ON TORSION ELEMENTS IN THE CHOW-GROUPS OF QUADRICS

A. VISHIK

0. Introduction

In this paper we present some new example of torsion elements in the Chow-groups of projective quadric. For hyperbolic projective quadric X and any 0 ⩽ p < dim(X)/2, the Chow-group of cycles of codimension p (modulo rational equevalence) is a free abelian group generated by the class h^{p} of plane section of codimension p,. By transfer arguments it follows, that for arbitrary (smooth) projective quadric, CH^{p}(X) = ℤ · h^{p} ⊕ T ors.(CH^{p}(X)).

On the other hand, for any projective quadric X of dimension more than 2, the Picard group CH^{1}(X) is isomorphic to ℤ·h, in other words, there is no torsion. The natural question arises: can we extend this result to higher Chow-groups.

Basic here is the following conjecture due to N. Karpenko: Let X_{φ} be a projective quadric, defined by quadratic form ϕ of di-

mension n over a field of characteristic not 2.

Conjecture 1 ([7], Conjecture 0.1). For any p, if n is sufficiently large, then CH^{p}(X_{φ}) = ℤ · h^{p}. This conjecture was supported by the following computations: Theorem ([6], Theorem 6.1; [8], Theorem 6.1, Theorem 8.5). Under above notations,

(a) CH^{2}(X_{φ}) = ℤ, for n > 8;

(b) CH^{3}(X_{φ}) = ℤ, for n > 12;

(c) CH^{4}(X_{φ}) = ℤ, for n > 24;

Moreover, the boundaries in (a) and (b) are exact ones, and more generally:

Theorem ([7], Theorem 2.4). For any p > 1 there exist a 4p-dimensional quadratic form φ (over a

suitable field F), such that T ors.(CH^{p}(X_{φ})) ≠ 0.

In [7] an attempt was made to make Conjecture 1 more precise: Conjecture 2 ([7], Conjecture 0.2). If n > 4p for some p, then CH^{p}(X_{φ}) = ℤ · h.

The principal aim of this note is to disprove this stronger version of the Conjecture - see Corollary 2.1 .

For any r ⩾ 2 we will construct the quadratic form φ of dimension 6 · 2^{r} (over suitable field), such that T ors.(CH^{2^{r}+1}(X_{φ})) ≠ 0. This clearly will disprove the Conjecture 2 .

The main ingredients of the construction are: the special pair of O.Izhboldin (see [3], Section 9), the computations of unramified cohomology by B.Kahn-M.Rost-R.J.Sujatha (including the motivic interpretation of the later)(see [5], and [4], Appendix), and the motivic cohomological operations of V.Voevodsky (see [17]). Although, the original problem is formulated in the classical Chow-motivic language, we have to work in the bigger triangulated category of mixed motives of V.Voevodsky (see [16]). In this category we can use the operations and the decomposition of the motive of a quadric from [14], Theorem 3.1.

Acknoledgements: First of all, I wanted to thank Vladimir Voevodsky, with whom we discussed the Conjecture above extensively few years ago, and whose insight into the problem proved to be extremely usefull. Also, I wanted to thank Oleg Izhboldin, for the very fruitfull discussions and the fact that his construction of the field with u-invariant 9 (see [3]) inspired the current article. Finally, I want to thank Nikita Karpenko for usefull comments. This work was done during my stay at the Max-Plank Institut für Mathematik, and I’d like to express my gratitude to the MPIM for the support and excellent working conditions.

1. Some preliminary computations

Under generalized Albert form we will understand the anisotropic part of the difference of two n-fold Pfister forms:

(⟨⟨a_{1}, …, a_{n}⟩⟩ − ⟨⟨b_{1}, …, b_{n}⟩⟩)_{an}.

Such forms were studied extensively in [1].

Proposition 1.1 .

For any r, over some field of given characteristic ≠ 2 there exists anisotropic generalized Albert form of dimension 6 · 2^{r}.

Proof of Proposition 1.1

Lemma 1.2 .

Let F be a field of characteristic not 2, and q/F be an anisotropic quadratic form. Let K := F (x_{1}, …, x_{r}) be a purely transcendental extension, and p := q × ⟨⟨x_{1} …, x_{r}⟩⟩ be a quadratic form over K. Then p is anisotropic.

Proof It is evidently sufficient to consider the case r = 1. Suppose that p := q × ⟨1, −x⟩ is isotropic over F (x). Restricting to

F ((x)), we get a contradiction with the Theorem of Springer (see [10], VI, Proposition 1.9). So, p is anisotropic.

