of the variety \underline{Q^{i}} can be described as follows:

Claim 3.2 .

\underline{Q^{i}} = Q × Q⟨1⟩ × ⋯ × Q⟨i⟩.

Proof of Claim 3.2

Induction on i. Case i = 0 is trivial. Since \underline{Q^{i}} × X_{Q^{i}} = \underline{Q^{i}} (by Theorem 2.3.6 , for example), and X_{Q^{r}} × X_{Q^{s}} = X_{Q^{max(r,s)}} (by Theorem 2.3.6 ), we have: \underline{Q^{i}} × Q is an extension of \underline{Q^{i}}(j)[2j], \underline{Q^{i}}(dim(Q) − j)[2 dim(Q)−2j], 0 ⩽ j ⩽ i, and \underline{Q^{i}}×Q⟨i+1⟩(i+1)[2i+2] (by the already proven part of Theorem 3.1 (since Q is an extension of X_{Q^{j}}(j)[2j], 0 ⩽ j ⩽ i, Q⟨i+1⟩(i+1)[2i+2], and X_{Q^{j}}(dim(Q) − j)[2 dim(Q) − 2j], 0 ⩽ j ⩽ i)). Certainly, \underline{Q^{i+1}} is a fibration over \underline{Q^{i}} with fibers - quadrics Q(i+1, x) (quadric of i+1 dimensional projective subspaces on Q, containing fixed i-dimensional space, determined by x ∈ \underline{Q^{i}}).

Over \underline{Q^{i}}, Q is (globally) i-times strongly isotropic (see Definition 2.4.24 ), and, so the motive of \underline{Q^{i}} × Q is a direct sum of \underline{Q^{i}}(j)[2j], \underline{Q^{i}}(dim(Q) − j)[2 dim(Q) − 2j], 0 ⩽ j ⩽ i, and \underline{Q^{i+1}}(i + 1)[2i + 2] (by Theorem 2.4.25 ).

Moreover, the maps \underline{Q^{i}} × Q → \underline{Q^{i}}(i)[2i] which define this decomposition are just id×α_{i} (see beginning of the proof of Theorem 3.1 for the definition of α_{i}). So, \underline{Q^{i}} ×Q will be also a direct sum of \underline{Q^{i}}(j)[2j], \underline{Q^{i}}(dim(Q) − j)[2 dim(Q) − 2j], 0 ⩽ j ⩽ i, and \underline{Q^{i}} × Q⟨i + 1⟩(i + 1)[2i + 2] (since the maps \underline{Q^{i}} × Q → \underline{Q^{i}}(i)[2i] there are the same). And we get an isomorphism \underline{Q^{i+1}} = \underline{Q^{i}} × Q⟨i + 1⟩.

2I would like to thank V.Voevodsky for suggesting me the use of later varieties

Claim 3.2 is proven.

Now, since \underline{Q^{n/2−1}} × X_{Q^{n/2−1}} = \underline{Q^{n/2−1}}, we have an exact triangle: Y × Q⟨n/2⟩ → \underline{Q^{n/2−1}} × Q⟨n/2⟩ → Q⟨n/2⟩ → Y [1] × Q⟨n/2⟩, where Y = Cone[−1](\underline{Q^{n/2−1}} → X\underline{Q^{n/2−1}}) (it is an extension of: \underline{Q^{n/2−1}} ×\underline{Q^{n/2−1}} ×Q⟨n/2⟩, \underline{Q^{n/2−1}} ×\underline{Q^{n/2−1}} ×\underline{Q^{n/2−1}} × Q⟨n/2⟩[1], etc. ... ). But from Claim 3.2 we know that \underline{Q^{n/2−1}} × Q⟨n/2⟩ = \underline{Q^{n/2}}, and also (since the last fibration \underline{Q^{n/2}} → \underline{Q^{n/2−1}} has fiber - 0-dimensional quadric k(\sqrt{det (Q)}), which is defined over k) we have: \underline{Q^{n/2−1}} × k(\sqrt{det (Q)}) = \underline{Q^{n/2}}. We have the commutative diagram :

\underline{Q^{n/2−1}} × \underline{Q^{n/2−1}} × Q⟨n/2⟩ ⟶ \underline{Q^{n/2−1}} × Q⟨n/2⟩ ⟶ Q⟨n/2⟩
∥
\underline{Q^{n/2−1}} × k(\sqrt{det (Q)}) ⟶ k(\sqrt{det (Q)})

The upper row is not an exact triangle, but the sequence Hom(Q⟨n/2⟩, k(\sqrt{det (Q)}))→ Hom(\underline{Q^{n/2−1}}×Q⟨n/2⟩, k(\sqrt{det (Q)})) → Hom(\underline{Q^{n/2−1}}×\underline{Q^{n/2−1}}×Q⟨n/2⟩, k(\sqrt{det (Q)})) is exact, since Y × Q⟨n/2⟩ → \underline{Q^{n/2−1}} × Q⟨n/2⟩ → Q⟨n/2⟩ is an exact triangle and Hom(\underline{Q^{n/2−1}}×⋯×\underline{Q^{n/2−1}}×Q⟨n/2⟩[j], k(\sqrt{det (Q)})) = Hom(\underline{Q^{n/2−1}}×⋯×\underline{Q^{n/2−1}}× k(\sqrt{det (Q)}) [j], k(\sqrt{det (Q)}))=0 for j > 0 by Theorem 2.1.19 and Theorem 2.1.23.

Since \underline{Q^{n/2−1}} ×\underline{Q^{n/2−1}} ×Q⟨n/2⟩ = \underline{Q^{n/2−1}} ×\underline{Q^{n/2−1}} ×k(\sqrt{det (Q)}), and the later is a smooth projective variety, it is enough to check that the composition map (not the projection): \underline{Q^{n/2−1}} × \underline{Q^{n/2−1}} × Q⟨n/2⟩ → k(\sqrt{det (Q)}) is zero over algebraic closure (for a smooth projective variety A Hom(A, ℤ) = CH^{0}(A) = ⊕_{con.comp.(A)}ℤ ↪ ⊕_{con.comp.(A|\overline{k})}ℤ = CH^{0}(A|\overline{k}) = Hom(A|\overline{k}, ℤ)) ,where it is evident since Q⟨n/2⟩|\overline{k} = k(\sqrt{det (Q)}) |\overline{k}, and we get a map Q⟨n/2⟩ → k(\sqrt{det (Q)}), and, hence, a morphism from Q⟨n/2⟩ to the X_{Q^{n/2}} × k (\sqrt{det (Q)}) which will be an isomorphism by the same arguments as before (multiply everything by X_{Q^{n/2−1}}, ...).

Theorem 3.1 is proven.

Remark The motives Q⟨i⟩ have the following geometric sense: as soon as our quadric Q is i-times isotropic (i.e.: Q^{i−1} has a rational point, and our quadric has i − 1-dimensional projective subspace l_{i−1}) then Q⟨i⟩ becomes just the dim(Q) − 2i - dimensional quadric of i-dimensional projective subspaces on Q, containing l_{i−1} (it follows from Theorem 2.4.23 , and Theorem 3.1 , since if Q is i-times isotropic, then X_{Q^{j}} = ℤ for all 0 ⩽ j < i (see Theorem 2.3.4 )). In the language of quadratic forms: we have q = i · ℍ ⊥ q′, and Q⟨i⟩ is just the quadric Q′.

Claim 3.3 . Suppose n is odd. Then Q⟨[n − 1/2]⟩ = Cone[−1](X_{Q^{[n−1/2]}} ⟶^{γ} X_{Q^{[n−1/2]}} (1) [3]), where γ ∈ Hom (X_{Q^{[n−1/2]}}, ℤ (1) [3]) (identification by Theorem 2.3.2 ) is equal to β ∘ τ^{−1} (c_{2} (Q)) (β is a bokstein, definition of τ is given in Definition 2.1.25 , and c_{2}(Q) is defined in Definition 2.4.13 , see also Theorem 2.3.7 ).

Proof of the Claim 3.3 From Theorem 2.3.7 we know, that 2 - torsion part in Hom (X_{Q^{[n−1/2]}}, ℤ (1) [3]) is equal to the Ker : K_{2}^{M} (k) /2 → K_{2}^{M} (k (Q^{[n−1/2]})) /2 (via β ∘ τ^{−1}). Since X\underline{Q^{[n−1/2]}} = X_{Q^{[n−1/2]}}, and \underline{Q^{[n−1/2]}} is a consecutive fibration over Q with fibers - quadrics of dimensions n − 2, n − 4, …, 1, we see that this kernel is ℤ/2, or 0 (since by Theorem 2.4.11 for quadric P of dimension > 2 there is no kernel at all, and for a conic C/E it consists of one pure 2-symbol {a, b} (corresponding to the conic: C has form ⟨1, −a, −b⟩)(see [13], exact sequence (1) on p.4). We have two cases: 1) c_{2} (Q) ≠ 0, 2) c_{2} (Q) = 0.

1) Since, certainly, c_{2} (Q) ∈ Ker : K_{2}^{M} (k) /2 → K_{2}^{M} (k (Q^{[n−1/2]})) /2 (Q becomes hyperbolic over k (Q^{[n−1/2]}); see Definition 2.4.12 ), and Ker : K_{2}^{M} (k) /2 → K_{2}^{M} (k (Q^{[n−3/2]})) /2 = 0 (since the later variety is a consecutive fibration with fibers - quadrics of dimension ⩾ 3 (see Theorem 2.4.11 )), we get that Q⟨[n − 1/2]⟩ × k (Q^{[n−3/2]}) is a conic without a point (since conic with a point, i.e.: ℙ^{1} do not have any kernel, but our conic does). So, γ × Id_{Q^{[n−3/2]}} ≠ 0, and, hence, γ ≠ 0. But the only nontrivial element in Hom (X_{Q^{[n−1/2]}}, ℤ (1) [3]) is β ∘ τ^{−1} (c_{2} (Q)).

2) In the same manner, get: Q⟨[n − 1/2]⟩ × k (Q^{[n−3]}) is a conic with trivial c_{2} (since c_{2} will be the same - see Definition 2.4.12 ), i.e. conic with a point. So, γ × Id_{Q^{[n−3/2]}} = 0, but from the commutative diagram:

Hom (X_{Q^{[n−1/2]}}, ℤ (1) [3])  ⟶^{τ∘β^{−1}}  K_{2}^{M} (k) /2
↓  ↓^{j}
Hom (X_{Q^{[n−1/2]}} × k (Q^{[n−3/2]}) , ℤ (1) [3]) ⟶^{τ∘β^{−1}} K_{2}^{M} (k (Q^{[n−3/2]})) /2

and injectivity of j (see Theorem 2.4.11 ) and τ ∘ β^{−1} (see Theorem 2.3.7 ), we have: γ = 0 (τ ∘ β^{−1} is an injection).

Claim 3.3 is proven.

From the proof of Theorem 3.1 we can deduce more, namely, since the morphisms β′_{i} : X_{Q^{i}} (n − i) [2n − 2i] → Q′⟨i⟩ are actually liftings of the compositions X_{Q^{i}} (n − i) [2n − 2i] ⟶ ℤ (n − i) [2n − 2i] ⟶^{β_{i}} Q, by Theorem 2.5.10 , we get the following

Proposition 3.4 .

There is the following exact triangle:

Q → P′ → ⊕_{i=0,...,[n−1/2]}X_{Q^{i}} (n − i) [2n − 2i + 1] → Q[1],

where in case n is odd, P′ = P, and in case n- even we have triangle

P [−1] → (X_{Q^{n/2}} × k\sqrt{det (Q)}) (n/2) [n] → P′ → P,

where P fits into the following Postnikov tower:

X_{Q}  X_{Q^{1}} (1) [2]  X_{Q^{[n−1/2]}} ([n − 1/2]) [2[n − 1/2]]
↗ ⋆ ↓ [1] ↗ ⋆ ↓ [1] …  ↗ ⋆  ↓ [1]
P ⟵P^{1} ⟵ P^{2}  P^{[n−1/2]} ⟵  0.

Remark

Unfortunately, in general, P is not equal to a direct sum ⊕_{j=0,...,[n−1/2]} X_{Q^{j}} (j) [2j], as already an example of a 3 - dimensional quadric shows. More precisely, we can state the following (for more general result - see Corollary 4.4 ):

Statement 3.5 . A) Suppose, Q is a 3 - dimensional quadric. Then the following is equivalent:

1) Q is a small Pfister quadric, i.e.: Q corresponds to a quadratic form ⟨1, −a, −b, ab, −c⟩ for some a, b, c ∈ k^{∗}.

2) c_{2} (Q) is a pure symbol (see Definition 2.4.12 ,Definition 2.4.4 ).

3) P = X_{Q} ⊕ X_{Q^{1}} (1) [2].

4) Q = Cone[−1] (X_{Q} → X_{Q} (3) [7]) ⊕ Cone[−1] (X_{Q^{1}} → X_{Q^{1}} (1) [3]) (1) [2]. B) Q is hyperbolic iff c_{2} (Q) = 0.

Proof of the statement

(1 ⇔ 2) Let Q corresponds to the quadratic form q, then consider an Albert quadric Q′ with quadratic form q′ = q + ⟨−det (q)⟩. Everything follows from: a) c_{2} (Q) = c_{2} (q′), which is sum of two pure symbols, b) isotropness of q′ is equivalent to the fact that this sum is a pure symbol (see Theorem 2.4.9 ), c) isotropness of q′ is equivalent to the fact that q represents it’s own determinant, i.e. there is a subform of dimension 4 and determinant 1 in it, i.e. q is a small Pfister form.

(2 ⇒ 4) Consider decomposition of Q from Theorem 3.1 .

X_{Q}  X_{Q^{1}} (1) [2]
↗ ⋆ ↓[1]↘[1] ↗ |
Q ⟵R^{1}⟶ Q′⟨1⟩ ⋆ [1]
↖ ↑ ⋆ ↙[1] ↖ ↓
X_{Q} (3) [6]  X_{Q^{1}} (2) [4]

It is well known, that in the case of a 3 - dimensional quadric, Q^{1} is a Severi-Brauer variety, corresponding to central simple algebra A of rank 4, which represents c_{2} (Q) ∈ K_{2}^{M} (k) /2 = H_{et}^{2} (k, ℤ/2) ⊂ H_{et}^{2} (k, O^{∗}) = Br (k) .

If c_{2} (Q) is a pure symbol {a, b}, then this central simple algebra will be just M_{2} (Quat ({a, b})) and, hence, X_{Q^{1}} = X_{C}, where C is a Severi-Brauer conic, corresponding to the c.s.a. Quat ({a, b}) (since existence of a rational point on these two varieties is equivalent (see Theorem 2.3.4 ). By the way, notice, that Hom (X_{Q^{1}}, X_{Q^{1}} (1) [3]) = Hom (X_{Q^{1}}, ℤ (1) [3]) = ℤ/2, with the only nontrivial element γ (see Theorem 2.3.2 and Theorem 2.3.7 ), and C = Cone[−1](X_{Q^{1}} ⟶^{γ} X_{Q^{1}} (1) [3]) (see Claim 3.3 ).

From the Claim 3.3 we know that Q⟨1⟩ = C.

Hom (X_{Q}, C(1)[3]) = Hom (C, ℤ (2) [5]) = 0, by duality (see Theorem 2.1.23 ), since X_{Q} × Q⟨i⟩ = Q⟨i⟩ and dim (C) = 1 < 3 (see Theorem 2.1.18 ). In the same way, since X_{C} = X_{Q^{1}}, we have (by duality, and Theorem 2.4.16 ): Hom (C, X_{Q} (2) [5]) = Hom (ℤ, C (1) [3]) = 0. So, Q = C(1)[2] ⊕ Cone[−1] (X_{Q}, X_{Q} (3) [7]) (since Q is an extension of X_{Q}, C(1)[2], and X_{Q}(3)[6] (by Theorem 3.1 ), and as we see, the middle part can’t be hooked to any other).

(4 ⇒ 3) Evident.

(3 ⇒ 2) Suppose P = X_{Q} ⊕ X_{Q^{1}} (1) [2],

Q = Cone[−1](P → X_{Q^{1}}(2)[5]⊕X_{Q}(3)[7]) (see Proposition 3.4 ). Consider the map δ : ℤ(1)[2] → Q given by some subconic C ⊂ Q (see Theorem 2.1.17 and Theorem 2.1.23 ). Then α_{1} ∘ δ : ℤ(1)[2] → ℤ(1)[2] (here α_{1} as in Theorem 3.1 is given by some subquadric R of dimension 2 (see Theorem 2.1.17 )) is just multiplication by 2=intersection number of R and C (everything can be considered in the category of Chow motives by Theorem 2.2.4 ).

Consider the composition ℤ(1)[2] →^{δ} Q → P. Since P = X_{Q} ⊕ X_{Q^{1}}(1)[2], we get a morphism δ′ : ℤ(1)[2] → X_{Q^{1}}(1)[2], s.t. it’s composition with the natural projection X_{Q^{1}}(1)[2] → ℤ(1)[2] is multiplication by 2 (enough to check over \overline{k}).

By Theorem 2.3.3 (2), we have a zero-cycle of degree 2 on Q^{1}, i.e. there exists field extension E/k of degree 4k + 2 s.t. Q^{1} has a point over E.

Consider again the quadric Q′, corresponding to the form q′ = q ⊥ − det(q). This is an Albert form corresponding to c_{2}(q′) = c_{2}(Q) (see Definition 2.4.7 ). (Q′|_{k(Q′)})_{anis.} is a 2-fold Pfister form. Hence (since Pfister form is hyperbolic over it’s generic point (by Theorem 2.4.5 )), we have h_{2}(Q′) = 2 (see Definition 2.4.19 ), i.e. X_{Q′}^{1} = X_{Q′}^{2} (see Theorem 2.4.20 ).

Since Q is a hyperplane section in Q′, we have: X_{Q′}^{1} ⩽ X_{Q^{1}} ⩽ X_{Q′}^{2}, and, consequently, X_{Q′}^{2} = X_{Q^{1}} (see the discussion after Theorem 2.4.20 ).

So, if Q^{1} has a point E of degree 4k + 2, then Q′^{2} has a point of degree 4k + 2 (see Theorem 2.4.18 ) (since X_{Q′}^{2}|_{E} = ℤ). Then by Theorem 2.4.9 we get: c_{2}(Q) is pure.

Statement 3.5 is proven.

In the same way as in Proposition 3.4 we see that α′_{i} × Id_{X_{Q^{j}}} is a lifting of

Q × X_{Q^{j}} ⟶^{α_{i}×Id} X_{Q^{j}} (i) [2i] = X_{Q^{i}} (i) [2i] × X_{Q^{j}}

for any j ⩾ i (since X_{Q^{i}} × X_{Q^{j}} = ℤ × X_{Q^{j}} (see Theorem 2.4.16 ), and morphisms Q⟨i + 1⟩ → X_{Q^{i}}(i)[2i] were obtained from the morphisms Q⟨i + 1⟩ → ℤ(i)[2i] (see the proof of Theorem 3.1 ), which, on their part, can be lifted to α_{i} : Q → ℤ(i)[2i] (see the beginning of the proof of Theorem 3.1 for the definition)).

On the other hand, X_{Q^{i}} ×X̃_{Q^{j}} = 0 for any j ⩽ i (by Theorem 2.4.16 )(see Definition 2.3.8 ).

Let n be odd. Denote X_{Q^{i}} × X̃_{Q^{j}} as X_{i,j}, X_{Q^{i}} as X_{i}, and put X_{[n+1/2]} = 0. Then we have the following Postnikov system (the maps in the lower row come from the maps X_{Q^{i+1}} → ℤ (multiplied by id_{X_{Q^{i}}}) via identification X_{Q^{i+1}} × X_{Q^{i}} = X_{Q^{i+1}} (see Theorem 2.4.16 )):

X_{0,1}  X_{1,2}  X_{[n−1/2],[n+1/2]}
[1]↗ ⋆ ↓ [1]↗ ⋆ ↓ … [1]↗ ⋆ ↓
X_{0} ⟵ X_{1} ⟵ X_{2}  X_{[n−1/2]} ← 0

Multiplying tower from Proposition 3.4 with this Postnikov system, and using arguments above and Theorem 2.5.10 , we get:

Proposition 3.6 .

Suppose n is odd. We have the following decomposition for Q:

Q × X_{0,1}  Q × X_{1,2}  Q × X_{[n−1/2],[n+1/2]}
[1]↗ ⋆ ↓ [1]↗ ⋆ ↓  …  [1]↗ ⋆  ↓
Q = Q × X_{0} ⟵ Q × X_{1} ⟵ Q × X_{2}  Q × X_{[n−1/2]} ← 0,

where each Q × X_{i,i+1} on it’s part has a decomposition:

Q × X_{i,i+1} → ⊕_{j=0,...,i}X_{i,i+1} (j) [2j] → ⊕_{j=0,...,i}X_{i,i+1} (n − j) [2n − 2j + 1] → Q × X_{i,i+1}[1].