Lemma 1.3 . Suppose the Pfister form ⟨⟨a, b⟩⟩ over the field F is anisotropic. Then

the Albert form ⟨a, b, −ab, yz, −y, −z⟩ is anisotropic over the field L := F (y, z).

Proof If our Albert form is isotropic over L, then it is isotropic over F ((y))((z)),

and by [10], VI, Proposition 1.9, we have that ⟨a, b, −ab, −y⟩ is isotropic over F ((y)) (since ⟨y, −1⟩ is evidently anisotropic over the last field). Again by the Theorem of Springer we get a contradiction.

Suppose K = k(a, b, c, d, x_{1}, …, x_{r}) be purely transcendental extension, generated by specified variables. Consider quadratic form φ := ρ × ⟨⟨x_{1}, …, x_{r}⟩⟩, where ρ := ⟨a, b, −ab, cd, −c, −d⟩.

Lemma 1.4 . The form φ is anisotropic. Proof By Lemma 1.2 , it is sufficient to prove that ⟨a, b, −ab, cd, −c, −d⟩ is

anisotropic over L := k(a, b, c, d). Take F := k(a, b). Then evidently ⟨⟨a, b⟩⟩ is anisotropic over F. By Lemma 1.3 we get what we need.

Proposition 1.1 is proven.

2. The Main Theorem

Main Theorem . Let φ is anisotropic generalized Albert form of dimension 6 · 2^{r} over

the field k of characteristic 0. Then Tors. CH^{2^{r}+1}(X_{φ}) ≠ 0.

Combining the Main Theorem with Proposition 1.1 , we get: Corollary 2.1 . Conjecture 2 is wrong. Proof of the Main Theorem Since φ is anisotropic generalized Albert form of dimension 6 · 2^{r},

we have: φ = ρ × ⟨⟨x_{1}, …, x_{r}⟩⟩, where ρ = ⟨a, b, −ab, −c, −d, cd⟩ is 6-dimensional Albert form, and ⟨⟨x_{1}, …, x_{r}⟩⟩ is some Pfiter form (see [1]).

Denote ψ := ⟨a, b, −ab, −c, −d⟩ × ⟨⟨x_{1}, …, x_{r}⟩⟩. Then (φ, ψ) is a special pair of Izhboldin (see [3], Section 9), and by condition, it is an anisotropic special pair.

By [3], Theorem 9.4 (3), we have that ψ|_{k(X_{φ})} is anisotropic Pfister neighbor, i.e., it is proportional to a subform (of dimension more than half) of anisotropic Pfister form π. This Pfister form corresponds to a nontrivial pure symbol α in K_{r+3}^{M}(k(X_{φ}))/2. Under the natural map K_{∗}^{M}(k(X_{φ}))/2 → H_{et.}^{∗}(k(X_{φ}), ℤ/2) (which is an isomorphism by “Milnor’s conjecture” - see [17]), α goes to the nontrivial element e^{r+3}(Pf(ψ|_{k(X_{φ})})) of H_{et.}^{r+3}(k(X_{φ}), ℤ/2) (we are following the notations of [3], Lemma 7.4). By [3], Corollary 7.3, e^{r+3}(Pf(ψ|_{k(X_{φ})})) actually belongs to the unramied part: H_{nr}^{r+3}(k(X_{φ})/k, ℤ/2) of H_{et.}^{r+3}(k(X_{φ}), ℤ/2).

We have natural map: η_{2}^{∗} : H_{et.}^{∗}(k, ℤ/2) → H_{nr}^{∗}(k(X_{φ})/k, ℤ/2). Let us denote H̃_{nr}^{∗}(k(X_{φ})/k, ℤ/2) := H_{nr}^{∗}(k(X_{φ})/k, ℤ/2)/image(η_{2}^{∗}). For arbitrary element x from H_{nr}^{∗}(k(X_{φ})/k, ℤ/2), we denote it’s image in H̃_{nr}^{∗}(k(X_{φ})/k, ℤ/2) as x̃.

Proposition 2.2 (cf. [3], Lemma 10.5, Lemma 7.4).

Under the above notations, ẽ^{r+3}(Pf(ψ|_{k(X_{φ})})) is nonzero.