Similar result can be formulated in the case n- even.

Remark Proposition 3.6 shows that if X_{0} = X_{[n−1/2]}, that is: existence of a rational point on Q is equivalent to the existence of an [n − 1/2]-dimensional plane on it, then there is the following exact triangle:

Q → ⊕_{j=0,...,[n−1/2]}X_{Q} (j) [2j] → ⊕_{j=0,...,[n−1/2]}X_{Q} (n − j) [2n − 2j + 1] → Q[1].

From this it is quite easy to see that each X_{Q} (j) [2j] is actually hooked to only one other (i.e. Q is a direct sum of pure motives (see Theorem 2.2.4 ), each of which is an extension of just two X’s), namely X_{Q} (j + [n − 1/2]) [2j + 2[n − 1/2] + 1]. This way we get Rost decomposition for Q - hyperplane section of a Pfister quadric.

Let us introduce the following set: J(Q) = { set of such j, that X_{Q^{[dim(Q)/2]−j+1}} ≠ X_{Q^{[dim(Q)/2]−j}} } = {j_{1}, j_{2}, …, j_{s}}. Certainly, j_{s} = [dim(Q)/2], and j_{s−k+1} − j_{s−k} = h_{k} - k-th Witt number (see Definition 2.4.19 and Theorem 2.4.20 ). We have X_{Q^{[dim(Q)/2]−jr}} = ⋯ = X_{Q^{[dim(Q)/2]−j_{r+1}+1}} (see Theorem 2.4.20 ), and we can denote it as X {j_{r}}.

The same methods as in Proposition 3.6 , and Theorem 2.5.10 (with X = Q⟨h_{1} + ⋯ + h_{t}⟩(h_{1} + ⋯ + h_{t})[2(h_{1} + ⋯ + h_{t})] and X_{m} = Q⟨h_{1} + ⋯ + h_{t+1}⟩(h_{1} + ⋯ + h_{t+1})[2(h_{1} + ⋯ + h_{t+1})]) give us the Postnikov tower for Q with graded parts (⊕_{i=j_{r},...,j_{r+1}−1}X {j_{r}}(∗)[2∗])_{j_{r}∈J(Q)}.

3.2. Ring of endomorphisms of M(Q).

Theorem 3.7 .

Postnikov tower from Theorem 3.1 is compatible with the endomorphisms of Q. More precisely, for any φ : Q → Q there exists unique set of morphisms: φ_{i} : X_{Q^{i}}(i)[2i] → X_{Q^{i}}(i)[2i], φ′_{i} : X_{Q^{i}}(n − i)[2n − 2i] → X_{Q^{i}}(n − i)[2n − 2i], φ⟨i⟩ : Q′⟨i⟩ → Q′⟨i⟩, and φ_{R^{i}} : R^{i} → R^{i}, which commute with the maps from the tower.

Proof of Theorem 3.7

Lemma 3.8 .

Hom(X_{Q^{i}}(n − i)[2n − 2i], Q′⟨i + 1⟩) = Hom(X_{Q^{i}}(n − i)[2n − 2i], Q′⟨i + 1⟩[−1]) = 0. Proof of Lemma 3.8 In the same way as in the Proposition 3.4 (using Theorem 2.5.10 ) we can obtain

the following decomposition for Q:

P_{L}  P_{L}
↗ ↑ ⋆ ↘[1]  ↗ ⋆ ↓[1]↘[1]
Q ⟶ R̃ ⟵ Q′⟨i + 1⟩ and Q ⟵R̃′ ⟶ Q′⟨i + 1⟩
↖ ⋆ ↓[1] ↙[1]  ↖ ↑ ⋆ ↙[1]
P_{U}  P_{U}

(which are two halves of the same octahedron), where P_{L} is an extension of: X_{Q}, X_{Q^{1}}(1)[2], …, X_{Q^{i}}(i)[2i], and P_{U} = X_{Q}(n)[2n]⊕X_{Q^{1}}(n−1)[2n−2]⊕⋯⊕X_{Q^{i}}(n− i)[2n − 2i].

It is evident, that the pair X_{Q^{i}} × (P_{U} → Q) is isomorphic to the pair X_{Q^{i}} ×

(⊕_{j=0,...,i}ℤ(n − j)[2n − 2j] ⟶^{⊕β_{j}} Q), and from the construction (see Theorem 3.1 ) we have, that the pair X_{Q^{i}} × (Q → P_{L}) is isomorphic to the pair X_{Q^{i}} × (Q ⟶^{⊕α_{j}} ⊕_{j=0,...,i}ℤ(j)[2j]). Denote Z_{U} := ⊕_{j=0,...,i}ℤ(n−j)[2n−2j] and Z_{L} := ⊕_{j=0,...,i}ℤ(j)[2j].

Since the pairs ⊕_{j=0,...,i}ℤ(n−j)[2n−2j] ⟶^{⊕β_{j}} Q and Q ⟶^{⊕α_{j}} ⊕_{j=0,...,i}ℤ(j)[2j] are dual

to each other via duality \underline{Hom}(−, ℤ(n)[2n]), we have the commutative diagram:

Hom(X_{Q^{i}}(n − i)[2n − 2i], P_{U}[b])  ⟶  Hom(X_{Q^{i}}(n − i)[2n − 2i], Q[b])
∥  ∥
Hom(X_{Q^{i}}(n − i)[2n − 2i], P_{U}[b] × X_{Q^{i}}) ⟶ Hom(X_{Q^{i}}(n − i)[2n − 2i], Q[b] × X_{Q^{i}})
∥  ∥
Hom(X_{Q^{i}}(n − i)[2n − 2i], Z_{U}[b] × X_{Q^{i}}) ⟶ Hom(X_{Q^{i}}(n − i)[2n − 2i], Q[b] × X_{Q^{i}})
∥  ∥
Hom(X_{Q^{i}}(n − i)[2n − 2i], Z_{U}[b]) ⟶ Hom(X_{Q^{i}}(n − i)[2n − 2i], Q[b])
∥  ∥
Hom(Z_{L} × X_{Q^{i}}, ℤ(i)[2i + b]) ⟶ Hom(Q × X_{Q^{i}}, ℤ(i)[2i + b])
∥  ∥
Hom(P_{L} × X_{Q^{i}}, ℤ(i)[2i + b]) ⟶ Hom(Q × X_{Q^{i}}, ℤ(i)[2i + b])

(first and third identifications - by Theorem 2.3.2 )

Consider two exact sequences, corresponding to Hom’s from P_{L} × X_{Q^{i}}(n − i)[2n − 2i] → Q × X_{Q^{i}}(n − i)[2n − 2i] → R̃′ × X_{Q^{i}}(n − i)[2n − 2i] to ℤ(n)[2n + b], and from X_{Q^{i}}(n − i)[2n − 2i] to P_{U}[b] → Q[b] → R̃[b].

Considering the first and the last row of the diagram above (together with the isomorphism between them), we see, that Hom(X_{Q^{i}}(n − i)[2n − 2i], R̃[b]) has as many elements as Hom(R̃′ × X_{Q^{i}}, ℤ(i)[2i + b]).

But: Hom(R̃′×X_{Q^{i}}, ℤ(i)[2i+b]) = 0. Really, R̃′×X_{Q^{i}} is an extension of: X_{Q^{j}}(j)[2j], X_{Q^{j}}(n − j)[2n − 2j] for i < j ⩽ [n/2], and X_{Q^{i}}(n − r)[2n − 2r] for 0 ⩽ r ⩽ i (by Theorem 3.1 ), and all “round numbers” (-) are > i, by Theorem 2.3.3 (1), we get what we need.

So, we have: Hom(X_{Q^{i}}(n − i)[2n − 2i], R̃[b]) = 0.

But since Hom(X_{Q^{i}}(n − i)[2n − 2i], P_{L}[b]) = Hom(X_{Q^{i}}(n − i)[2n − 2i], P_{L}[b] × X_{Q^{i}}) = Hom(X_{Q^{i}}(n − i)[2n − 2i], (⊕_{0⩽j⩽i}ℤ(j)[2j])[b] × X_{Q^{i}})) (by Theorem 2.3.2 ) = Hom(X_{Q^{i}}(n − i)[2n − 2i], (⊕_{0⩽j⩽i}ℤ(j)[2j])[b]) = 0, (since n − i > j, by the same arguments as above), we have: Hom(X_{Q^{i}}(n − i)[2n − 2i], Q′⟨i + 1⟩[b]) = 0 for all b.

Lemma 3.8 is proven.

Suppose we already found φ_{j}, φ′_{j}, φ_{R^{j+1}} and φ⟨j + 1⟩ for all j < i.

Consider the diagram:

R^{i+1} ⟶ Q′⟨i⟩ ⟶ X_{Q^{i}}(i)[2i]
↓^{φ⟨i⟩}
R^{i+1} ⟶ Q′⟨i⟩ ⟶ X_{Q^{i}}(i)[2i]

Since Hom(R^{i+1}, X_{Q^{i}}(i)[2i]) = Hom(R^{i+1}, X_{Q^{i}}(i)[2i − 1]) = 0 (since R^{i+1} is an extension of X_{Q^{m}}(m)[2m] and X_{Q^{m}}(n − m)[2n − 2m] , m > i and X_{Q^{i}}(n − i)[2n − 2i]), and Hom(X_{Q^{m}}(∗)[∗′], X_{Q^{i}}(i)[2i+b]) = Hom(X_{Q^{m}}(∗)[∗′], ℤ(i)[2i+b]) (by Theorem 2.3.2 ) = 0 (since ∗ > i, and for any smooth projective variety S, Hom(X_{S}, ℤ(−t)[∗]) = 0 for all t > 0 (see Theorem 2.3.3 )), we get unique morphisms φ_{i} and φ_{R^{i+1}}.

Now, consider the diagram:

X_{Q^{i}}(n − i)[2n − 2i] ⟶ R^{i+1} ⟶ Q′⟨i + 1⟩
↓^{φ_{R^{i+1}}}
X_{Q^{i}}(n − i)[2n − 2i] ⟶ R^{i+1} ⟶ Q′⟨i + 1⟩

By Lemma 3.8 we get unique morphisms φ⟨i + 1⟩ and φ′_{i} (which make it commutative).

Theorem 3.7 is proven.

From the uniqueness it follows that the maps : φ ↦ φ_{i}, φ′_{i}, φ_{R^{i+1}}, φ⟨i⟩ (see Theorem 3.7 for the definition) are homomorphisms of rings.

The same considerations as in the proof of Theorem 3.7 give the following:

Corollary 3.9 . The Postnikov tower from Theorem 3.1 is defined up to unique isomorphism by the quadric Q. In particular, the motives Q⟨i⟩, R^{i} are defined uniquely by the quadric Q.

Since Hom(X_{Q^{i}}(i)[2i], X_{Q^{i}}(i)[2i]) = Hom(X_{Q^{i}}, ℤ) = ℤ (see Theorem 2.3.2 and the end of the proof of Theorem 2.3.4 ), we see that φ_{i} and φ′_{i}, for i ⩽ [(n − 1)/2] are, actually, just integers. For even n, φ⟨n/2⟩ can be considered as an element of M_{2}(ℤ) (We fix some algebraic closure k ⊂ \overline{k}, and a basis in Q′⟨n/2⟩|\overline{k} = ℤ(n/2)[n] ⊕ ℤ(n/2)[n] consisting of planes of middle dimension (via identification: CH^{n/2}(Q|\overline{k}) = Hom(ℤ(n/2)[n], Q|\overline{k}) = Hom(ℤ(n/2)[n], Q′⟨n/2⟩|\overline{k}) = ℤ ⊕ ℤ) (it is unique up to permutation); Gal(\overline{k}/k) acts on M_{2}(ℤ) through Gal(k\sqrt{det(Q)}/k), and if det(Q) ≠ 1,

the generator of the Gal(k\sqrt{det(Q)}/k) acts on M_{2}(ℤ) via conjugation with (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})).

We have a natural map α : End(Q) → ×_{i=0,...,[n/2]} (End(X_{Q^{i}}(i)[2i]) × End(X_{Q^{i}}(n − i)[2n − 2i])) = End(Q|k̄) in the case n-odd (equality - since both groups are isomorphic to ∏_{l=1,...,2[n/2]+2} ℤ - by Theorem 2.4.22 , and the fact that Hom(ℤ, ℤ(i)[2i]) = 0 for any i ≠ 0, and Hom(ℤ, ℤ) = ℤ (see Theorem 2.1.18 , Theorem 2.1.20 )), and α : End(Q) → ×_{i=0,...,n/2−1} (End(X_{Q^{i}}(i)[2i]) × End(X_{Q^{i}}(n − i)[2n − 2i])) ×

× End(X_{Q^{n/2}}(n/2)[n] × k (\sqrt{det(Q)})) ⊂ End(Q|k̄) in the case n-even (the same references).

Lemma 3.10 .

(End(Q))_{tors} = Ker(End(Q) → End(Q|k̄))

Moreover, 2^{[n/2]+1} · (End(Q))_{tors} = 0.

1) If q ∉ I^{2}(W (k)), then (End(Q))_{nilp} = Ker(End(Q) → End(Q|k̄))

Moreover, ((End(Q))_{nilp})^{2[n/2]+1} = 0.

2) If q ∈ I^{2}(W (k)), then (Ker(End(Q) → M_{2}(ℤ)))_{nilp} = Ker(α), and, moreover, Ker(α)^{2[n/2]+1} = 0.

Proof of Lemma 3.10

We will consider only the case n-odd, other cases are similar.

Since End(Q′) is torsion free for any hyperbolic Q′, and on Q^{[n/2]} there exists point of degree 2^{[n/2]+1}, from the existence of transfers we have that (End(Q))_{2^{[n/2]+1}} = Ker(End(Q) → End(Q|k̄)).

Let φ ∈ Ker(α). Since φ acts trivially on each “elementary” piece X_{Q^{i}}(∗)[2∗], and Q is an extension of 2[n/2] + 2 such “elementary” pieces (see Definition 2.5.6 ), we have (by Theorem 2.5.7 ): φ^{2[n/2]+1} = 0 (we just need to mention that on Q⟨[n/2]⟩ φ will also act as 0 (to improve number by 1)).

Lemma 3.10 is proven.

Corollary 3.11 .

For any nontrivial idempotent u ∈ End(Q), α(u) is again a nontrivial idempotent of End(Q|k̄).

Proof Evident.

Remark Using transfer arguments, one can easily show, that 2^{k} · (Ker(α))^{l} = 0, if k + \frac{1}{2}l > [n/2]; then also: 2^{k} · (Ker(α))^{2^{l}} = 0, if k + l > [n/2].

Lemma 3.12 . α maps idempotents of End(Q) surjectively onto idempotents of End(Q|k̄)∩(Image(α)).

Proof of Lemma 3.12

Suppose, \overline{u} is some idempotent in the image of α, then there exists u ∈ End(Q), such that u^{2} − u = x ∈ Ker(α). Since x commutes with u, we have that u^{2^{[n/2]+2}} = (u + x)^{2^{[n/2]+1}} = u^{2^{[n/2]+1}} by remark above. So, u^{2^{[n/2]+1}} will be an idempotent, and α(u^{2^{[n/2]+1}}) = \overline{u}. Lemma 3.12 is proven.

Lemma 3.13 .

For any 0 ⩽ i ⩽ [(n − 1)/2] there exist φ(i), φ′(i) ∈ End(Q), s.t. φ(i)_{j} = 2δ_{ij}, φ(i)′_{j} = 0, φ′(i)′_{j} = 2δ_{ij}, φ′(i)_{j} = 0.

For n-even there exists φ, ψ ∈ End(Q), s.t. φ⟨n/2⟩ = (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}) , ψ⟨n/2⟩ = (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}),

and φ_{i} = φ′_{i} = ψ_{i} = ψ′_{i} = 0.

Proof of Lemma 3.13

Evidently, φ(i) = h^{dim(Q)−i} × h^{i}, φ′(i) = h^{i} × h^{dim(Q)−i}, ψ = h^{dim(Q)/2} × h^{dim(Q)/2} and φ = 2Δ − ∑_{i<n/2}(φ(i) + φ′(i)), where h^{i} is a plane section of Q of codimension i, will work (we use identification: End(Q) = CH^{dim(Q)}(Q × Q) - see Theorem 2.1.23 , Theorem 2.1.17 ).

Since the category Chow(k) is closed under taking kernels and cokernels of projectors (see Definition 2.2.1 ), and Chow(k) → DM_{gm}(k) is a full embedding (see Theorem 2.2.4 ), we see that Q is decomposable (i.e. can be represented as a direct sum of two nonzero motives in DM_{gm}(k) (which will be automatically in Chow(k) by Theorem 2.2.4 )), if and only if End(Q) contains a nontrivial idempotent.

Corollary 3.14 .

For n-odd: the motive of Q is decomposable into a direct sum iff the image of α/2 : End(Q) → ℤ/2 × ⋯ × ℤ/2, is bigger than the diagonal ℤ/2 (generated by id).

Proof

From Lemma 3.13 it follows that the image of α contains 2 · End(Q|\overline{k}), so, each element of image(α/2) ⊂ End(Q|\overline{k})/2 · End(Q|\overline{k}) corresponds to an idempotent of End(Q|\overline{k}) ∩ image(α), and, hence, to an idempotent of End(Q).

Remark The same is true for dim(Q) > 0 - even by Lemma 3.23 , Proposition 4.2 and the fact that if Q⟨n/2⟩ × X_{Q^{n/2}}(n/2)[n] is a direct summand in Q, then Q is hyperbolic over the quadratic extension k\sqrt{det(Q)}.

Corollary 3.15 .

Consider two fields: K_{1} and K_{2} over k. Call K_{1} ∼ K_{2}, if each of them can be represented as an extension of odd degree of the other. Let dim(Q_{1}) = dim(Q_{2})-odd, and M(Q_{1}) is undecomposable. Then the following conditions are equivalent:

1) M(Q_{1}) = M(Q_{2})

2) k(Q_{1}) ∼ k(Q_{2})

3) X_{Q_{1}} = X_{Q_{2}}

Proof (2 ⇒ 1) If k(Q_{1}) ∼ k(Q_{2}), then we have morphism f : Q_{1} → Q_{2} which is of odd degree on the generic cycle. Consider the closure of it’s graph in Q_{1} × Q_{2}. We get some morphism φ : M(Q_{1}) → M(Q_{2}) (via identification: Hom(Q_{1}, Q_{2}) = CH^{dim(Q_{i})}(Q_{1} × Q_{2}), see Theorem 2.1.23 , Theorem 2.1.17 ); analogously, we have ψ : M(Q_{2}) → M(Q_{1}). Now, we have (ψ ∘ φ)_{0} and (ψ ∘ φ)′_{0} are odd (for definition of φ_{j}, φ′_{j} see Theorem 3.7 ). If for some i we would have (ψ ∘ φ)_{i}, or (ψ ∘ φ)′_{i} - even, then we would have a nontrivial idempotent in the M(Q_{1}) (by Corollary 3.14 ). So, all of them are odd. Now, using Lemma 3.13 , we can correct φ, to make all φ_{i} = φ′_{i} = 1, and φ itself - an isomorphism (by Lemma 3.10 (consider 1 − φ)).

(1 ⇒ 2) If M(Q_{1}) = M(Q_{2}), then we have some cycles P_{1,2} on Q_{1}×Q_{2} of dimension = dim(Q_{i}), which have an odd degree over corresponding factors. Now we need to use the Springer’s theorem (see Theorem 2.4.14 ), to get rational morphism Q_{1} → Q_{2} and back. This morphisms should give us desired extension by the same arguments as above.

(3 ⇒ 2) If X_{Q_{1}} = X_{Q_{2}}, then we have rational morphism from Q_{1} to Q_{2} and back (by Springer’s theorem), which should cover the generic point “oddly” (or we will have an idempotent).

(2 ⇒ 3) By Theorem 2.3.4 and Springer’s theorem.

Remark 0 The same is true for dim(Q_{i})-even - by Lemma 3.25 .

Remark 1 In particular, we see that if k(Q_{1}) ∼ k(Q_{2}), then M(Q_{1}) is decomposable iff M(Q_{2}) is.

Remark 2 For decomposable quadrics it is completely wrong. For example, for two small Pfister subquadrics of the same big Pfister quadric we have: k(Q_{1}) ∼ k(Q_{2}), though, in general, M(Q_{1}) ≠ M(Q_{2}); more trivially, for two isotropic quadrics we have k(Q_{1}) = k(Q_{2})-purely transcendental/k, but motives can be different (see Theorem 2.4.23). For general statement (independent of decomposability) see Proposition 5.1.

Corollary 3.16 .

Let h_{1}(Q) > 1 (i.e. X_{Q} = X_{Q^{1}}). Then M(Q) is decomposable.

Proof Consider the odd-dimensional case first.

Over k(Q) Q has a line, so we have rational map Q → Q^{1}, which gives us some cycle of dimension dim(Q) + 1 on Q × Q (“graph”), and, hence a morphism π : M(Q) → M(Q)(−1)[−2] (see Theorem 2.1.23 , Theorem 2.1.17 ). Then the composition: h ∘ π (where h is a hyperplane section embedded diagonally) is a morphism φ : Q → Q, s.t. φ′_{0} = 0 and φ_{0} = 1 (see Theorem 3.7 for the definition of φ_{i}) (to compute this numbers you should change k by \overline{k}, and then compute the intersection numbers of cycle corresponding to φ with Q × Spec(\overline{k}) and h × l_{1} (h here is a hyperplane section, and l_{1} is some line on Q|\overline{k})). By Corollary 3.14 M(Q) is decomposable.

In the case of even-dimensional quadric we have to add that in the basis of CH^{n/2}(Q|\overline{k}) consisting of n/2 - dimensional planes T, T′ the matrix of φ (which is just φ⟨n/2⟩, see

discussion after Corollary 3.9 ) has the form: (\begin{matrix}a & c \\ a & c \end{matrix}) for some a, c (since CH^{n/2−1}(Q|\overline{k}) = ℤ is generated by the class of subquadric S of dimension n/2 + 1, and h(S) = T + T′) So, we can use Lemma 3.13 , to correct φ (by subtracting [(a+c)/2]·ψ (notations from Lemma 3.13 )), and make either φ⟨n/2⟩ - an idempotent in M_{2}(ℤ), or (φ⟨n/2⟩)^{2} = 0 (in which case we can substitute φ by φ^{2}). Now, again using Lemma 3.13 , we can correct φ, to make all φ_{i}, φ′_{j} equal to 1, or 0, which means that φ|\overline{k} - is an idempotent. By Lemma 3.12 we have a nontrivial idempotent in End(Q) (it will be equal to φ^{2^{[n/2]+1}} (see the proof of Lemma 3.12 )).

Corollary 3.17 . For any automorphism φ ∈ Ker(Aut(M(Q)) → GL_{2}(ℤ)) : φ^{2^{[n/2]+2}} = 1.

Proof Since φ ∈ Aut(Q), φ_{i}, φ′_{i} = ±1, for 0 ⩽ i ⩽ [(n − 1)/2] (see Theorem 3.7 for the definition of φ_{i} and φ′_{i}). So, (φ^{2})_{i} = (φ′^{2})_{i} = 1. Hence, (since φ ∈ Ker) φ^{2} = 1 + y, where y ∈ Ker(α). Now, the same arguments as in the proof of Lemma 3.12 above show, that (1 + y)^{2^{[n/2]+1}} = 1, and φ^{2^{[n/2]+2}} = 1.

We should mention that usual “algebro-geometric” automorphisms of a quadric almost don’t act on it’s motive.

Observation 3.18 .

The map SO(q) → Aut(M(Q)) is trivial.

Proof

By the theorem of Cartan-Deudonne (see [9], p.27) we know that O(q) is generated by reflections τ_{y} : x ↦ x − \frac{B(x,y)}{q(y)} · y (q(y) ≠ 0). In particular, SO(q) is generated by pairs of reflections. It is easy to see that graphs of τ_{y} and τ_{z} are rationally equivalent: just consider the graph of τ_{λ_{0}·y+λ_{1}·z}. So SO(q) does not act on M(Q) (since τ_{y}^{2} = 1).

Remark The group ℤ/2 = O(q)/ SO(q) acts nontrivially on the motive of the even-dimensional quadric Q, even when Q is hyperbolic (for the generator τ we have

τ⟨dim(Q)/2⟩ = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})). On the other hand, evidently, this ℤ/2 does not act on the

motive of odd-dimensional hyperbolic quadric.

If we consider natural Lie algebra structure on End(Q) and denote as D^{k}(End(Q)) the k-th derivative of End(Q), then we have:

Corollary 3.19 .

1) In the case q ∉ I^{2}(W (k)): D^{2+[log_{2}(n)]}(End(Q)) = 0, and D^{[n/2]}(Aut(Q)) = 1.

2) In the case q ∈ I^{2}(W (k)): D^{2+[log_{2}(n)]}(Ker(End(Q) → M_{2}(ℤ))) = 0, and D^{[n/2]}(Ker(Aut(Q) → GL_{2}(ℤ))) = 1.

More generally, D^{[n/2]−i}(Ker(End(Q⟨i⟩)→M_{2}(ℤ))) = 0, D^{2+[log_{2}(n−2i)]}(Ker(Aut(Q⟨i⟩) → GL_{2}(ℤ))) = 1.

Proof Consider the case q ∉ I^{2}(W (k)). D(End(Q)) ⊂ Ker(α), now, the fact that D^{2+[log_{2}(n)]}(Ker(α)) = 0 is evident (since for any elements x_{1}, …, x_{2[n/2]+1} from Ker(α) we have ∏ x_{i} = 0 (see the proof of Lemma 3.10 )). Analogously in the case of Aut(Q).

3.3. Some unpleasant computations. In this section we will prove that, roughly speaking, it is enough to have reasonable (motivic) morphism between two quadrics, in order to show that they contain isomorphic direct summands. We have so many computations here because of the even-dimensional case.

Let Q_{1} and Q_{2} be two quadrics of the same dimension n. We can fix an algebraic closure k ⊂ \overline{k} and bases in CH^{i}(Q_{1}|\overline{k}), CH^{i}(Q_{2}|\overline{k}), consisting of: projective subspace l_{n−i}(1, 2) of dimension n − i for n ⩾ i > n/2, plane section h^{i}(1, 2) of codimension i for 0 ⩽ i < n/2, and of two projective subspaces l_{n/2}(1, 2)_{1}, l_{n/2}(1, 2)_{2} of dimension n/2 from different families (i.e. whose intersection is 1) for i = n/2 (if n is even).

For an element φ ∈ Hom(Q_{1}, Q_{2}) we can define integers φ_{i}, φ′_{i}, and φ⟨n/2⟩ ∈ M_{2}(ℤ) in the same way as in Theorem 3.7 . I.e.: if φ is represented by a cycle F ⊂ Q × Q, then φ_{i} = (F, h^{i}(1)×l_{i}(2)), φ′_{i} = (F, l_{i}(1)×h^{i}(2)) for all 0 ⩽ i < n/2, and φ⟨n/2⟩_{a,b} = (F, l_{n/2}(1)_{a} × l_{n/2}(2)_{b}), where (A, B) for two cycles A, B of complementary dimension is their intersection number.

The generator of Gal(k\sqrt{det(Q_{1})}/k) acts on M_{2}(ℤ) via multiplication by (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})

on the right (if det(Q_{1}) ≠ 1), and generator of Gal(k\sqrt{det(Q_{2})}/k) acts on M_{2}(ℤ) via multiplication by the same matrix on the left (since such a generator σ_{1,2} intercanges the l_{n/2}(1, 2)_{1} and l_{n/2}(1, 2)_{2}). Gal(\overline{k}/k) does not act on those ℤ’s, where φ_{i}’s and φ′_{i}’s live (since it acts trivially on corresponding basis elements).

Up to duality (see Theorem 2.1.23 ), we have 4 cases:

1) det(Q_{1}) = det(Q_{2}) = 1;

2) det(Q_{1}) = 1, det(Q_{2}) ≠ 1;

3) det(Q_{1}) = det(Q_{2}) ≠ 1;

4) 1 ≠ det(Q_{1}) ≠ det(Q_{2}) ≠ 1.

Let U_{2} be the subgroup of M_{2}(ℤ), consisting of matrices (\begin{matrix}a & b \\ a & b \end{matrix}). Analogously, let

U_{1} be a subgroup of M_{2}(ℤ), consisting of matrices (\begin{matrix}a & a \\ c & c \end{matrix}). Let V be a subgroup of

M_{2}(ℤ), consisting of matrices (\begin{matrix}a & b \\ b & a \end{matrix}).

Let \overline{⟨n/2⟩} be the natural map Hom(Q_{1}, Q_{2}) → M_{2}(ℤ).

We have the following analog of Lemma 3.13 .

Lemma 3.20 .

A) For any 0 ⩽ i < n/2 there exist elements u_{i}, u′_{i} in Hom(Q_{1}, Q_{2}), s.t. (u_{i})_{j} = 2δ_{ij}, (u_{i})′_{j} = 0, (u′_{i})_{j} = 0, (u′_{i})′_{j} = 2δ_{ij}, and u_{i}⟨n/2⟩ = u′_{i}⟨n/2⟩ = 0.

There exists w_{n/2} ∈ Hom(Q_{1}, Q_{2}), s.t. (w_{n/2})_{i} = (w_{n/2})′_{i} = 0, and w_{n/2}⟨n/2⟩ =

(\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}).

B) In the four cases specified above we have the following:

1) 2^{r} · M_{2}(ℤ) ⊂ image(\overline{⟨n/2⟩}), for some large r.

2) 2^{r} · U_{2} ⊂ image(\overline{⟨n/2⟩}) ⊂ U_{2}, for some large r.

3) 2^{r} · V ⊂ image(\overline{⟨n/2⟩}) ⊂ V, for some large r.

4) image(\overline{⟨n/2⟩}) = ℤ · w_{n/2}.

Proof

A) u_{i} corresponds to the cycle h^{n−i} × h^{i} ⊂ Q_{1} × Q_{2}, u′_{i} - to h^{i} × h^{n−i} ⊂ Q_{1} × Q_{2}, and w_{n/2} - to h^{n/2} × h^{n/2} ⊂ Q_{1} × Q_{2}.

B) Since φ is defined over k, φ⟨n/2⟩ will be stable under Gal(\overline{k}/k), so we get inclusions: in the second case - into U_{2}, in the third - into V, and in the fourth - into ℤ · w_{n/2}.

Since Q_{1}, Q_{2} are hyperbolic over some field extension E of degree 2^{r} over k, and if Q_{1}|_{E}, Q_{2}|_{E} are hyperbolic, then image(\overline{⟨n/2⟩}|_{E}) = M_{2}(ℤ), we get the first inclusions in all 3 cases: consider the composition ψ̃ : Q_{1} → Q_{1} × E →^{ψ} Q_{2} × E → Q_{2}, then ψ̃⟨n/2⟩ = ∑_{σ∈Gal(E/k)} σ(ψ⟨n/2⟩). In the fourth case we have an equality, since w_{n/2} ∈ image(\overline{⟨n/2⟩}) by A).

Let Q_{1}, Q_{2} be two quadrics of the same dimension n. Let J^{d}, J^{u} be subsets of [0, 1, …, [n − 1/2]. For a subset I of [0, 1, …, [n − 1/2]] put δ_{I}(j) = 1, if j ∈ I, and 0, if j ∉ I, for all 0 ⩽ j < n/2. Let J = (J^{d}, J^{u}). Denote Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩ the subset of M_{2}(ℤ), consisting of the images of maps φ ∈ Hom(Q_{1}, Q_{2}), s.t. φ_{i} = δ_{J^{d}}(i), φ′_{j} = δ_{J^{u}}(j). Denote Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩ the subgroup of M_{2}(ℤ), generated by the images of maps φ ∈ Hom(Q_{1}, Q_{2}), s.t. φ_{i} = 0, φ′_{j} = 0 for all 0 ⩽ i, j < n/2.

Lemma 3.21 .

Let N, N′ be undecomposable direct summands in Q, s.t. for corresponding idempotents p_{N}, p_{N′} we have (p_{N})_{i} = (p_{N′})_{i} = 1 for some i (see above, or Theorem 3.7 for the definition). Then there are morphisms u : N → N′ and v : N′ → N defining the isomorphism N = N′.

In particular, (p_{N})_{j} = (p_{N′})_{j} and (p_{N})′_{j} = (p_{N′})′_{j} for all j.

Proof

Let j_{N} : N → Q, π_{N} : Q → N, and j_{N′} : N′ → Q, π_{N′} : Q → N′ be maps defining the decompositions (in particular, j_{N} ∘ π_{N} = p_{N} and j_{N′} ∘ π_{N′} = p_{N′}).

Let dim(Q) = n is odd. Then End(N)|\overline{k} is (∏_{l:(p_{N})_{l}=1} ℤ) × (∏_{j:(p_{N})′_{j}=1} ℤ). Consider u = π_{N′} ∘ j_{N} and v = π_{N} ∘ j_{N′}. Then φ = π_{N} ∘ p_{N′} ∘ j_{N} ∈ End(N) has the property that: φ_{l} and φ′_{j} is either 0, or 1 (since p_{N}, p_{N′} have such property). Hence, φ|\overline{k} is an idempotent. By the proof of Lemma 3.12 , φ^{r} will be an idempotent in End(N) ⊂ End(Q). Since N is undecomposable, it should be p_{N} (since φ_{i} = 1 by condition). Hence φ^{r} is an isomorphism, and φ is an isomorphism. In the same way, considering ψ = π_{N′} ∘ p_{N} ∘ j_{N′} ∈ End(N′), we get that π_{N′} ∘ j_{N} and π_{N} ∘ j_{N′} perform an isomorphism N = N′.

Let dim(Q) = n is even. We have 3 cases:

1) rank(p_{N}⟨n/2⟩) = 0, or rank(p_{N′}⟨n/2⟩) = 0;

2) rank(p_{N}⟨n/2⟩) = 2, or rank(p_{N′}⟨n/2⟩) = 2;

3) rank(p_{N}⟨n/2⟩) = rank(p_{N′}⟨n/2⟩) = 1.

1) Consider u = π_{N′} ∘ j_{N} and v = π_{N} ∘ j_{N′}. Can assume that rank(p_{N}⟨n/2⟩) = 0. We have: p_{N}⟨n/2⟩ = 0. Then End(N)|\overline{k} = (∏_{l:(p_{N})_{l}=1; 0⩽l<n/2} ℤ) × (∏_{j:(p_{N})′_{j}=1; 0⩽j<n/2} ℤ). Let φ = v ∘ u : N → N, and ψ = u∘v : N′ → N′. Since φ_{l}, φ′_{j} are 0, or 1 (since p_{N}, p_{N′} have such a property), we have that φ|\overline{k} will be an idempotent in End(N)|\overline{k} ⊂ End(Q)|\overline{k}. In the same way as before, we get that φ : N → N is an isomorphism.

We have 3 choices for the rank(p_{N′}⟨n/2⟩): a) 0; b) 1; c) 2.

a) In the same way as before we get that ψ = u ∘ v is an isomorphism N′ → N′. So, u, v are isomorphisms.

b) We have: End(N′)|\overline{k} = (∏_{l:(p_{N′})_{l}=1; 0⩽l<n/2} ℤ)×(∏_{j:(p_{N′})′_{j}=1; 0⩽j<n/2} ℤ)×ℤ, where the last term corresponds to the endomorphisms of ℤ(n/2)[n] as a direct summand of N′|\overline{k} (and the generator is represented by p_{N′}⟨n/2⟩ ∈ M_{2}(ℤ) = End(Q⟨n/2⟩|\overline{k})). ψ_{l}, ψ′_{j} are 0, or 1 as before, and ψ⟨n/2⟩ = 0 (since p_{N}⟨n/2⟩ = 0). So, ψ|\overline{k} is an idempotent in End(N′|\overline{k}) ⊂ End(Q|\overline{k}), and it can be lifted to a nontrivial idempotent in End(N′) (nontrivial, since ψ_{i} = 1 and ψ⟨n/2⟩ = 0, while p_{N′}⟨n/2⟩ ≠ 0). Contradiction (N′ is undecomposable).

c) We have p_{N′}⟨n/2⟩ = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}). Hence, ψ|\overline{k} is an idempotent in End(N|\overline{k}). An it

should be nontrivial, since ψ⟨n/2⟩ = 0 and ψ_{i} = 1, but p_{N′}⟨n/2⟩ = Id_{2}. Contradiction.

2) Consider, again, u = π_{N′} ∘ j_{N} and v = π_{N} ∘ j_{N′}.

We can assume that rank(p_{N}⟨n/2⟩) = 2, i.e.: p_{N}⟨n/2⟩ = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}).

We have 3 choices for the rank(p_{N′}⟨n/2⟩): a) 0; b) 1; c) 2. a) The same as 1)c).

b) We have End(N)|\overline{k} = (∏_{l:(p_{N})_{l}=1; 0⩽l<n/2} ℤ) × (∏_{j:(p_{N})′_{j}=1; 0⩽j<n/2} ℤ) × M_{2}(ℤ), where the last term corresponds to the endomorphisms of ℤ(n/2)[n] ⊕ ℤ(n/2)[n] as a direct summand of N|\overline{k}. φ_{l}, φ′_{j} are 0, or 1 as before, and φ⟨n/2⟩ = p_{N′}. Using Lemma 3.12 we can lift it to the idempotent on N (equal to φ^{r}, see the proof of Lemma 3.12 ), which should be nontrivial, since φ_{i} = 1, rank(φ⟨n/2⟩) = 1, but rank(p_{N}⟨n/2⟩) = 2.

c) The same considerations as above shows that φ, ψ will be the isomorphisms, and so will be u and v.

3) We have: End(N)|\overline{k} = (∏_{l:(p_{N})_{l}=1; 0⩽l<n/2} ℤ) × (∏_{j:(p_{N})′_{j}=1; 0⩽j<n/2} ℤ) × ℤ, where the last term corresponds to the endomorphisms of ℤ(n/2)[n] as a direct summand of N|\overline{k} (and the generator is represented by p_{N}⟨n/2⟩ ∈ M_{2}(ℤ) = End(Q⟨n/2⟩|\overline{k})), and End(N′)|\overline{k} = (∏_{l:(p_{N′})_{l}=1; 0⩽l<n/2} ℤ) × (∏_{j:(p_{N′})′_{j}=1; 0⩽j<n/2} ℤ) × ℤ, where the last term corresponds to the endomorphisms of ℤ(n/2)[n] as a direct summand of N′|\overline{k} (and the generator is represented by p_{N′}⟨n/2⟩ ∈ M_{2}(ℤ) = End(Q⟨n/2⟩|\overline{k})).

Take φ = p_{N} ∘ p_{N′} ∘ p_{N}, and φ′ = p_{N′} ∘ p_{N} ∘ p_{N′}. We have: p_{N}⟨n/2⟩ ∘ p_{N′}⟨n.2⟩ ∘ p_{N}⟨n/2⟩ = λ · p_{N}⟨n/2⟩, and p_{N′}⟨n/2⟩ ∘ p_{N}⟨n/2⟩ ∘ p_{N′}⟨n/2⟩ = λ · p_{N′}⟨n/2⟩. By Lemma 3.13 we can correct φ into φ̃ (without changing φ_{l} and φ′_{j}), and make λ = 1, or 0 (we can’t get 0 by 1)b), so, λ is odd).

We have: φ̃|\overline{k} an idempotent. Using Lemma 3.12 we can lift it to the idempotent on N (equal to φ̃^{r}, see the proof of Lemma 3.12 ), which should be an identity, since φ_{i} = 1 and N is undecomposable. Then φ̃ will be an isomorphism N → N. The same can be done with N′ instead of N.

In particular, we see that (p_{N′})_{l} = (p_{N})_{l} and (p_{N′})′_{j} = (p_{N})′_{j} for all 0 ⩽ l, j < n/2. Denote J^{d} - the set of such i, that (p_{N})_{i} = 1, and J^{u} - the set of such j, that (p_{N})′_{j} = 1, and J = (J^{d}, J^{u}).

Let (p_{N})⟨n/2⟩ = (\begin{matrix}a & b \\ c & d \end{matrix}). Since rank(p_{N}⟨n/2⟩) = 1 (and p_{N} is an idempotent in

M_{2}(ℤ)), we have: a + d = 1.

By Lemma 3.13 we have element ρ ∈ End(Q), s.t. ρ_{i} = ρ′_{i} = 1, for all 0 ⩽ i < n/2,

and ρ⟨n/2⟩ = (\begin{matrix}0 & −1 \\ −1 & 0 \end{matrix}).

Since a − d is odd, we have: either a − c + b − d, or a − b + c − d is not divisible by 4. Suppose that the first one is not divisible (the other case is completely symmetric).

We have: (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ End(Q)_{J}⟨n/2⟩. Then (multiplying by ρ) we have: (\begin{matrix}−b & −a \\ −d & −c \end{matrix})∈ End(Q)_{J}⟨n/2⟩. Subtracting, we get: (\begin{matrix}a + b & a + b \\ c + d & c + d \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩. Using Lemma

3.13 , we get: (\begin{matrix}a + b − c − d & a + b − c − d \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩.

Since rank(p_{N}) = 1, we have that det(Q) = 1 (really, if det(Q) ≠ 1, then End(Q)_{J}⟨n/2⟩ is contained in the subgroup of M_{2}(ℤ), generated by (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) and (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (by Lemma 3.20 )). Since det(Q) = 1, by Lemma 3.20 , we have that

(\begin{matrix}2^{r} & 2^{r} \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩, for large r.

Since a + b − c − d is not divisible by 4, we have: (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩. Then (\begin{matrix}2 & 0 \\ 2 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩ (by duality), and so, (\begin{matrix}4 & 0 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩.

We can assume that a is odd and d is even. If b − c is even, we can use Lemma 3.13 , and our matrices (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ End(Q)_{J}⟨n/2⟩, and (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩, to find some matrix of the type (\begin{matrix}f & 0 \\ 0 & 0 \end{matrix}) in End(Q)_{J}⟨n/2⟩. Multiplying by ρ, and considering square, we get that (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{J}⟨n/2⟩. Since, by Lemma 3.12 , we can lift the corresponding element to an idempotent, we get a contradiction by 1)b).

So, b − c is odd, and without loss of generality we can assume that a, b are odd, and c, d are even.

Using matrices: (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}), (\begin{matrix}4 & 0 \\ 0 & 0 \end{matrix}), p_{N}, and Lemma 3.13 (or Lemma 3.20 ), we get that End(Q)_{J}⟨n/2⟩ contains one of the following matrices: (\begin{matrix}1 & 1 \\ 0 & 0 \end{matrix}), (\begin{matrix}1 & −1 \\ 0 & 0 \end{matrix}), (\begin{matrix}−1 & 1 \\ 0 & 0 \end{matrix}), or (\begin{matrix}−1 & −1 \\ 0 & 0 \end{matrix}). Considering squares, we reduce the choice to the first two cases. Consider the second case (the first one is similar). Let ψ ∈ End(Q) be the corresponding representative.

Consider ψ ∘ ψ^{∨} ∈ End(Q). We have ψ ∘ ψ^{∨}⟨n/2⟩ = (\begin{matrix}2 & 0 \\ 0 & 0 \end{matrix}). I.e.: (\begin{matrix}2 & 0 \\ 0 & 0 \end{matrix}) ∈

End(Q)_{I}⟨n/2⟩, where I = (J^{d} ∩ J^{u}, J^{d} ∩ J^{u}). Multiplying by ρ, we get: (\begin{matrix}0 & 2 \\ 0 & 0 \end{matrix}) ∈

End(Q)_{I}⟨n/2⟩, and considering square: (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{I}⟨n/2⟩. Considering pow-

ers of the corresponding representative, we get an idempotent γ ∈ End(Q), s.t. γ_{i} = γ′_{i} = δ_{J^{d}∩J^{u}}(i). Let M be corresponding direct summand.

If J^{d} ∩ J^{u} is nonempty, and j ∈ J^{d} ∩ J^{u}, then considering N and undecomposable direct summand M′ in M “containing j”, we get (by what was proven above ((p_{N})_{l} = (p_{M′})_{l} and (p_{N})′_{s} = (p_{M′})′_{s})) that J^{d} ∩J^{u} = J^{d} = J^{u}. This is impossible by 1)b), since rank(γ) = 0.

So, J^{d} ∩J^{u} is empty, which means that (\begin{matrix}2 & 0 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{0}⟨n/2⟩. Then 2·M_{2}(ℤ) ⊂

End(Q)_{0}⟨n/2⟩. Then (\begin{matrix}1 & 1 \\ 0 & 0 \end{matrix}) ∈ End(Q)_{J}⟨n/2⟩. Let β be corresponding representa-

tive, which we can assume to be an idempotent (by Lemma 3.12 ); let N_{1} be the corresponding direct summand.

Since (\begin{matrix}a & b \\ c & d \end{matrix}) is an idempotent of rank = 1 in M_{2}(ℤ), we have that (\begin{matrix}a & b \\ c & d \end{matrix}) =

± (\begin{matrix}x & y \\ z & t \end{matrix}) · (\begin{matrix}1 & 1 \\ 0 & 0 \end{matrix}) · (\begin{matrix}t & −y \\ −z & x \end{matrix}), for some matrix (\begin{matrix}x & y \\ z & t \end{matrix}) with xt − yz = ±1. I.e.:

a = ±x(t − z), b = ±x(x − y), c = ±z(t − z), d = ±z(x − y). Since a, b are odd, and c, d are even, we get: x, t are odd, and y, z are even.

Since 2 · M_{2}(ℤ) ⊂ End(Q)_{0}⟨n/2⟩, and we have an identity element in End(Q), we

get an automorphism ε : Q → Q, s.t. ε⟨n/2⟩ = (\begin{matrix}x & y \\ z & t \end{matrix}) (use Lemma 3.12 ).

ε and ε^{−1} gives us an isomorphism between N and N_{1}.

In the same way, we can construct an isomorphism between N′ and some direct

summand N′_{1}, given by an idempotent β′, where β′⟨n/2⟩ is (\begin{matrix}1 & 1 \\ 0 & 0 \end{matrix}), or (\begin{matrix}1 & 0 \\ 1 & 0 \end{matrix}) (de-

pending on: is a′ − b′ even or odd). The last case can’t happen, since in this case β + β′ would provide us (using Lemma 3.13 ) with an idempotent α ∈ End(Q), s.t. α_{i} = α′_{i} = 0 for all i, and α⟨n/2⟩ = Id. Then p_{N} ∘(1−α) would give us an idempotent p_{N′′} in Q, s.t. (p_{N′′})_{i} = 1, but rank(p_{N′′}⟨n/2⟩) = 0. This contradicts 1)b).

So, N_{1} and N′_{1} are isomorphic, and hence, N and N′ are isomorphic.

Lemma above explains the following definition.

Definition 3.22 .

Let 0 ⩽ i, j < n/2. We say that X_{Q^{i}}(i)[2i] is hanged to X_{Q^{j}}(j)[2j] (respectively, X_{Q^{j}}(n − j)[2n − 2j]), if for some undecomposable direct summand N of Q we have for corresponding idempotent p_{N}: (p_{N})_{i} = 1 and (p_{N})_{j} = 1 (resp. (p_{N})′_{j} = 1).

Let we have some idempotent p_{N} ∈ End(Q). Then by Theorem 3.7 it gives us the set of idempotents: (p_{N})_{i}, (p_{N})′_{i}, (p_{N})_{R^{j}}, (p_{N})⟨j⟩ for all 0 ⩽ i < n/2, 1 ⩽ j ⩽ n/2. And our whole Postnikov tower is a direct sum of two towers, corresponding to the decomposition: Q = N ⊕ N^{⊥}.

This gives us that N is an extension of “elementary pieces”: (p_{N})_{i} · X_{Q^{i}}(i)[2i], (p_{N})′_{i} · X_{Q^{i}}(n − i)[2n − 2i], and (p_{N})⟨n/2⟩ · Q′⟨n/2⟩ (if n is even) in the same sense as Q is an extension of X_{Q^{i}}(i)[2i], X_{Q^{i}}(n − i)[2n − 2i], and Q′⟨n/2⟩, see Theorem 3.1 .

Let J^{d}(N) be the set of such 0 ⩽ i < n/2, that (p_{N})_{i} = 1, and J^{u}(N) be the set of such 0 ⩽ i < n/2, that (p_{N})′_{i} = 1.

Since all idempotents in X_{Q^{i}}(i)[2i] and X_{Q^{i}}(n − i)[2n − 2i] are either 0, or identity maps, we have that N is an extension of: X_{Q^{i}}(i)[2i], for i ∈ J^{d}(N), X_{Q^{i}}(n−i)[2n−2i], for i ∈ J^{u}(N), and p_{N}⟨n/2⟩ · Q′⟨n/2⟩.

We have three cases:

1) rank(p_{N}⟨n/2⟩) = 0; 2) rank(p_{N}⟨n/2⟩) = 1; 3) rank(p_{N}⟨n/2⟩) = 2.

The considerations above give us the following

Lemma 3.23 .

Let Q be projective quadric of dimension n, and N be a direct summand in Q, then N is an extension (in the same sense as in Theorem 3.1 ) of X_{Q^{i}}(i)[2i], for i ∈ J^{d}(N), X_{Q^{j}}(n − j)[2n − 2j], for j ∈ J^{u}(N) in the first case specified above; the same objects + X_{Q^{n/2}}(n/2)[n] in the second case (det(Q) is necessarily 1 in such case); and the same objects + X_{Q^{n/2}}(n/2)[n] × k\sqrt{det(Q)} in the third case.

Remark We could just say that N is an extension of it’s “elementary pieces” in the sense of Definition 2.5.6 . The picture from Theorem 3.1 just clarify: in which order elementary pieces appear.

Let J be a subset of [0, 1, …, n/2 − 1]. We will denote the pair (J, J) in the same way: J.

Lemma 3.24 .

Let Q_{1}, Q_{2} be projective quadrics, s.t.: dim(Q_{1}) = dim(Q_{2}) = n.

Let φ ∈ Hom(Q_{1}, Q_{2}) be such element, that φ_{i} = φ′_{i} = δ_{J}(i) (see the definition before Lemma 3.21 ). Then there exist direct summands: N_{1} in Q_{1}, and N_{2} in Q_{2}, s.t. for corresponding idempotents p_{N_{1}}, p_{N_{2}} we have: (p_{N_{1}})_{i} = (p_{N_{1}})′_{i} = (p_{N_{2}})_{i} = (p_{N_{2}})′_{i} = δ_{J}(i). And there exist isomorphisms u : N_{1} → N_{2}, v : N_{2} → N_{1}.

Proof

The case n - odd is completely trivial: by Lemma 3.12 , (φ ∘ φ^{∨})^{2^{s}} and (φ^{∨} ∘ φ)^{2^{s}} will be the idempotents for large s, and φ, φ^{∨} will be isomorphisms of corresponding direct summands.

Let now n is even.

By Lemma 3.13 , in End(Q_{1,2}) we have element ρ_{1,2}, s.t. (ρ_{1,2})_{i} = (ρ_{1,2})′_{i} = 1, for

all 0 ⩽ i < n/2, and (ρ_{1,2})⟨n/2⟩ = (\begin{matrix}0 & −1 \\ −1 & 0 \end{matrix}).

Up to duality we have four cases: A) 1 ≠ det(Q_{1}) ≠ det(Q_{2}) ≠ 1; B) det(Q_{1}) = det(Q_{2}) ≠ 1; C) det(Q_{1}) = 1, det(Q_{2}) ≠ 1; D) det(Q_{1}) = det(Q_{2}) = 1.

A) By Lemma 3.20 we can correct φ to make φ⟨n/2⟩ = 0, and then p_{N_{2}} := (φ∘φ^{∨})^{2^{s}} and p_{N_{1}} := (φ^{∨} ∘ φ)^{2^{s}} will be the idempotents for large s (by Lemma 3.12 ), with (p_{N_{j}})_{i} = (p_{N_{j}})′_{i} = δ_{J}(i), (p_{N_{j}})⟨n/2⟩ = 0, and corresponding direct summands will be isomorphic via φ and φ^{∨}.

B) Using Lemma 3.20 , we can assume that φ⟨n/2⟩ = (\begin{matrix}a & 0 \\ 0 & a \end{matrix}). (φ ∘ φ^{∨})⟨n/2⟩ =

(\begin{matrix}a^{2} & 0 \\ 0 & a^{2} \end{matrix}). If a is even then, by Lemma 3.13 , we have 0 ∈ End(Q_{2})_{J}⟨n/2⟩ (the same

with Q_{1}). So, by Lemma 3.12 , we have idempotents p_{N_{1}} ∈ End(Q_{1}), p_{N_{2}} ∈ End(Q_{2}), s.t. (p_{N_{j}})_{i} = (p_{N_{j}})′_{i} = δ_{J}(i), (p_{N_{j}})⟨n/2⟩ = 0. And p_{N_{2}} ∘ φ ∘ p_{N_{1}}, p_{N_{1}} ∘ φ^{∨} ∘ p_{N_{2}} will give an isomorphism between N_{1} and N_{2}.

If a is odd, then since (by Lemma 3.13 ) 2 · (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩, by

Lemma 3.20 we have: (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix})∈Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩. So,(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})∈Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩.

If ψ is corresponding representative, then by Lemma 3.12 p_{N_{2}} := (ψ ∘ ψ^{∨})^{2^{s}} and p_{N_{1}} := (ψ^{∨}∘ψ)^{2^{s}}, for large s, will be idempotents, and corresponding direct summands will be isomorphic.

C) By Lemma 3.20 we can assume that φ⟨n/2⟩ = (\begin{matrix}a & 0 \\ a & 0 \end{matrix}).

Then (φ^{∨} ∘ φ)⟨n/2⟩ = (\begin{matrix}2a^{2} & 0 \\ 0 & 0 \end{matrix}) ∈ End(Q_{1})_{J}⟨n/2⟩. Then, using ρ and con-

sidering square, we have 0 ∈ End(Q_{1})_{J}⟨n/2⟩. Then 0 ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩, and 0 ∈ End(Q_{2})_{J}⟨n/2⟩. In the same way as before this gives us isomorphic direct summands N_{1,2} in Q_{1,2}.

D) Let φ⟨n/2⟩ = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩.

Since det(Q_{1})=det(Q_{2})=1, by Lemma 3.20 we have: 2^{r}·M_{2}(ℤ)⊂Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Considering φ⟨n/2⟩ ( mod 2 ) ∈ M_{2}(ℤ/2), and using ρ_{1,2}, and the fact above, we get that either φ⟨n/2⟩ ( mod 2 ) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}), or = (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}), or we have: (\begin{matrix}0 & 0 \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩ ⊂ M_{2}(ℤ).

In the last case p_{N_{2}} := (φ ∘ φ^{∨})^{2^{s}} and p_{N_{1}} := (φ^{∨} ∘ φ)^{2^{s}} will be idempotents in End(Q_{2}) and End(Q_{1}) for large s (see the proof of Lemma 3.12 ), and φ, φ^{∨} will give us an isomorphism of corresponding direct summands, and (p_{N_{1}})_{i} = (p_{N_{1}})′_{i} = (p_{N_{2}})_{i} = (p_{N_{2}})′_{i} = δ_{J}(i).

Because of the ρ_{1,2}, it is enough to consider the case φ⟨n/2⟩ ( mod 2 ) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}). In particular, a − c is odd. We have two cases: 1) a + b − c − d is not divisible by 4, 2) a − b − c + d is not

divisible by 4.

1) Considering φ − φ ∘ ρ_{1}, we have: (\begin{matrix}a + b & a + b \\ c + d & c + d \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩. Then,

by Lemma 3.20 , (\begin{matrix}a + b − c − d & a + b − c − d \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Since det(Q_{1,2}) = 1, by Lemma 3.20 we have that (\begin{matrix}2^{r} & 2^{r} \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Hence, (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Using matrices: (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}), (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩, and the fact that a − c and b−d are odd, we can correct φ, to make φ⟨n/2⟩ = (\begin{matrix}g & 0 \\ 0 & f \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩ for some odd g and f. Then also, φ^{∨}⟨n/2⟩ = (\begin{matrix}g & 0 \\ 0 & f \end{matrix}) ∈ Hom(Q_{2}, Q_{1})_{J}⟨n/2⟩.

We have (\begin{matrix}2g & 2g \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{1})_{0}⟨n/2⟩. Again, by Lemma 3.20 , (\begin{matrix}2 & 2 \\ 0 & 0 \end{matrix}) ∈

Hom(Q_{1}, Q_{1})_{0}⟨n/2⟩. By duality, (\begin{matrix}2 & 0 \\ 2 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{1})_{0}⟨n/2⟩. And then: (\begin{matrix}4 & 0 \\ 0 & 0 \end{matrix}) ∈

Hom(Q_{1}, Q_{1})_{0}⟨n/2⟩. Then (\begin{matrix}4g & 0 \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩. And by Lemma 3.20 ,

(\begin{matrix}4 & 0 \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Then we can reduce everything to: g = ±1, f = ±1. Denoting corresponding element as u, and dual as v, we get: (u ∘ v)_{i} = (u ∘ v)′_{i} =

(v ∘ u)_{i} = (v ∘ u)′_{i} = δ_{J}(i), and (u ∘ v)⟨n/2⟩ = (v ∘ u)⟨n/2⟩ = Id. So (by the proof of Lemma 3.12 ), u ∘ v and v ∘ u define direct summands in Q_{2} and Q_{1}, and u, v give an isomorphism between them.

2) Since (φ + ρ_{2} ∘ φ ∘ ρ_{1})⟨n/2⟩ = (\begin{matrix}a + d & b + c \\ b + c & a + d \end{matrix}), by Lemma 3.20 we have that

(\begin{matrix}a + d − b − c & 0 \\ 0 & a + d − b − c \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Since 2^{r} · M_{2}(ℤ) ⊂ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩, we have: (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

Using the later matrix, ρ_{1,2}, and the fact that (\begin{matrix}a & b \\ c & d \end{matrix}) ≡ (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ( mod 2 ), we can assume that (\begin{matrix}1 + x & y \\ 0 & 1 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩, where x and y are even.

Then (using ρ_{1,2} and duality) (\begin{matrix}1 & y \\ 0 & 1 + x \end{matrix}) ∈ Hom(Q_{2}, Q_{1})_{J}⟨n/2⟩. Then (multiply-

ing) (\begin{matrix}1 + x & 2y(1 + x) \\ 0 & 1 + x \end{matrix}) ∈ End(Q_{2})_{J}⟨n/2⟩, and (since 2^{r} · M_{2}(ℤ) ⊂ End(Q_{2})_{0}⟨n/2⟩)

(\begin{matrix}1 & 2y \\ 0 & 1 \end{matrix}) ∈ End(Q_{2})_{J}⟨n/2⟩.

Then: (\begin{matrix}1 + x & 3y \\ 0 & 1 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩, and (\begin{matrix}0 & 2y \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩.

In the same way, since (using ρ and Lemma 3.20 ) (\begin{matrix}1 + y & x \\ 0 & 1 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩,

we can get: (\begin{matrix}0 & 2x \\ 0 & 0 \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩. Since 2^{r} · M_{2}(ℤ) ⊂ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩,

we get that 2^{k+1} · M_{2}(ℤ) ⊂ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩, where 2^{k} is the maximal power of 2, which divides both x and y.

So, everything can be reduced to the following 3 cases: 1) x = 2^{k}, y = 0; 2) x = 0, y = 2^{k}; 3) x = 2^{k}, y = 2^{k}.

First two cases are equivalent (using ρ_{1,2} and matrices: (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}), (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix})).

Consider the second case. If φ⟨n/2⟩ = (\begin{matrix}1 & 2^{k} \\ 0 & 1 \end{matrix}), then (using duality, ρ_{1,2}, and ma-

trix (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix})) we get some ψ ∈ Q_{2} → Q_{1}: ψ⟨n/2⟩ = (\begin{matrix}1 & −2^{k} \\ 0 & 1 \end{matrix}) ∈ Hom(Q_{2}, Q_{1})_{J}⟨n/2⟩.

Then (φ ∘ ψ)⟨n/2⟩ = Id, and (ψ ∘ φ)⟨n/2⟩ = Id. So, p_{N_{1}} := (ψ ∘ φ)^{2^{s}} and p_{N_{2}} := (φ ∘ ψ)^{2^{s}}, for large s, will be the idempotents in Q_{1} and Q_{2} with desired properties. φ, ψ will give an isomorphism of corresponding direct summands.

In the third case (by Lemma 3.20 ), we get (\begin{matrix}1+2^{k−1} & 2^{k−1} \\ −2^{k−1} & 1−2^{k−1} \end{matrix})∈Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩.

On the other hand (using (\begin{matrix}2 & 0 \\ 0 & 2 \end{matrix})), we have (\begin{matrix}1 & 2^{k} \\ 0 & 1 − 2^{k} \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩.

Since (\begin{matrix}0 & 0 \\ 0 & 2^{k+1} \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{0}⟨n/2⟩, we get (\begin{matrix}1 & 2^{k} \\ 0 & 1 + 2^{k} \end{matrix}) ∈ Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩.

Then, again using Lemma 3.20 , we have (\begin{matrix}1 − 2^{k−1} & 2^{k−1} \\ −2^{k−1} & 1 + 2^{k−1} \end{matrix})∈Hom(Q_{1}, Q_{2})_{J}⟨n/2⟩,

which means that there is some ψ ∈ Hom(Q_{2}, Q_{1}), s.t. ψ⟨n/2⟩=(\begin{matrix}1−2^{k−1} & −2^{k−1} \\ 2^{k−1} & 1+2^{k−1} \end{matrix})

∈ Hom(Q_{2}, Q_{1})_{J}⟨n/2⟩.

Since ψ⟨n/2⟩ = φ⟨n/2⟩^{−1}, we get that p_{N_{1}} := (ψ ∘ φ)^{2^{s}} and p_{N_{2}} := (φ ∘ ψ)^{2^{s}}, for large s, will be the idempotents in Q_{1} and Q_{2} with (p_{N_{1}})_{i} = (p_{N_{1}})′_{i} = (p_{N_{2}})_{i} = (p_{N_{2}})′_{i} = δ_{J}(i), and φ, ψ perform isomorphism of corresponding direct summands.

Lemma 3.25 .

Let Q_{1}, Q_{2} be projective quadric, s.t. dim(Q_{1}) = dim(Q_{2}). Suppose we have φ ∈ Hom(Q_{1}, Q_{2}), ψ = Hom(Q_{2}, Q_{1}), s.t. φ_{i} = ψ_{i} = 1 for some 0 ⩽ i < n/2 (see the beginning of this subsection). Then there exist direct summands N_{1} in Q_{1}, and N_{2} in Q_{2}, s.t. N_{t} contains X_{Q_{t}^{i}}(i)[2i] (in the sense of Lemma 3.23 ), and there exist an isomorphism: N_{1} = N_{2}.

Proof

By Lemma 3.20 we can assume that φ_{i} and φ′_{i} are either 0, or 1. The same about ψ. As usually, we will denote: J^{d}(φ) - the set of such 0 ⩽ l < n/2, that φ_{l} = 1; J^{u}(φ)

- the set of such 0 ⩽ j < n/2, that φ′_{j} = 1.

We, certainly, have i ∈ J^{d}(φ) ∩ J^{d}(ψ)

We have two cases: 1) i ∈ J^{d}(φ) ∩ J^{d}(ψ) ∩ J^{u}(φ) ∩ J^{u}(ψ), 2) i ∉ J^{d}(φ) ∩ J^{d}(ψ) ∩ J^{u}(φ) ∩ J^{u}(ψ)

1) Take ε := φ ∘ ψ ∘ φ ∘ φ^{∨} ∘ ψ^{∨}, then i ∈ J^{d}(ε) = J^{u}(ε), and so, ε satisfies the conditions of Lemma 3.24 .

2) Take ε̃:= φ ∘ ψ ∘ φ + ψ^{∨} ∘ φ^{∨} ∘ ψ^{∨}. By Lemma 3.20 we can correct ε̃ into ε, to make ε_{l} and ε′_{j} be 0 or 1 (instead of 0, 1 and 2). Then i ∈ J^{d}(ε) = J^{u}(ε), and again, ε satisfies the conditions of Lemma 3.24 .

We will also need some generalization of the above result.

Lemma 3.26 .

Let Q_{1}, Q_{2} be two projective quadrics. Suppose we have two morphisms: φ : Q_{1} → Q_{2}(l)[2l], and ψ : Q_{2}(l)[2l] → Q_{1}, s.t. (ψ ∘ φ)_{i} = 1 for some i ≠ n_{1,2}/2, where n_{1} = dim(Q_{1}), n_{2} = dim(Q_{2}). Then there are direct summands N_{1} in Q_{1}, N_{2} in Q_{2}, s.t. (p_{N_{1}})_{i} = 1, and N_{1} is isomorphic to N_{2}(l)[2l].

Proof

Using Theorem 2.4.23 , we can change q_{1,2} by d_{1,2} ·ℍ ⊥ q_{1,2}, and assume that l = 0. We have two cases: 1) dim(Q_{1}) = dim(Q_{2}); 2) dim(Q_{1}) ≠ dim(Q_{2}).

1) In this case everything follows from Lemma 3.25 .

2) Let us choose, as usually, some algebraic closure k ⊂\overline{k}, and bases in CH^{n_{t}−p}(Q_{t}|\overline{k}) (consisting of plane section h^{n_{t}−p} of dimension p, if n_{t} ⩾ p > n_{t}/2; of projective subspace l_{p} of dimension p, if 0 ⩽ p < n_{t}/2; and of two projective subspaces l′_{n_{t}/2}, l′′_{n_{t}/2} of dimension n_{t}/2 from different families, if p = n_{t}/2).

Let f : Q_{1} → Q_{2} will be some morphism. Denote as f_{p} the matrix of f, corresponding to the bases above in CH^{n_{1}−p}(Q_{1}|\overline{k}) and CH^{n_{2}−p}(Q_{2}|\overline{k}). The matrix f_{p} will be an integer, if 0 ⩽ p ⩽ min(n_{1}, n_{2}), and p ≠ n_{1,2}; and it will be the matrix 2 by 1 (resp. 1 by 2), if p = n_{1}/2 (resp. n_{2}/2). Since n_{1} ≠ n_{2} we will not have other cases.

Let n_{1} > n_{2}.

In the same way, as in Lemma 3.20 , we have elements u_{l} in Hom(Q_{1}, Q_{2}), for all 0 ⩽ l ⩽ n_{2}, l ≠ n_{1,2} s.t. u_{l} acts trivially on all CH^{n_{1}−p}(Q_{1}|\overline{k}) with p ≠ l, and the matrix of the action of u_{i} on CH^{n_{1}−l}(Q_{1}|\overline{k}) is 4 if n_{2}/2 < l < n_{1}/2, and 2 in all other cases. Also, we have elements w_{n_{1}/2}, w_{n_{2}/2} (if n_{1} and n_{2} are even), s.t. w_{n_{t}/2} acts trivially on CH^{n_{1}−l}(Q_{1}|\overline{k}) for 0 ⩽ l ⩽ n_{2}, l ≠ n_{t}, and the matrices of the action of

w_{n_{t}/2} on CH^{n_{1}−n_{t}/2}(Q_{1}|\overline{k}) are (2 2) (if n_{2} ⩾ n_{1}/2, and 0 if not), for t = 1, and (\begin{matrix}1 \\ 1 \end{matrix}),

for t = 2. Really, u_{l} and w_{l} are represented by the cycles h^{l} × h^{n_{2}−l} ⊂ Q_{1} × Q_{2}.

In the same way, we have elements v_{l} in Hom(Q_{2}, Q_{1}), for all 0 ⩽ l ⩽ n_{2}, l ≠ n_{1,2} s.t. v_{l} acts trivially on all CH^{n_{2}−p}(Q_{2}|\overline{k}) with p ≠ l, and the matrix of the action of v_{l} on CH^{n_{2}−l}(Q_{2}|\overline{k}) is 1 if n_{2}/2 < l < n_{1}/2, and 2 in all other cases; and we have y_{n_{1}/2}, y_{n_{2}/2}, s.t. y_{n_{t}/2} acts trivially on CH^{n_{2}−l}(Q_{2}|\overline{k}) for 0 ⩽ l ⩽ n_{2}, l ≠ n_{t}, and the

matrices of the action of y_{n_{t}/2} on CH^{n_{2}−n_{t}/2}(Q_{2}|\overline{k}) are (\begin{matrix}1 \\ 1 \end{matrix}) (if n_{2} ⩾ n_{1}/2, and 0 if

not), for t = 1, and (1 1), for t = 2.

Using Lemma 3.13 we can find π ∈ End(Q_{1}), s.t. π_{j} = 1, for all j ≠ n_{1,2}, s.t. (φ ∘ ψ)_{j} is odd, and π_{j} = 0, for all j ≠ n_{1,2}, s.t. (φ ∘ ψ)_{j} is even. Changing φ by φ ∘ π, we can assume, that φ_{j} is either odd, or 0, for all j ≠ n_{1,2}.

Using the elements u_{l} and v_{l} above, we can correct φ and ψ, to make φ_{j} be ±1, or 0, and ψ_{j} be 1, or 0 (depending on the evenness of ψ_{j}) for all 0 ⩽ j ⩽ n_{2}, j ≠ n_{1,2}. Changing φ by φ ∘ ψ ∘ φ, and ψ by ψ ∘ φ ∘ ψ ∘ φ ∘ ψ, we can assume that φ_{j} = ψ_{j} and = 0, or 1, for all 0 ⩽ j ⩽ n_{2}, j ≠ n_{1,2}.

Let φ_{n_{2}/2} = (\begin{matrix}α \\ β \end{matrix}), and ψ_{n_{2}/2} = (γ δ). Using w_{n_{2}/2} and y_{n_{2}/2}, we can correct φ

and ψ, to make α = 0, and δ = 0. Then (ψ ∘ φ)_{n_{2}/2} = 0, and changing φ by φ ∘ ψ ∘ φ and ψ by ψ ∘ φ ∘ ψ, we can assume that φ_{n_{2}/2} = 0, and ψ_{n_{2}/2} = 0.

Let φ_{n_{1}/2} = (a b), and ψ_{n_{1}/2} = (\begin{matrix}c \\ d \end{matrix}).

Using y_{n_{1}/2}, we can assume that d = 0, and because of w_{n_{1}/2}, we can assume that a is 0, or 1.

If a = 0, or c is even, using (φ ∘ ψ) and Lemma 3.13 , we can find η ∈ End(Q_{2}), s.t. η_{j} = (φ ∘ ψ)_{j} = 1, or 0, for all 0 ⩽ j ⩽ n_{2}, j ≠ n_{1}/2, and η_{n_{1}/2} = 0. Then changing φ by η ∘ φ, and ψ by ψ ∘ η, we can assume that φ_{n_{1}/2} = 0, and ψ_{n_{1}/2} = 0.

If a = 1, and c is odd, we have that (ψ ∘ φ ∘ ψ − ψ)_{n_{1}/2} = (\begin{matrix}c(c − 1) \\ 0 \end{matrix}), and

(ψ ∘φ∘ψ −ψ)_{l} = 0, for all l ≠ n_{1}/2. Since c is odd (and d = 0), we have: det(Q_{1}) = 0

(since if det(Q_{1}) ≠ 1, then (\begin{matrix}c \\ d \end{matrix}) should be stable under Gal(k\sqrt{det(Q)}/k), which

acts via: (\begin{matrix}c \\ d \end{matrix}) ↦ (\begin{matrix}d \\ c \end{matrix})). Then by the same arguments as in Lemma 3.20 , we

have μ ∈ Hom(Q_{2}, Q_{1}), s.t. μ_{n_{1}/2} = (\begin{matrix}(c − 1) · 2^{r} \\ 0 \end{matrix}), for large r, and μ_{l} = 0 for all

l ≠ n_{1}/2. Then we have ν ∈ Hom(Q_{2}, Q_{1}), s.t. ν_{n_{1}/2} = (\begin{matrix}c − 1 \\ 0 \end{matrix}), and ν_{l} = 0 for all

l ≠ n_{1}/2. Then we can correct ψ, to make ψ_{n_{1}/2} = (\begin{matrix}1 \\ 0 \end{matrix}). Since a = 1, we will have:

(φ ∘ ψ)_{n_{1}/2} = 1, and (ψ ∘ φ)_{n_{1}/2} = (\begin{matrix}1 & b \\ 0 & 0 \end{matrix}).

So, we get (φ ∘ ψ)_{l} and (ψ ∘ φ)_{l} are idempotents for all l. So, by Lemma 3.12 we have direct summands N_{1,2} in Q_{1} and Q_{2}, which will be isomorphic via φ and ψ. And (p_{N_{1}})_{i} = (ψ ∘ φ)_{i} = 1 (since we did not change the evenness of this number).

Lemma is proven.

4. Higher forms - generalization of Rost motives

It is well known that if quadratic form r is divisible by the Pfister form ⟨⟨a_{1}, …, a_{n}⟩⟩, then the Witt numbers of r are all divisible by 2^{n} (it follows from the fact that if q is divisible by ⟨⟨a_{1}, …, a_{n}⟩⟩, then (q|_{k(Q)})|_{anis.} is also divisible by ⟨⟨a_{1}, …, a_{n}⟩⟩ (by the theorem of Pfister, q is divisible by ⟨⟨a_{1}, …, a_{n}⟩⟩ if and only if q|_{k(Q_{{a_{1},...,an}})} is hyperbolic (follows from [9], IX, Corollary 2.9, VII, Theorem 3.2, since the Pfister form is hyperbolic over it’s generic point (see Theorem 2.4.5 ))). So, X_{Q} =…=X_{Q^{2^{n}−1}}, X_{Q^{2^{n}}} =…=X_{Q^{2·2^{n}−1}}, etc ... (see Theorem 2.4.20 ).

In this section we will show that this fact reflects the motivic structure of such a quadrics. Namely, the motive of a quadric R with quadratic form r = ⟨⟨a_{1}, …, a_{n}⟩⟩ · q can be decomposed (in DM_{gm}(k) as well as in Chow(k) (see Theorem 2.2.4 )) into a direct sum of 2^{n} motives if dim(q)-even, and 2^{n} + 1 motives if dim(q) is odd, s.t. for each 0 ⩽ d ⩽ [dim(q)/2] − 1 all X’s from the set X_{Q^{d·2^{n}}}, …, X_{Q^{(d+1)·2^{n}−1}} belong to 2^{n} different isomorphic (up to shift) summands (for dim(q)-odd “in the middle” we will have also the motive of a Pfister quadric Q_{{a_{1},...,a_{n}}}).

More precisely, the following is true:

Theorem 4.1 .

Let quadric R corresponds to the quadratic form r = q × ⟨⟨a_{1}, …, a_{n}⟩⟩ for some form q. Then the motive of R can be decomposed into a direct sum:

R = ⊕_{i=1,...,2^{n}}F_{α}(Q)(i)[2i] in the case dim(q) is even, and
R = (⊕_{i=1,...,2^{n}}F_{α}(Q)(i)[2i]) ⊕ Q_{α}(dim(R)/2 − 2^{n−1} + 1)[dim(R) − 2^{n} + 2],

if dim(q) is odd (here Q_{α} is the Pfister quadric, corresponding to the symbol α = {a_{1}, …, a_{n}} ∈ K_{n}^{M}(k)/2), for some pure motive (see Theorem 2.2.4 ) F_{α}(Q). Moreover, F_{α}(Q) is an extension of: X_{R^{i}}(i)[2i], X_{R^{l}}(dim(R) − l)[2dim(R) − 2l], where l = i + 2^{n} − 1, and i runs through all numbers of the form k · 2^{n}, 0 ⩽ k ⩽ [dim(Q)/2].

So, we get some operations F_{α} which act on quadrics, producing higher forms of them. Notice, that higher form F_{α}(Q) consists of the same number of “elementary” pieces (X_{Q^{i}}(∗)[2∗]’s) as Q itself (but they are 2^{n}-times as apart as they were in Q (I mean difference between round (-), or square [-] numbers)).

This generalizes the Rost motives, since M_{\{a_{1},...,a_{n}\}} = F_{\{a_{2},...,a_{n}\}} (k (\sqrt{a_{1}})).

We expect that F_{α}’s are, actually, functors on the pairs Q → ℙ^{dim(Q)}, and, in particular, they preserve direct sum decomposition, and so, act on all pure motives coming from quadrics.

We will begin with the case of 1-symbol {a} ∈ K_{1}^{M}(k)/2.

Proposition 4.2 .

Suppose R ⊂ ℙ^{m+1} - is a quadric of dimension m. Then the following conditions are equivalent:

(1) There exists some quadratic extension k(\sqrt{a}), s.t. R is hyperbolic over it.

(2) r = q × ⟨1, −a⟩ for some form q.

(3) There exists φ ∈ PGL_{m+1} - automorphism of order 2, which acts on ℙ^{m+1} without rational fixed points, and an operator r′(x, y) := r(x, φ(y)) is skew-symmetric.

Any of the above conditions implies the following one :

(4) The motive of R can be decomposed into a direct sum: R = R_{1}⊕R_{1}(1)[2] in the case m ≡ 2 (mod 4), and R = R_{1} ⊕ k (\sqrt{det(R)}) (dim(R)/2)[dim(R)] ⊕ R_{1}(1)[2], if m ≡ 0 (mod 4), where R_{1} is an extension of: X_{R^{i}}(i)[2i], X_{R^{l}}(dim(R)−l)[2dim(R)−2l], where i ( l ) run through all even ( respectively, odd ) numbers ⩽ 2[dim(R)/4].

Proof of Proposition 4.2

(1 ⇔ 2) This is classical (see, for example [9], VII, Theorem 3.2).

(2 ⇒ 3) If r = q × ⟨1, −a⟩ for some quadratic form q, take φ(v_{1}, v_{2}) = (a · v_{2}, v_{1}). (3 ⇒ 2) Since φ^{2} = a·Id, we have that m+2 is even and φ has eigenvalues: \sqrt{a} and

−\sqrt{a} - (m + 2)/2 of each. Let A_{1} and A_{2} are corresponding eigenspaces (defined over k(\sqrt{a})). Since r′ is skew-symmetric, we can always find an r′-isotropic subspace V_{1} of dimension equal to (m + 1)/2, such that V_{1} ∩ A_{j} = 0. Then V_{1} ∩ φ(V_{1}) = 0. And we have V = V_{1} ⊕φ(V_{1}). Evidently, r(V_{1}, φ(V_{1})) = 0, and r(φ(v_{1}), φ(v′_{1})) = −a·r(v_{1}, v′_{1}). Now it is enough to take q = r|_{V_{1}}.

(3 ⇒ 4)

Consider the following cycle P ⊂ R × R: (x, y) ∈ P iff x and y are contained in some line, stable under φ. dim(P) = dim(R) + 1, so it defines the morphism p : R → R(−1)[−2] (by Theorem 2.1.23 , Theorem 2.1.17 ).

Since the condition on x and y above is symmetric, we have p^{∨}(−1)[−2] = p, where p^{∨} ∈ Hom(R(1)[2], R) is the element dual to p via duality: R = \underline{Hom}(R, ℤ(dim(R))[2 dim(R)]).

Let H be a cycle in R×R, represented by a hyperplane in R, embedded diagonally, and h : R → R(1)[2] - the corresponding morphism (see Theorem 2.1.17 ).

Consider the following endomorphism of R: π = p(1)[2] ∘ h. Let us compute numbers: π_{i}, π′_{i} (see Theorem 3.7 ). All the computations can be, evidently, performed over \overline{k}. (to compute these numbers we have to find the intersection of our cycle with the cycles of the type : h^{i} × l_{i}, l^{j} × h^{j}, and l′_{dim(R)/2} × l′′_{dim(R)/2} over \overline{k}, see the proof of Theorem 2.4.22 and the discussion before Lemma 3.10 ).

Notice, that p = p^{∨}(−1)[−2], and h^{∨} = h(−1)[−2] (if morphism ψ : R → R(m)[2m] is represented by a cycle Ψ, then the dual morphism ψ^{∨} : R(−m)[−2m] → R will be represented by the cycle Ψ^{∨} obtained from Ψ by the reflection in the diagonal).

Then we have: π^{∨} = h(−1)[−2] ∘ p, and, hence, π_{i} = (π^{∨})′_{i} = π′_{i+1}, for i <

dim(R)/2−1, and for i = dim(R)/2−1: 2·π_{dim(R)/2−1} = s(π⟨n/2⟩) (here s ((\begin{matrix}a & b \\ c & d \end{matrix}))=

a + b + c + d) (shift by 1 in numbers, since the h is situated from the different sides of p in π and π^{∨}).

So, it is enough to compute π′_{i} (and π⟨dim(R)/2⟩).

Consider the locus of fixed points of φ: A_{1} ∪ A_{2}, where A_{1} and A_{2} are projective spaces in ℙ^{m+1} of dimension dim(R)/2 (I will denote them in the same way as corresponding linear spaces). Evidently, A_{j} ∩R is A_{j}. Since A_{1} ∩A_{2} = ∅, we can find space T of dimension dim(R)/2 on Q if dim(R) ≡ 2 (mod 4), and of dimension dim(R)/2−1 if dim(R) ≡ 0 (mod 4) on R, such that T ∩ A_{j} = ∅ and (consequently) φ(T) ∩ A_{j} = ∅. Then T ∩ φ(T) = ∅, i.e. corresponding affine spaces are linearly independent. We have a complete flag of subspaces T_{0} ⊂ T_{1} ⊂ ⋯ ⊂ T in T.

Since T_{i−1} does not intersect A_{j}, π′_{i} is equal to the degree of the variety S_{i−1}, consisting of all lines l(t, φ(t)), t ∈ T_{i−1} (this degree is the same as the intersection number of the cycle representing π with the cycle h^{i} × l_{i}).

Consider arbitrary point t_{i−1} ∈ T_{i−1}\T_{i−2}. Then projection of S_{i−1} from the point t_{i−1} will be equal to the cone over S_{i−2}, from which it follows, that the degree of S_{i−1} minus the degree of S_{i−2} is equal to 1. Since, evidently, π′_{0} = 0, π′_{1} = 1, we have that π′_{i} = i, for all i < dim(R)/2, and in the basis of CH^{dim(R)/2}(R|\overline{k}) = ℤ⊕ℤ, consisting of

planes, π⟨dim(R)/2⟩ looks as (\begin{matrix}a & a \\ c & c \end{matrix}), where a+c = dim(R)/2; moreover, in the case

dim(R) ≡ 2 (mod 4), the same considerations give us that for some half-dimensional plane T in R (the one which does not intersect A_{j}), the cycle p(T) (i.e: the union of all lines l(t, φ(t)) for t ∈ T) has the degree dim(R)/2 + 1.

Using Lemma 3.13 and our π, we can find ψ ∈ End(R), s.t.: ψ′_{i} = (1 + (−1)^{i+1})/2,

ψ_{i} = (1 + (−1)^{i})/2, for all i < dim(R)/2, and ψ⟨dim(R)/2⟩ looks as: (\begin{matrix}b & b \\ 1 − b & 1 − b \end{matrix}),

if dim(R)≡2 (mod 4), for some b, and as: (\begin{matrix}b & b \\ −b & −b \end{matrix}) if dim(R)≡0 (mod 4) (b here is

equal to (a − c + 1)/2). Since in the last case image(\overline{⟨dim(R)/2⟩}) (see the definition of \overline{⟨dim(R)/2⟩} before Lemma 3.20 ) is contained in the subring of M_{2}(ℤ), generated

by id and (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) (det(R) ≠ 1, and we can go first to the field where anisotropic

part of q is equal to λ · ⟨1, det(R)⟩, before passing to the algebraic closure, but for 0-dimensional anisotropic quadric P/E, End(P) is ℤ ⊕ ℤ, additively generated by the matrices specified above (in End(P |\overline{E}))), the corresponding matrix must be 0.

Now, using Lemma 3.13 and Lemma 3.12 , we can correct ψ to make it an idempotent (ψ⟨n/2⟩ is already an idempotent, and we need only to make ψ_{i}, ψ′_{i} be equal 0, or 1, then use Lemma 3.12 ). We get a decomposition: R = R_{1} ⊕ R_{2}(1)[2], or R = R_{1} ⊕ k(\sqrt{a})(dim(R)/2)[dim(R)] ⊕ R_{2}(1)[2] (in the second case we have: ψ⟨n/2⟩ = 0 (see above), and 1 − ψ − ψ^{∨} will give us the idempotent for the middle summand (in both cases: ψ_{i} + ψ_{i}^{∨} = 1 for all 0 ⩽ i < n/2)). Let us prove, that R_{1} = R_{2}. Let T′, T -basis of CH^{dim(R)/2}(R|\overline{k}) consisting of planes, and T_{dim(R)/2−1} and h^{dim(R)/2−1} are bases of CH^{dim(R)/2−1}(R|\overline{k}) and CH^{dim(R)/2+1}(R|\overline{k}), consisting of a plane and a subquadric, correspondingly. Let us find the matrices of p : CH^{dim(R)/2}(R|\overline{k}) → CH^{dim(R)/2+1}(R|\overline{k}), and p : CH^{dim(R)/2−1}(R|\overline{k}) → CH^{dim(R)/2}(R|\overline{k}) in those bases.

Since cycle p is symmetric in R × R, both matrices will be the same. We know

that one of them will be (\begin{matrix}a \\ c \end{matrix}), where a + c = dim(R)/2. But in the case dim(R) ≡

2 (mod 4), we also have 2 · c = dim(R)/2 + 1 (since for some plane T we have degree p(T) = dim(R)/2 + 1, see above), i.e. c − a = 1, and b = 0. So, b = 0 always.

We will consider the case: dim(R) ≡ 2 (mod 4). Consider the following map α : R → R(−1)[−2]. α = p − ∑_{i=0,...,dim(R)/2−1}(v(i) + v′(i + 1)) · ([i + 1/2])), where v(i), v′(i) are defined by the cycles h^{dim(R)−i} × h^{i+1} and h^{i} × h^{dim(R)−i+1}, respectively.

We can construct two maps: β : R_{1} → R_{1} and γ : R_{2}(1)[2] → R_{2}(1)[2] in the following way. β = ψ ∘α(1)[2]∘(1−ψ)(1)[2]∘h∘ψ, γ = (1−ψ)∘h(−1)[−2]∘ψ(−1)[−2]∘α ∘(1−ψ). (β ⊕ γ) defines a map R = R_{1} ⊕ R_{2}(1)[2] → R_{1} ⊕ R_{2}(1)[2], s.t. (β ⊕ γ)_{i} = (β ⊕ γ)′_{i} = 1 for all i, and it is not difficult to check that (β ⊕ γ)⟨dim(R)/2⟩ = id, and, hence, it is an isomorphism (since 1 − (β ⊕ γ) is a nilpotent by Lemma 3.10 ). Then (1 − ψ)(1)[2] ∘ h ∘ ψ : R_{1} → R_{2} is also an isomorphism.

Proposition is proven.

Higher forms have some kind of relatives. Namely, we have the following:

Lemma 4.3 .

Let r′ = q′ · ⟨1, −a⟩ + ⟨x⟩, and q = q′ + ⟨x⟩. Then we have a decomposition: R′ = R′_{1} ⊕ R′_{2}, where R′_{1} = F_{a}(Q) and R′_{2} = F_{a}(Q′)(1)[2]. Here R′_{1} is an extension of X_{R′}^{i}(i)[2i], X_{R′}^{i}(dim(R′) − i)[2 dim(R′) − 2i] for all even i, and R′_{2} - for all odd i.

Proof

We will consider the case dim(r′) ≡ 3 (mod 4), other one is similar. Let r = q · ⟨1, −a⟩ (as usually). Then the idempotent ψ : R → R, which gives the decomposition R = R_{1} ⊕ R_{1}(1)[2], is actually a composition R →^{h} R′(1)[2] → R(1)[2] →^{α} R, where α was defined in the end of the proof of Proposition 4.2 , and the middle map is induced by the embedding of the subquadric R′ into R. (since the map R →^{h} R(1)[2] has such a decomposition, because the map h is given by a cycle Δ_{R′} ⊂ R × R, which lives in R × R′ (we can take arbitrary hyperplane section here)).

This gives us a map g : R′ → R′ (equal to a composition R′ → R →^{α(−1)[−2]}

R(−1)[−2] →^{h(−1)[−2]} R′), and it is easy to see that g_{i} = g′_{i} = 1 if i-even, and 0otherwise (g_{i}, g′_{i} here are numbers associated with the morphism g (see Theorem 3.7 ), they are equal to the intersection numbers of the cycle representing our morphism (see Theorem 2.1.23 (4) and Theorem 2.1.17 ) with the cycles h^{i} × l_{i} and l_{i} × h^{i} (see the proof of Theorem 2.4.22 )) (since (h ∘ α)_{i} = (h ∘ α)′_{i+1} = 1 for odd i, and = 0 for even i).

So, we get a decomposition: R′ = R′_{1} ⊕ R′_{2}.

From the above it is clear that R′_{1} = R_{1} = F_{a}(Q) (via the maps t : R′_{1} → R′ →

R → F_{a}(Q), and s : F_{a}(Q) → R →^{α(−1)[−2]} R(−1)[−2] →^{h(−1)[−2]} R′ → R′_{1}). Really,

the two composition maps u := s ∘ t : R′_{1} → R′_{1}, and v := t ∘ s : F_{a}(Q) → F_{a}(Q) are automorphisms of R′_{1} and F_{a}(Q) respectively (consider the morphisms u ⊕ id_{R′_{2}} : R′ → R′, and v ⊕ id_{F_{a}(Q)(1)[2]} : R → R, they are isomorphisms by Lemma 3.10 (they are equal to the identity maps modulo nilpotents)).

Consider r′′ = q′ · ⟨1, −a⟩. Then the composition R′ →^{h} R′′(1)[2] →^{α′′(1)[2]} R′′ → R′, where α′′ is morphism constructed in the same way as α but for R′′, is the same as 1 − g modulo nilpotents (we need just to compute numbers). Hence, in the same way as above: R′_{2} = F_{a}(Q′)(1)[2].

Results above permit us to clarify the picture in the Proposition 3.4 :

Corollary 4.4 .

Suppose n-dimensional quadric R is such that (in the notations of Proposition 3.4) P = ⊕_{i=0,...,[dim(R)−1/2]}X_{R^{i}}(i)[2i]. Then:

1) If r ∈ I^{2}(W (k)), then R = R_{1} ⊕ R_{1}(1)[2], where R_{1} is an extension of the same X’s as in Proposition 4.2 .

2) If r ∉ I^{1}(W (k)), then R has a decomposition as in Lemma 4.3 .

Proof

We remind that P′ := Cone((⊕_{i=0,...,[n−1/2]}X_{R^{i}}(n−i)[2n−2i]) → R), where the maps X_{R^{i}}(n−i)[2n−2i] → R are the compositions: X_{R^{i}}(n−i)[2n−2i] → ℤ(n−i)[2n−2i] → R, where the first map is the natural projection X_{R^{i}} → ℤ shifted by (n−i)[2n−2i], and the second map correspond to the plane section of codimension i via identification: Hom(ℤ(n − i)[2n − 2i], R) = CH^{i}(R) (see Theorem 2.1.17 , Theorem 2.1.23 (4)), and: P = P′ for n-odd, and P = Cone(X_{R^{n/2}} × k\sqrt{det(R)}(n/2)[n] →^{η} P′), for some morphism η when n is even.

1) Consider hyperplane section of dimension dim(R)/2 − 1. It defines some morphism ρ : ℤ(dim(R)/2 − 1)[dim(R) − 2] → R → P.

Since P = ⊕_{i=0,...,[dim(R)−1/2]}X_{R^{i}}(i)[2i], we get some morphism ε : ℤ(dim(R)/2 − 1)[dim(R) − 2] → X_{R^{dim(R)/2−1}}(dim(R)/2 − 1)[dim(R) − 2]. This morphism, composed with the standard map X_{R^{dim(R)/2−1}}(dim(R)/2 − 1)[dim(R) − 2] → ℤ(dim(R)/2 − 1)[dim(R) − 2], gives 2 : ℤ → ℤ (shifted).

Really, this can be checked over algebraic closure, where X_{R^{i}}|\overline{k}(i)[2i]) = ℤ(i)[2i], and we don’t have any maps ℤ(dim(R)/2 − 1)[dim(R) − 2] → ℤ(i)[2i] for i < dim(R)/2 − 1 by Theorem 2.1.18 , so ε|\overline{k} is the only nontrivial component of ρ. But the composition R|\overline{k} → P |\overline{k} → X_{R^{dim(R)/2−1}}|\overline{k}(dim(R)/2 − 1)[dim(R) − 2] → ℤ|\overline{k}(dim(R)/2 − 1)[dim(R) − 2] is just the map, corresponding to the plane section of codimension dim(R)/2 − 1 (via the identification: Hom(R, ℤ(i)[2i]) = CH^{i}(R), see Theorem 2.1.17 )(see the definition of P above, and the proof of Theorem 2.4.22 ). So, the composition: ℤ|\overline{k}(dim(R)/2−1)[dim(R)−2] → R|\overline{k} → P |\overline{k} → X_{R^{dim(R)/2−1}}|\overline{k}(dim(R)/2−1)[dim(R)− 2] = ℤ|\overline{k}(dim(R)/2 − 1)[dim(R) − 2] is the multiplication by the intersection number of subquadrics of dimensions dim(R)/2 − 1 and dim(R)/2 + 1, which is 2.

That means that R^{dim(R)/2−1} contains a point E of degree 2 · d with d-odd (by Theorem 2.3.3 (2)); i.e. over E R becomes hyperbolic (since it becomes dim(R)/2−1times isotropic, and since det(R) = 1).

We have tower of fields: k ⊂ F ⊂ E, where [F : k] = d, and [E : F] = 2. So, over F we have a decomposition (by Proposition 4.2 ): R|_{F} = R̃_{1} ⊕ R̃_{1}(1)[2] (since det(R) = 1, we must have dim(R) ≡ 2 (mod 4) (since our form over F appears to be divisible by binary one (see equivalence (1 ⇔ 2) in Proposition 4.2 ))).

I.e.: we have ψ ∈ End(R|_{F}) that ψ_{i} = ψ′_{i−1} = 1, for odd i, and 0 for even i,

and ψ⟨dim(R)/2⟩ = (\begin{matrix}1 & 1 \\ 0 & 0 \end{matrix}) (see the proof of Proposition 4.2 (notice, b = 0 there

(notations from there))).

But, now, the existence of transfers from F to k gives us that in the End(R) there is element φ, s.t. φ_{i} = φ′_{i−1} = d, for odd i, and = 0 for even i; and φ⟨dim(R)/2⟩:

φ⟨dim(R)/2⟩ = (\begin{matrix}d & d \\ 0 & 0 \end{matrix}). Using Lemma 3.13 and Lemma 3.12 we can correct φ to

make it an idempotent. We get: R = R_{1} ⊕ R_{1}(1)[2].

2) The same considerations give us that over some E of degree 2 · d with d-odd R becomes hyperbolic. Again, using transfers, we get that R = R_{1} ⊕ R_{2}.

Remark In particular, in the case 1) above all Witt numbers are divisible by 2.

Consider now the case of a pure symbol {a_{1}, …, a_{n}} (see Definition 2.4.4 ) of higher dimension.

Proof of Theorem 4.1

Let us first prove the following:

Lemma 4.5 .

Suppose for some quadric R we have undecomposable direct summand N(i_{0})[2i_{0}] which is contained in R′⟨i_{0}⟩ (i.e.: for the corresponding idempotent g we have g_{j} = g′_{j} = 0 for all j < i_{0} (see Theorem 3.7 for the definition of g_{j}, g′_{j})), “the smallest elementary piece” of N(i_{0})[2i_{0}] is X_{R^{i_{0}}}(i_{0})[2i_{0}] (i.e.: g_{j} = 0 for j < i_{0}), and X_{R^{i_{0}+t}} = X_{R^{i_{0}}} (see in this connection: Definition 2.4.19 , Theorem 2.4.20 , and discussion after it).

Then in R we have also direct summands isomorphic to N(i_{0} + t)[2i_{0} + 2t].

Proof

Consider the projection: π : \underline{R^{i_{0}}} → \underline{R^{i_{0}−1}}. It’s fibers are quadrics R(i_{0}, x) of i_{0}- dimensional planes on R, containing fixed i_{0} − 1-dimensional plane on R (defined by the point x from \underline{R^{i_{0}−1}}). Let F be the preimage of Δ\underline{R^{i_{0}−1}} ⊂ \underline{R^{i_{0}−1}} × \underline{R^{i_{0}−1}} of the diagonal, under the projection: π × π : \underline{R^{i_{0}}} × \underline{R^{i_{0}}} → \underline{R^{i_{0}−1}} × \underline{R^{i_{0}−1}}. F = \underline{R^{i_{0}}} ×\underline{R^{i_{0}−1}} \underline{R^{i_{0}}} has natural projection ψ : F → \underline{R^{i_{0}−1}} with fibers R(i_{0}, x) × R(i_{0}, x).

Since X_{R^{i_{0}+t}} = X_{R^{i_{0}}}, we have that each R(i_{0}, x) has a rational t-dimensional plane as soon as it has a rational point (see remark before Claim 3.3 , and also: Definition 2.4.19 , Theorem 2.4.20 , and discussion after it), hence, there is a rational map R(i_{0}, x) → R(i_{0}, x)^{t}, so we have cycle P_{t} of dimension dim(\underline{R^{i_{0}}}) + t on F, and, hence on \underline{R^{i_{0}}} × \underline{R^{i_{0}}}, which over generic point of \underline{R^{i_{0}−1}} corresponds to the rational map φ : R(i_{0}, k(\underline{R^{i_{0}−1}})) → R(i_{0}, k(\underline{R^{i_{0}−1}}))^{t} (this cycle is a closure of the cycle over k(\underline{R^{i_{0}−1}}), consisting of pairs (x, y) ∈ R(i_{0}, k(\underline{R^{i_{0}−1}})) × R(i_{0}, k(\underline{R^{i_{0}−1}})), s.t. y ∈ φ(x), where φ(x) is a t-dimensional plane on R(i_{0}, k(\underline{R^{i_{0}−1}})), obtained via map φ).

This gives us the map p_{t} : \underline{R^{i_{0}}} → \underline{R^{i_{0}}}(−t)[−2t] (by Theorem 2.1.26 (4), Theorem 2.1.17 ).

By the Claim 3.2 we know that \underline{R^{i_{0}}} = \underline{R^{i_{0}−1}} × R⟨i_{0}⟩, and \underline{R^{i_{0}−1}} × R⟨i_{0}⟩ is a direct summand in \underline{R^{i_{0}−1}} × R(−i_{0})[−2i_{0}], in particular, we have map φ : \underline{R^{i_{0}−1}} × R⟨i_{0}⟩ → \underline{R^{i_{0}−1}} × R(−i_{0})[−2i_{0}], given by the cycle: Φ ⊂ \underline{R^{i_{0}}} × \underline{R^{i_{0}−1}} × R, consisting of triples: (F_{i_{0}}^{.}, F_{i_{0}}^{⩽i_{0}−1}, y), where F_{i_{0}}^{.} is a point of \underline{R^{i_{0}}}, and y ∈ F_{i_{0}}^{i_{0}}. Over \overline{k}, φ maps ℤ × ℤ(t)[2t] to ℤ × ℤ(t)[2t] identically.

Let ψ : \underline{R^{i_{0}−1}} → ℤ denote the natural projection; h^{t} : R → R(t)[2t] denotes the morphism, corresponding (by Theorem 2.1.17 and Theorem 2.1.23 ) to the plane section of codimension t embedded diagonally into R × R; and π : R(−i_{0})[−2i_{0}] → N comes from N(i_{0})[2i_{0}] being a direct summand in R.

Let d := dim(\underline{R^{i_{0}−1}}), and, as usually, n := dim(R).

Sublemma 4.6.

Let N be undecomposable direct summand of R, contained in R′⟨i_{0}⟩. Then the natural maps: \underline{R^{i_{0}−1}} × N → N and N(d)[2d] → \underline{R^{i_{0}−1}} × N have splittings: ρ and ν.

Really, first of all, since N(i_{0} + t)[2i_{0} + 2t] is a direct summand in R, it will be also a direct summand in R⟨1⟩(1)[2], ..., R⟨i_{0}⟩(i_{0})[2i_{0}] (by Theorem 3.7 the corresponding idempotent of End(R) will give us idempotents in End(R⟨1⟩), …, End(R⟨i_{0}⟩)).

Consider R⟨j⟩×N for 0 ⩽ j ⩽ i_{0}. It is an extension on X_{R^{j}} ×N, R⟨j +1⟩(1)[2]×N and X_{R^{j}}(n − 2j)[2n − 4j] × N (in the sense specified in Theorem 3.1 ).

But N itself is an extension of X_{R^{m}}(∗)[2∗]’s with m ⩾ i_{0}. Hence, X_{R^{j}} × N = N (by Theorem 2.4.16 ).

Motive N is pure (as a direct summand of R). So, Hom(N, X_{P} (k)[2k + 1]) = 0 for any smooth projective P.

Really, it is enough to prove that Hom(R, X_{P} (k)[2k + 1]) = 0. We can consider exact triangle SK^{r}(X_{P}) → X_{P} → Z_{r}, where SK^{r}(X_{P}) is the motive of the r-th skeleton of X_{P}. For large r, Z_{r} as a complex of sheaves with transfers in Nisnevich topology (see Definition 2.1.8 , and Definition 2.3.1 (and text before it)), does not have cohomology sheaves with numbers > −T, for any given T. From [17], Proposition 3.1.8 it follows that for large r Hom(R, Z_{r}(k)[2k + 1]) = 0. So, we can change X_{P} by it’s finite piece SK^{r}(X_{P}), which is an extension (see Definition 2.5.6 ) of: P, P × P [1], …, P × ⋯ × P [r − 1]. But Hom(R, P × ⋯ × P [s](k)[2k + 1]) = 0 for s ⩾ 0 (for k < 0 - by Theorem 2.1.18 (1) and duality (Theorem 2.1.23 ), for k ⩾ 0 - by Theorem 2.1.18 (3) and duality). So, Hom(R, SK^{r}(X_{P})(k)[2k + 1]) = 0, and Hom(N, X_{P} (k)[2k + 1]) = 0.

Since R⟨j + 1⟩(1)[2] × N and X_{R}(n − 2j)[2n − 4j] × N are extensions of X_{P} (∗)[2∗]’s (use Theorem 2.4.16 ), and there are no 1-st ext’s between N and X_{P} (k)[2k] for any smooth projective P and any k (see above), we have that the map R⟨j⟩ × N → N has a splitting, and N is a direct summand in R⟨j⟩ × N.

Now, using the fact, that \underline{R^{i_{0}−1}} = R × R⟨1⟩ × ⋯ × R⟨i_{0} − 1⟩, we get that the natural map \underline{R^{i_{0}−1}} × N → N has a splitting.

By duality: \underline{Hom}(\underline{R^{i_{0}−1}}, ℤ(d)[2d]) = \underline{R^{i_{0}−1}}, we get a splitting for the natural map: N(d)[2d] → \underline{R^{i_{0}−1}} × N.

Let us consider the following diagram:

\underline{R^{i_{0}−1}} × R(−i_{0} − t)[−2i_{0} − 2t] ⟶^{ψ×id} R(−i_{0} − t)[−2i_{0} − 2t]
↑^{φ(−t)[−2t]}  ↓^{h^{t}(−i_{0}−t)[−2i_{0}−2t]}
\underline{R^{i_{0}−1}} × R⟨i_{0}⟩(−t)[−2t]  R(−i_{0})[−2i_{0}]
↑^{p_{t}}  ↓^{π}
\underline{R^{i_{0}−1}} × R⟨i_{0}⟩  ⟵^{ρ}  N

It is not difficult to see, that over \overline{k} the maps: α := (ψ×id)∘φ(−t)[−2t]∘p_{t}∘ρ : N → R(−i_{0} − t)[−2i_{0} − 2t], and β := π ∘ h^{t}(−i_{0} − t)[−2i_{0} − 2t] : R(−i_{0} − t)[−2i_{0} − 2t] → N map ℤ to ℤ identically.

By Lemma 3.26 we have a direct summand in R(−t − i_{0})[−2t − 2i_{0}], isomorphic to N.

Lemma is proven.

Now we can prove Theorem 4.1 .

Let quadric R corresponds to the quadratic form r = q×⟨⟨a_{1}, …, a_{n}⟩⟩. By induction we can assume that

R = ⊕_{i=1,...,2^{n−1}}F_{α_{n−1}}(P)(i)[2i],

where α_{n−1} = {a_{1}, …, a_{n−1}} and quadric P corresponds to quadratic form p = ⟨1, −a_{n}⟩ × q (dimension of P is even).

Since all Witt numbers of r are divisible by 2^{n} (see the discussion before Theorem 4.1 ) we have that for even number of k’s : X_{R} = X_{R^{k·2^{n−1}}}. Let N_{1} be an undecomposable direct summand in F_{α_{n−1}}(P), containing X_{R}. By Lemma 4.5 we have the even number of undecomposable direct summands in F_{α_{n−1}}(P), each isomorphic to N_{1}(k · 2^{n−1})[k · 2^{n}], and containing X_{R^{k·2^{n−1}}}(k · 2^{n−1})[k · 2^{n}].

By duality, we have also the even number of undecomposable direct summands in F_{α_{n−1}}(P), isomorphic to N_{1}^{up}(−k · 2^{n−1})[−k · 2^{n}], where N_{1}^{up} is an undecomposable direct summand in F_{α_{n−1}}(P), containing the “upper elementary piece” X_{R}(dim(R) − 2^{n−1} + 1)[2 dim(R) − 2^{n} + 2].

It could be that either each of N_{1}(∗)[2∗]’s coincide with some of N^{up}(∗)[2∗]’s, or they are all different (actually, only the first case is possible), but in any case there are even number of such identifications, since Witt numbers are divisible by 2^{n}.

Let k_{1} be the smallest number such that X_{R^{k_{1}·2^{n−1}}}(k_{1} · 2^{n−1})[k_{1} · 2^{n}] is not contained in any of N_{1}(∗)[2∗]’s or N^{up}(∗)[2∗]’s. By considerations above, it should be even.

Then we can consider the undecomposable direct summand N_{k_{1}} in F_{α_{n−1}}(P), containing X_{R^{k_{1}·2^{n−1}}}(k_{1} · 2^{n−1})[k_{1} · 2^{n}]. It will be evidently contained in R⟨k_{1} · 2^{n−1}⟩.

We can again apply Lemma 4.5 , etc ... .

If dim(q)-even in this way we get that for each 0 ⩽ k ⩽ [dim(q)/2], there are undecomposable direct summands N_{k} and N_{k}(2^{n−1})[2^{n}], and M_{k} and M_{k}(2^{n−1})[2^{n}] in R, s.t. N contains X_{R^{k·2^{n}}}(k · 2^{n})[2k · 2^{n}] (in the sense of Lemma 3.23 ), and M_{k} contains X_{R^{(k+1)·2^{n}−1}}(dim(R) − (k + 1) · 2^{n} + 1)[2 dim(R) − (k + 1) · 2^{n+1} + 2]. By Lemma 3.21 , two such direct summands either coincide, or do not “intersect” (in the sense of Lemma 3.23 ). Then F_{α_{n−1}}(P) = S ⊕ S(2^{n−1})[2^{n}], and S is an extension of X_{R^{k·2^{n}}}(k·2^{n})[2k·2^{n}] and X_{R^{(k+1)·2^{n}−1}}(dim(R)−(k+1)·2^{n}+1)[2 dim(R)−(k+1)·2^{n+1}+2] for 0 ⩽ k ⩽ [dim(q)/2].

If dim(q)-odd in the same way we get F_{α_{n−1}}(P) = S ⊕ S(2^{n−1})[2^{n}] ⊕ A, where A is an extension of X_{R^{dim(R)/2−2^{n−1}+1}}(dim(R)/2 − 2^{n−1} + 1)[dim(R) − 2^{n} + 2] and X_{R^{dim(R)/2}}(dim(R)/2)[dim(R)].

But we have X_{dim(R)/2−j} = X_{Q_{α}} for all 0 ⩽ j ⩽ 2^{n−1} − 1. Really, q · ⟨⟨a_{1}, …, a_{n}⟩⟩ − ⟨⟨a_{1}, …, a_{n}⟩⟩ ∈ I^{n+1}(W (k)) (see Definition 2.4.2 ). So, q·⟨⟨a_{1}, …, a_{n}⟩⟩ ∈ I^{n+1}(W (k)) ⇔ ⟨⟨a_{1}, …, a_{n}⟩⟩ ∈ I^{n+1}(W (k)) ⇔ Q_{α} has a rational point (by Theorem 2.4.5 ). So, if q ·⟨⟨a_{1}, …, a_{n}⟩⟩ is hyperbolic, then Q_{α} has a rational point. And, conversely, if Q_{α} has a rational point, then it is hyperbolic (by Theorem 2.4.5 ), and so is q · ⟨⟨a_{1}, …, a_{n}⟩⟩. So, by Theorem 2.4.18 , X_{dim(R)/2−j} = X_{Q_{α}}.

A should be a nontrivial extension of the specified above objects, or X_{R^{dim(R)/2}}(dim(R)/2)[dim(R)] will be a direct summand in the motive of R, but we know that Hom(X_{Q_{α}}, ℤ(2^{n−1} −1)[2^{n} −1]) = ℤ/2 (easily follows from the fact that M_{α} is an extension of X_{Q_{α}} and X_{Q_{α}}(2^{n−1} −1)[2^{n} −2] (see [20], Theorem 4.4), see the cited place), and pure motives (i.e.: direct summands in the motives of smooth projective varieties) do not have such cohomology groups (by Theorem 2.1.18 ). And as I mentioned, there is only one nontrivial element in Hom(X_{Q_{α}}, X_{Q_{α}}(2^{n−1} −1)[2^{n} −1]) = Hom(X_{Q_{α}}, ℤ(2^{n−1} − 1)[2^{n} − 1]) (by Theorem 2.3.2 ).

Hence, A = M_{α}(dim(R)/2 − 2^{n−1} + 1)[dim(R) − 2^{n} + 2].

Denote F_{{a_{1},...,a_{n}}}(Q) := S.

Theorem 4.1 is proven.

It seems to me that the following question could have the positive answer (at least, I don’t have any counterexample at the moment).

Question 4.7 .

Is it true that any undecomposable direct summand in the motive of a quadric is of one of the two following types:

1) Tate motive: ℤ(i)[2i];

2) Higher form of a quadric (shifted): F_{α}(Q)(j)[2j], where α ∈ K_{n}^{M}(k)/2 is pure and Q is some undecomposable quadric (certainly, these conditions do not guarantee that higher form will be undecomposable, but they are necessary).

?

5. Back from X’s to quadratic forms

Using the same methods as in Corollary 3.15 , we can generalize it to the case of arbitrary quadric. This shows that “elementary pieces” from Theorem 3.1 define the motive of a quadric uniquely. Since those “elementary pieces” contain just the same information as Universal Splitting Tower of Manfred Knebusch (see Definition 2.4.19 ), we get that the motive of a quadric is defined by the equivalence class of UST (Q) (see Definition 2.4.19 for the definition of equivalence). In many interesting cases we can compare UST for two quadrics, which gives isomorphism for their motives.

Proposition 5.1 .

Let Q_{1}, Q_{2} be projective quadrics, s.t. dim(Q_{1}) = dim(Q_{2}). Then the following conditions are equivalent:

1) M(Q_{1}) = M(Q_{2}).

2) X_{Q_{1}} = X_{Q_{2}}, X_{Q_{1}^{1}} = X_{Q_{2}^{1}}, …, X_{Q_{1}^{[dim(Q_{1})/2]}} = X_{Q_{2}^{[dim(Q_{2})/2]}}.

3) Universal splitting towers for Q_{1} and Q_{2} are equivalent (see Definition 2.4.19 ).

Proof

(1 ⇒ 2) It follows from the proof of Theorem 3.7 .

(2 ⇔ 3) Follows from Theorem 2.4.18 .

(2 ⇒ 1)

Let dim(Q) = n. Denote as I the set of such 0 ⩽ i < n/2 that there exist no direct summands in M(Q_{1}), which are isomorphic to some direct summand in M(Q_{2}) and which contain X_{Q_{1}^{i}}(i)[2i] (as usually, we say that some direct summand N of Q contains X_{Q^{i}}(i)[2i] if for the corresponding idempotent p_{N} we have (p_{N})_{i} = 1 (see Theorem 3.7 for the definition), see also Lemma 3.23 ). By duality, I will be also the set of such j that there exist no direct summands in M(Q_{1}), which are isomorphic to some direct summand in M(Q_{2}) and which contain X_{Q_{1}^{j}}(dim(Q_{1})−j)[2 dim(Q_{1})−2j] (in the sense of Lemma 3.23 ). We want to show that I is empty. Let i_{0} will be smallest element of I.

For any j ∉ I there exist idempotents u_{j}{1}, u′_{j}{1} ∈ End(Q_{1}), s.t. (u_{j}{1})_{j} = 1 and (u′_{j}{1})′_{j} = 1, and corresponding direct summands are isomorphic to some direct summands in M(Q_{2}). Denote the set of such u_{j}{1}, u′_{j}{1}’s as S. So, we have u_{s}{1}, s ∈ S.

Consider U{1} = ∏_{s∈S}(1 − u_{s}{1}) (we choose some order on S (in which we multiply u_{s})). Evidently, U{1}_{j} = U{1}′_{j} = 0, if j ∉ I, and U{1}_{i} = U{1}′_{i} = 1 for all i ∈ I.

So, by Lemma 3.24 , we have a direct summand M_{1} in Q_{1} (I will write Q instead of M(Q) from this point), which is an extension of all X_{Q_{1}^{i}}(i)[2i], X_{Q_{1}^{i}}(dim(Q_{1}) − i)[2 dim(Q_{1}) − 2i], i ∈ I, and may be something from Q_{1}⟨n/2⟩, see Lemma 3.23 .

The same can be done with Q_{2}.

Denote N = M(−i_{0})[−2i_{0}], and dim(N) := dim(Q) − 2i_{0}.

Since X\underline{Q}_{1}^{i_{0}} = X\underline{Q}_{2}^{i_{0}}, we have morphisms: v : \underline{Q}_{1}^{i_{0}} → \underline{Q}_{2}^{i_{0}} and w : \underline{Q}_{2}^{i_{0}} → \underline{Q}_{1}^{i_{0}} which are of degree 1 on zero-dimensional cycle ℤ × ⋯ × ℤ (by Theorem 2.4.18 and duality).

From Claim 3.2 we know that \underline{Q}^{i_{0}} = Q × Q⟨1⟩ × ⋯ × Q⟨i_{0}⟩.

By Sublemma 4.6we get morphisms ρ : N → Q×Q⟨1⟩×⋯×Q⟨i_{0} −1⟩×Q⟨i_{0}⟩, s.t. over \overline{k} ℤ goes to ℤ×⋯×ℤ identically, and ν : Q×Q⟨1⟩×⋯×Q⟨i_{0}−1⟩×Q⟨i_{0}⟩ → N, which makes just the opposite.

Consider the following morphism from N_{1} to N_{2}:

φ_{1→2} : N_{1} →^{ρ_{1}} Q_{1} × Q_{1}⟨1⟩ × ⋯ × Q_{1}⟨i_{0}⟩ →^{v} Q_{2} × Q_{2}⟨1⟩ × ⋯ × Q_{2}⟨i_{0}⟩ →^{ν_{2}} N_{2}.

Analogously, one can construct the opposite morphism φ_{2→1}.

We, evidently, have: (φ_{2→1} ∘ φ_{1→2})_{0} = (φ_{1→2} ∘ φ_{2→1})_{0} = 1. This gives us that some direct summand of N_{1} is isomorphic to some direct summand of N_{2}(by Lemma 3.25). We get a contradiction if I is not empty.

So, I is empty, and S = [0, …, [n − 1/2]] ∪ [[n − 1/2]′, …, 0′]. Let (in the notations as above) u_{s}\{1\} = u\overleftarrow{s} ∘ u\overrightarrow{s}; u_{s}\{2\} = u\overrightarrow{s} ∘ u\overleftarrow{s}.

For every subset J ⊂ S choose some s_{J} ∈ J, and let J′ = J\s_{J}. Choose some order on each J′.

Put U^{→} = ∑_{J⊂S}(−1)^{#(J)−1}u\overrightarrow{s_{J}} ∏_{s∈J′} u_{s}\{1\}, and U^{←} = ∑_{J⊂S}(−1)^{#(J)−1}u\overleftarrow{s_{J}} ∏_{s∈J′} u_{s}\{2\} (product in the order chosen).

Evidently, (U^{→} ∘ U^{←})_{i} = (U^{→} ∘ U^{←})′_{i} = (U^{←} ∘ U^{→})_{i} = (U^{←} ∘ U^{→})′_{i} = 1, for all 0 ⩽ i < n/2.

Hence, by Lemma 3.24 , we have that U^{→} and U^{←} give us an isomorphism of some direct summands L_{1} in Q_{1} and L_{2} in Q_{2}, where L_{1} contains X_{Q_{1}^{i}}(i)[2i] and X_{Q_{1}^{i}}(n − i)[2n − 2i] for all 0 ⩽ i < n/2 (in the sense of Lemma 3.23 ), and L_{2} contains X_{Q_{2}^{i}}(i)[2i] and X_{Q_{2}^{i}}(n − i)[2n − 2i] for all 0 ⩽ i < n/2.

The complement summands: L_{1}^{⊥} and L_{2}^{⊥} could be nontrivial only if n is even, and then they could contain only X_{Q_{1}^{n/2}}(n/2)[n], or X_{Q_{1}^{n/2}}(n/2)[n] × k\sqrt{det(Q_{1})}, and

X_{Q_{2}^{n/2}}(n/2)[n], or X_{Q_{2}^{n/2}}(n/2)[n] × k\sqrt{det(Q_{2})}, respectively. Since det(Q_{1}) = det(Q_{2})

(because k\sqrt{det(Q_{t})} = k(Q_{t}^{n/2})∩\overline{k}), and X_{Q_{1}^{n/2}} = X_{Q_{2}^{n/2}}, those summands will be also isomorphic.

So, M(Q_{1}) = M(Q_{2}).

Proposition 5.1 is proven.

Remark Using almost the same arguments as in the proof above one can get Rost decomposition for a motive of a Pfister neighbor (i.e. subquadric of dimension ⩾ 1/2 in a big Pfister quadric) (see [14], Proposition 4).

Corollary 5.2 .

a) Let q_{1} and q_{2} are Pfister half-neighbors (see Definition 2.4.6 ), i.e. such forms that q_{1} ⊥ q_{2} = q_{α} and dim(q_{1}) = dim(q_{2}) (α ∈ K_{n}^{M}(k)/2-pure). Then M(Q_{1}) = M(Q_{2}). (This is, in some sense, the extreme case of the Remark above.)

b) Let for some h ∈ K_{n}^{M}(k)/2 of length 2 we have two representations: h = α_{1}+β_{1} = α_{2} + β_{2} with α_{i}, β_{j}-pure; and A_{1}, A_{2} - corresponding Albert quadrics (i.e. quadrics corresponding to the forms q_{α_{i}} ⊥ −q_{β_{i}}, see Definition 2.4.7 ). Then M(A_{1}) = M(A_{2}).

Proof a) Over k(Q_{i}) the Pfister form q_{α} get a rational point, and so, becomes hyperbolic, which means that q_{1}|_{k(Q_{i})} = −q_{2}|_{k(Q_{i})}, and Universal splitting towers (see Definition 2.4.19 ) for q_{1} and q_{2} are equivalent.

b) By result of R.Elman and T.Y.Lam (see Theorem 2.4.8 ): q_{i} is 2^{s}-times isotropic iff α_{i} and β_{i} have common pure divisor of degree s (and these are the only possible values of isotropity index (i.e.: the number of elementary hyperbolic summands in A_{i})), and iff h itself is divisible by pure symbol of degree s (which does not depend on the choice of i, and is itself very interesting fact, expressing some outstanding properties of the ring K_{∗}^{M}(k)/2). Hence, A_{1}^{j} has a rational point iff A_{2}^{j} does.

U.s.towers are equivalent, and motives are isomorphic.

Remark If we would be able to prove that the motive of a quadric defines quadric itself up to isomorphism, then we would have affirmative answer to the question of T.Y.Lam (see [10], (6.6) on p.28): Are two Albert forms, corresponding to the same symbol of length 2 proportional?

6. Few remarks about the Witt numbers

Let q/k be an anisotropic quadratic form. We have: q|_{k(Q)} = h_{1} · H ⊥ q_{1}, where q_{1} is some anisotropic form defined over k(Q); q_{1}|_{k(Q)(Q_{1})} = h_{2} · H ⊥ q_{2}; etc. ... Continuing this way we get Witt numbers: h_{1}, h_{2}, …, h_{s}.

This numbers contain the same information as our set J(Q) (introduced in the very end of Section 1.1).

We can apply methods from previous sections to get some information on the possible behavior of the Witt numbers.

Statement 6.1 .

Suppose our field k contains \sqrt{−1}.

Suppose that h_{1} > h_{i} for every i > 1. Then X_{Q} is hooked only to X_{Q}(dim(Q) − h_{1} + 1)[2 dim(Q) − 2h_{1} + 2] (i.e. there is a direct summand in the motive of Q, which is an extension of X_{Q} and X_{Q}(dim(Q) − h_{1} + 1)[2 dim(Q) − 2h_{1} + 2]), and dim(Q) − h_{1} = 2^{m} − 2 for some m .

Proof

From Lemma 4.5 it follows that for each 0 ⩽ j < h_{1} we have an undecomposable direct summand N_{j} in Q, which contains X_{Q}(j)[2j] (in the sense of Lemma 3.23 ) and does not contain any X_{Q}(j′)[2j′], for any 0 ⩽ j′ < h_{1}, j′ ≠ j. Moreover, N_{j} = N_{0}(j)[2j]. Since h_{1} > h_{i} for every i > 1, the only way it could be possible is: N does not contain any X_{Q^{l}}(l)[2l] and X_{Q^{l}}(dim(Q)−l)[2 dim(Q)−2l] for l ⩾ h_{1}. So, X_{Q} is hooked (in the sense of Definition 3.22 ) only to something from outer shell (I mean to some X_{Q^{j}}(∗)[2∗] with X_{Q^{j}} = X_{Q}); and using again the fact that N_{j} = N_{0}(j)[2j], we get that it is hooked only to X_{Q}(dim(Q)−h_{1}+1)[2 dim(Q)−2h_{1}+2] (it should be hooked to something if Q is anisotropic (if the natural morphism Q → X_{Q} has splitting, then the projection Q → ℤ has splitting as well (by Theorem 2.3.2 ), which means that Q has a zero-cycle of degree 1, then Q has a rational point by Springer’s theorem)). So, we get some pure motive N = Cone[−1](X_{Q} → X_{Q}(dim(Q)−h_{1} +1)[2 dim(Q)−2h_{1} +3]). Denote n := dim(Q) − h_{1} + 1.

N := Cone[−1](X_{Q} ⟶^{μ} X_{Q}(n)[2n + 1]), where μ is the only nontrivial element from Hom(X_{Q}, X_{Q}(n)[2n + 1]) (consider Hom’s from our exact triangle to X_{Q}(n)[2n + 1], and take into account that N is a direct summand in Q(−h_{1} + 1)[−2h_{1} + 2]); we have that N is a direct summand in Q. In particular, Hom(N, ℤ(a)[b]) = 0, for b > 2a, or b − a > n (here we need inclusion into Q(−h_{1} + 1)[−2h_{1} + 2]). Considering an exact triangle N → X_{Q} → X_{Q}(n)[2n + 1], it is easy to see that multiplication by μ performs an isomorphism: Hom(X_{Q}, ℤ(c)[d]) → Hom(X_{Q}, ℤ(c + n)[d + 2n + 1]) for any c > 0, d − c > 0 (since Hom(Q, ℤ(i)[j]) = 0 for j − i > dim(Q) (see Theorem 2.1.18 ), and N is a direct summand in Q(−h_{1} + 1)[−2h_{1} + 2]).

For finite field extension E/k we have action of transfers on motivic cohomology. Transfer is a map H_{M}^{b,a}(X|_{E}, ℤ) = H_{M}^{b,a}(X, E) → H_{M}^{b,a}(X, ℤ), where the first equality is duality (see Theorem 2.1.26 ), and the second is induced by the projection π : E → ℤ (E-finite field extension)(see also Definition 2.1.24 ); the main property of transfer is: Tr ∘j = ·[E : k], where j : H_{M}^{b,a}(X, ℤ) → H_{M}^{b,a}(X, E) = H_{M}^{b,a}(X|_{E}, ℤ) is induced by the map j : ℤ → E dual to π via duality (see Theorem 2.1.26 ).

Consider X̃_{Q} := Cone[−1](X_{Q} → ℤ) (see Definition 2.3.8 ).

Quadric has a point E of degree 2, and over E, X_{Q} becomes ℤ (see Theorem 2.3.4 ), we have by Theorem 2.1.18 (see also Definition 2.1.24 ) that Hom(X_{Q}|_{E}, ℤ(a)[b]) = 0 for b − a > 0.

Also we have that X̃_{Q}|_{E} = 0, and so, Hom(X̃_{Q}|_{E}, ℤ(a)[b]) = 0 for all a, b.

Now, considering the composition Tr ∘j = ·[E : k] we get that Hom(X_{Q}, ℤ(a)[b]) is a 2-torsion group for b > a, and Hom(X̃_{Q}, ℤ(a)[b]) is a 2-torsion group for all a and b.

In particular, we get that Hom(X_{Q}, ℤ(a)[b]) embeds into Hom(X_{Q}, ℤ/2(a)[b]) for b > a, and Hom(X̃_{Q}, ℤ(a)[b]) embeds into Hom(X̃_{Q}, ℤ/2(a)[b]) for all a and b.

Moreover, for b > a Hom(X_{Q}, ℤ/2(a)[b]) coincides with Hom(X̃_{Q}, ℤ/2(a)[b]) (by Theorem 2.1.18 ).

From Theorem 2.3.11 we know, that the differential Q_{i} acts without cohomology on Hom(X̃_{Q}, ℤ/2(∗)[∗′]) for any i ⩽ [log_{2}(n + 1)] (see Theorem 2.1.26 ), Denote η := μ(mod2), i.e. the image of μ in cohomology with ℤ/2 coefficients.

Q_{i}(η) = 0, for all i ⩽ [log_{2}(n)]. Really, Q_{i}(η) ∈ Hom(X_{Q}, ℤ/2(n+2^{i}−1)[2n+2^{i+1}]), and the later group is an extension of 2-cotorsion in Hom(X_{Q}, ℤ(n+2^{i}−1)[2n+2^{i+1}]), and 2-torsion in Hom(X_{Q}, ℤ(n + 2^{i} − 1)[2n + 2^{i+1} + 1]).

But as was shown above, μ performs a surjections Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1} − 1]) → Hom(X_{Q}, ℤ(n + 2^{i} − 1)[2n + 2^{i+1}]), and Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1}]) → Hom(X_{Q}, ℤ(n + 2^{i} − 1)[2n + 2^{i+1} + 1]).

And the groups: Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1} − 1]), Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1}]) are zero. Really, from the exact triangle N → X_{Q} → X_{Q}(n)[2n + 1], we have exact sequence:

Hom(N, ℤ(2^{i} − 1)[2^{i+1} − 1]) ← Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1} − 1]) ← Hom(X_{Q}, ℤ(2^{i} − 1 − n)[2^{i+1} − 2n]) The first group is zero since N is a pure motive (i.e.: a direct summand in the motive of some smooth projective variety), and (consequently) Hom(X_{Q}, ℤ(a)[b]) = 0 for b > 2a (see Theorem 2.1.18 ). The third group is zero by Theorem 2.3.3 (since n > 2^{i} − 1). So, the second is zero as well. Analogously, in the case of Hom(X_{Q}, ℤ(2^{i} − 1)[2^{i+1}]).

Denote k = [log_{2}(n)]. We have η = Q_{i}(x_{i}), for all 1 ⩽ i ⩽ k. Let us prove that there exists such γ ∈ Hom(X_{Q}, ℤ/2(n − 2^{k+1} + 2 + k)[2n − 2^{k+2} + 4 + k]), that η = Q_{k} ∘ Q_{k−1} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}(γ).

Multiplication by μ is an isomorphism on “positive” diagonals (“number of the diagonal” is the difference between “square” [-] and “round” (-) number)(see above); μ(mod2) = η = Q_{i}(x_{i}); and Q_{j}’s does not act from under the first diagonal to above it (since by Theorem 2.3.9 , the natural projection X_{Q} → ℤ defines an isomorphism Hom(ℤ, ℤ/2(a)[b]) → Hom(X_{Q}, ℤ/2(a)[b]), and since Q_{j} is functorial with respect to the morphisms of simplicial schemes, we have that the images of such an action would come from Hom(ℤ, ℤ/2(c)[d]) with d > c; but such groups are trivial (by Theorem 2.1.18 )). From all this: Q_{j} is a monomorphism on the diagonals from n + 2 to n + 2^{j} + 1. Really, if Q_{j}(\overline{y}) = 0 for \overline{y} from such a diagonal, then \overline{y} is in the image of Q_{j}, but by Theorem 2.1.26 (6), in such case \overline{y} comes from y - element of integral cohomology, then y = μ · z, where z ∈ Hom(X_{Q}, ℤ(a)[b]) with 1 ⩽ b − a ⩽ 2^{j}, and Q_{j}(\overline{μ · z}) = η ·Q_{j}(\overline{z}) (since η = Q_{j}(x_{j}), and by Theorem 2.1.26 (5) (we should remind that ρ is a class of {−1} ∈ K_{1}^{M}(k)/2 = Hom(ℤ, ℤ/2(1)[1]), see Definition 2.1.25 , and it is 0 in our case)). But on the diagonals from 1-st to 2^{j}-th the Q_{j} is injective (since it does not act from under the first diagonal to above it, as was explained above). And multiplication by η = \overline{μ} is monomorphic on the diagonals beginning from the 1-st. Contradiction.

Let us prove by induction on m that η = Q_{m} ∘ Q_{m−1} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}η_{m}, for some η_{m}. The first step is trivial since Q_{0}η = bokst(η) = 0, since η comes from integral cohomology. But now, Q_{m}(η_{m−1}) lives on the n+2-nd diagonal, and, hence, is equal to 0 (since Q_{m} is monomorphic on such a diagonal and (Q_{m})^{2} = 0). So, η_{m−1} = Q_{m}(η_{m}).

Finally, we get: η = Q_{k} ∘ ⋯ ∘ Q_{1} ∘ Q_{0}(γ), and γ ∈ Hom(X_{Q}, ℤ/2(n − 2 − 2^{k+1} + 2 + k)[2n − 2^{k+2} + 4 + k]).

But, (2n − 2^{k+2} + 4 + k) − (n − 2^{k+1} + 2 + k) = n − 2^{k+1} + 2, and k = [log_{2}(n)], so: 2^{k} ⩽ n < 2^{k+1}. Since we know, that Q_{i}’s don’t act from under the first diagonal to above it, the only possible choice for n is n = 2^{k+1} − 1.

Remark 0 The condition \sqrt{−1} ∈ k is added only because of the lack of the appropriate reference (in Theorem 2.1.26 it is not specified that φ_{j}, ψ_{j} are expressible in terms of Q_{t}’s, though they are).

Remark 1

I think, one can prove that the condition: h_{1} > h_{i}, ∀i > 1 in Statement 6.1 can be replaced by: h_{i} is not divisible by h_{1} ∀i > 1.

Another observation is a some step on the road paved by the J-filtration conjecture. The J-filtration conjecture states basically, that for q ∈ I^{n}\I^{n+1}, j_{1}(Q) = 2^{n−1}. This conjecture was settled in [13], Statement 2 of Section 3.3. Now I can move a bit further and make some attempt to describe: what j_{2}(Q) stays for (see the definition of j_{i} in the very end of Section 3.1 ). Consider the natural map π : I^{n}(W (k)) → I^{n}(W (k))/I^{n+1}(W (k)) = K_{n}^{M}(k)/2 (by the result of V.Voevodsky (Milnor’s conjecture on quadratic forms), see [13], p.14).

Statement 6.2 .

Suppose our field k contains \sqrt{−1}.

Let q ∈ I^{n}\I^{n+1} (i.e.: j_{1}(Q) = 2^{n−1}). Then j_{2}(Q) ∈ {2^{r} − 2^{n−1}, r > n; 2^{n}; 2^{n−1} + 2^{m}, m < n − 1} (see the very end of Section 3.1 for the definition of j_{i}).

Moreover, for j_{2}(Q) < 2^{n} we have: π(q) is not pure (see Definition 2.4.4 ).

Proof

Since we are interested only in j_{1} and j_{2}, we can assume that #J(Q) = 2 (just change k by k(Q^{dim(Q)/2−j_{2}}) and q by (q|_{k(Q^{dim(Q)/2−j_{2}})})_{anis.}). Now, j_{2} − j_{1} = h_{1}. So, by Statement 6.1 we have that if j_{2} > 2^{n}, then dim(Q) − j_{2} + j_{1} = 2^{r} − 2. Since dim(Q) = 2j_{2} − 2 (because #J(Q) = 2), we get: j_{2}(Q) = 2^{r} − j_{1} = 2^{r} − 2^{n−1} (since j_{1} = 2^{n−1} by the J-filtration conjecture (see discussion above)).

In the case j_{2} < 2^{n}, using Lemma 4.5 , we get that if h_{2} = j_{1} is not divisible by h_{1} = j_{2} − j_{1}, then we will have a direct summand in the motive of Q, consisting of X’s only from the inner shell, (i.e. of X_{Q^{i}}(∗)[2∗]’s with X_{Q^{i}} = X_{Q^{dim(Q)/2}}).

Evidently, if such summand contains X_{Q^{i}}(i)[2i], then it contains X_{Q^{dim(Q)−(i+j_{1}−1)}}(i+ j_{1} −1)[2i+2j_{1} −2] as well (since the same holds over k(Q^{dim(Q)−j_{1}}), where anisotropic part of Q is just Pfister quadric).

Again, using Lemma 4.5 , we get that the entire inner shell will be a direct summand in Q, and, so, outer shell will be a direct summand (i.e. there is a direct summand, consisting (in the sense of Lemma 3.23 ) of X_{Q^{i}}(∗)[2∗]’s with X_{Q^{i}} = X_{Q}) and a pure motive as well. The proof of Statement 6.1 shows that in this case dim(Q) − h_{1} + 1 = 2^{m} − 2, which is impossible if j_{2} < 2^{n}.

So, h_{2} is divisible by h_{1}, and we get desired choice for j_{2}(Q).

For the last question, changing k by k(Q^{dim(Q)/2−j_{2}}), we can again assume that #J(Q) = 2 (if h ∈ K_{n}^{M}(k)/2 and for some extension E/k: h|_{E} is non-pure, then h is non-pure).

Suppose for j_{2}(Q) < 2^{n} we would have π(q) = α is pure symbol, then, X_{Q^{dim(Q)/2}} = X_{Q_{α}}, where Q_{α} is a Pfister quadric, corresponding to α.

Really, it follows from the fact that J(Q) does not contain any number of the form 2^{s} except j_{1} = 2^{n−1}, since by J-filtration conjecture: π(q)|_{E} = 0 ⇔ j_{1}(Q|_{E}) = 2^{s}, for some s > n − 1, or q|_{E} is hyperbolic (one could say: s = ∞ in this case), and since J(Q|_{E}) ⊂ J(Q), in our situation only the last case is possible, so we get: q is hyperbolic over some extension E/k iff π(q)|_{E} = 0. Existence of a rational point on Q^{dim(Q)/2} is equivalent to the fact that Q is hyperbolic. So, by definition we have: X_{Q^{dim(Q)/2}} = X_{Q_{α}}.

Then we would have that the inner shell is a direct summand. Really, we know that Q⟨h_{1}⟩ = Cone[−1](⊕_{i=0,...,2^{n−1}−1}X_{dim(Q)/2}(i)[2i] → ⊕_{i=0,...,2^{n−1}−1}X_{dim(Q)/2}(2^{n}−2− i)[2^{n+1} − 3 − 2i]) (by Theorem 3.1 and Theorem 2.5.10 (with X = Q⟨h_{1}⟩(h_{1})[2h_{1}], X_{m} = 0)) = Cone[−1](⊕_{i=0,...,2^{n−1}−1}X_{Q_{α}}(i)[2i] → ⊕_{i=0,...,2^{n−1}−1}X_{Q_{α}}(2^{n} − 2 − i)[2^{n+1} − 3 − 2i]) (by what was just proven).

Let α = π(q) - our pure symbol. We have a decomposition of the Rost motive M_{α}: M_{α} = Cone[−1](X_{Q_{α}} →^{μ} X_{Q_{α}}(2^{n−1} − 1)[2^{n} − 1]), for some morphism μ ∈ Hom(X_{Q_{α}}, X_{Q_{α}}(2^{n−1} − 1)[2^{n} − 1]) = Hom(X_{Q_{α}}, ℤ(2^{n−1} − 1)[2^{n} − 1]) (equality by Theorem 2.3.2 ) (see [14], Proposition 4 and [18] Theorem 4.5). And M_{α} is a direct summand in the motive of small Pfister quadric of dimension 2^{n−1} −1 (see [14], Proposition 4; and Definition 2.4.6 ).

Considering the Hom’s from the exact triangle M_{α} = Cone[−1](X_{Q_{α}} →^{μ} X_{Q_{α}}(2^{n−1} − 1)[2^{n} −1]) to ℤ(k)[2k +1], and using the fact that Hom(M_{α}, ℤ(k)[2k +1]) = 0 for all k (since M_{α} is pure motive (i.e.: a direct summand in the motive of a smooth projective variety)) (see Theorem 2.1.18 ), we get that the map: Hom(X_{Q_{α}}, ℤ(k −2^{n−1} +1)[2k − 2^{n} + 2]) → Hom(X_{Q_{α}}, ℤ(k)[2k + 1]) is surjective. In particular, Hom(X_{Q_{α}}, ℤ(k)[2k + 1]) = 0 for k < 2^{n−1} −1 (by Theorem 2.3.3 ). That means that X_{Q_{α}}(2^{n−1} −1)[2^{n} −2] - the most upper summand of the lower part of Q⟨h_{1}⟩ can be hanged only to X_{Q_{α}}(2^{n} − 2)[2^{n+1} − 4], and the extension is given by μ, which is the only nontrivial element of the group Hom(X_{Q_{α}}, ℤ(2^{n−1} − 1)[2^{n} − 1]) (see surjection above + Theorem 2.3.2 + Theorem 2.1.20 + the fact that our group has exponent 2 (by transfer arguments (see the proof of Statement 6.1 for the definition of transfer)). But then the Cone[−1] of this extension will be just Rost motive (shifted by (2^{n−1}−1)[2^{n}−2])(see above), so, it is pure, and consequently, a direct summand in Q⟨h_{1}⟩ (since there are no 1-st Ext’s from M_{α} to X_{Q_{α}}(k)[2k], as well as from X_{Q_{α}} to M_{α}(l)[2l]; really, Hom(M_{α}, X_{Q_{α}}(k)[2k+1]) = Hom(M_{α}, ℤ(k)[2k + 1]) (by Theorem 2.3.2 ) = 0 since M_{α} is pure by Theorem 2.1.18 , and Hom(X_{Q_{α}}, M_{α}(l)[2l + 1]) ⊂ Hom(X_{Q_{α}}, Q_{small}(l)[2l + 1]) (since M_{α} is a direct summand in Q_{small}) = Hom(X_{Q_{α}} × Q_{small}, ℤ(l + 2^{n−1} − 1)[2l + 2^{n} − 1]) (by duality Theorem 2.1.23 ) = Hom(Q_{small}, ℤ(l + 2^{n−1} − 1)[2l + 2^{n} − 1]) (since X_{Q_{small}} = X_{Q_{α}} by Theorem 2.4.21 ) = 0 (by Theorem 2.1.18 )).

So, by Lemma 4.5 Q⟨h_{1}⟩ is a direct sum of M_{α}(i)[2i], i = 0, …, 2^{n−1} − 1.

So, our inner shell Q⟨h_{1}⟩(h_{1})[2h_{1}] is a pure motive and a direct summand in Q, and the complement summand will be outer shell.

Hence, we get that outer shell is a direct summand, which is impossible if j_{2}(Q) < 2^{n} (in the proof of Statement 6.1 we use just the fact that outer shell is a direct summand).

Statement is proven.

Remark Actually, quadrics with j_{2}(Q) = 2^{n−1} + 2^{m} with m < n − 2 should not exist, but it is not proven.

The j_{2} analog of the J-filtration conjecture would be the following question: Question 6.3 . Let q ∈ I^{n}\I^{n+1}, then:

a) Are the following conditions equivalent ? 1. π(q) is not pure.

2. j_{2}(Q) = 3 · 2^{n−2}.

b) Are the following conditions equivalent ? 1. π(q) is pure.

2. j_{2}(Q) = 2^{n}, or j_{2}(Q) = 2^{k} − 2^{n−1}, k > n.

In connection with the above question we can state the following hypothetical description of quadrics with #J(Q) = 2 (It seems that M.Knebusch understood it, and was able to prove it in some cases (see [7])):

Question 6.4 . Is it true that any quadric Q with #J(Q) = 2 is of one of the three following types:

1) q = ⟨⟨a_{1}, …, a_{n−2}⟩⟩ · ⟨b, c, −bc, −d, −e, de⟩, where π(q)={a_{1}, …, a_{n−2}} · ({b, c} + {d, e}) ∈ K_{n}^{M}(k)/2 is not pure (length of π(q) is 2). j_{2}(Q) = 3 · 2^{n−2};

2) q = ⟨⟨a_{1}, …, a_{n−1}⟩⟩ · ⟨b, −c, −d, cd⟩, where π(q) = {a_{1}, …, a_{n−1}, b} ≠ 0, and {a_{1}, …, a_{n−1}, c, d} is not divisible by {a_{1}, …, a_{n−1}, b} (π(q) here is pure). j_{2}(Q) = 2^{n};

3) q = ⟨⟨a_{1} …, a_{n}⟩⟩ · (⟨⟨b_{1}, …, b_{k}⟩⟩ − ⟨1⟩) (here, again, π(q) = {a_{1}, …, a_{n}} is pure). j_{2}(Q) = 2^{n+k−1} − 2^{n−1}.

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