Proof

Suppose, ẽ^{r+3}(Pf(ψ|_{k(X_{φ})})) = 0. That means that e^{r+3}(Pf(ψ|_{k(X_{φ})})) = η_{2}^{r+3}(λ) for some λ ∈ H_{et.}^{r+3}(k, ℤ/2). By the results of V.Voevodsky, we have the natural identification:

K_{r+3}^{M}(k)/2 = H_{et.}^{r+3}(k, ℤ/2) = I^{r+3}(W (k))/I^{r+4}(W (k)),

where I ⊂ W (k) is the ideal of even-dimensional form in the Wittring W (k) (see [17] and [11], Section 3.1). So, there exists quadratic form q ⊂ I^{r+3}(W (k))\I^{r+4}(W (k)), s.t. \overline{q} ∈ I^{r+3}(W (k))/I^{r+4}(W (k)) corresponds to λ under the identifications above. By “J-filtration Conjecture” we have I^{r+3}(W (k)) = J^{r+3}(W (k)) (see [11], Section 3.3, Statement 2). That means that the degree (see [9]) of q is r + 3, and there exists a Generalized Splitting Tower of Manfred Knebusch (see [9]) k = k_{0} ⊂ k_{1} ⊂ ⋯ ⊂ k_{s−1} ⊂ k_{s}, s.t. for all 0 ⩽ l ⩽ s − 2, k_{l+1} = k_{l}(Q_{l}), where Q_{l} is quadric (over k_{l}) of dimension greater than 2^{r+3} −2, and (q|_{k_{s−1}})_{anis.} is proportional to anisotropic r+3-fold Pfister form over k_{s−1}. That means that \overline{q}|_{k_{s−1}} = λ|_{k_{s−1}} ∈ K_{r+3}^{M}(k_{s−1})/2 is a nonzero pure symbol.

Since dim(q_{l}) > 2^{r+3} > 6 · 2^{r} = dim(φ), by the result of Detlew Hoffmann (see [2], Theorem 1), we have that φ|_{k_{s−1}} is anisotropic.

Let us denote F := k_{s−1}. We have: φ|_{F} is anisotropic, and (if you want, again by [3], Theorem 9.4 (3)) ψ|_{F (X_{φ})} is a neighbor of anisotropic Pfister form ⟨⟨λ|_{F} ⟩⟩|_{F (X_{φ})} (notice that λ|_{F} is a pure symbol). In particular, λ|_{F (X_{φ})(X_{ψ})} = 0. Since ψ ⊂ φ, we have λ|_{F (X_{ψ})} = 0. That means that ψ|_{F} is a Pfister neighbour of ⟨⟨λ|_{F} ⟩⟩ By [3], Lemma 9.4 (2), we get: φ|_{F} is isotropic - a contradiction. So, ẽ^{r+3}(Pf(ψ|_{k(X_{φ})})) ≠ 0

Let γ := ẽ^{r+3}(Pf(ψ|_{k(X_{φ})})) ∈ H̃_{nr}^{r+3}(k(X_{φ})/k, ℤ/2). By Proposition 2.2 , γ ≠ 0.

Together with the map η_{2}^{m} : H_{et.}^{m}(k, ℤ/2) → H_{nr.}^{m}(k(X_{φ})/k, ℤ/2) we can consider the map: η^{m} : H_{et.}^{m}(k, ℚ/ℤ(m−1)) → H_{nr.}^{m}(k(X_{φ})/k, ℚ/ℤ(m− 1)).

From [5], Theorem 7.4 and Remark after it (and from “Milnor’s conjecture”, see [17]), we have an exact sequence:

0 → (Ker(η_{2}^{m}))_{0} → coker(η_{2}^{m}) → coker(η^{m}),

where (Ker(η_{2}^{m}))_{0} = {y ∈ Ker(η_{2}^{m}) |{−1} · y = 0}.

Lemma 2.3 .

The group Ker(η_{2}^{r+3} : H_{et.}^{r+3}(k, ℤ/2) → H_{nr.}^{r+3}(k(X_{φ})/k, ℤ/2)) is zero. Consequently, (Ker(η_{2}^{r+3}))_{0} = 0.

Proof

Suppose, our Ker is nontrivial. Since H_{nr.}^{r+3}(k(X_{φ})/k, ℤ/2)) is a subgroup in H_{et.}^{r+3}(k(X_{φ}), ℤ/2), in this case we get a nontrivial element h ∈ Ker(H_{et.}^{r+3}(k, ℤ/2) → H_{et.}^{r+3}(k(X_{φ}), ℤ/2).

By [11], Lemma from Section 3, there exists field extension L/k, s.t. h|_{L} is nonzero pure symbol {a_{1}, …, a_{r+3}} for some a_{i} ∈ L. Then φ|_{L} (up to coefficient) is a subform in the anisotropic Pfister form ⟨⟨a_{1}, …, a_{r+3}⟩⟩. Suppose ρ is a complementary form, i.e.: φ|_{L} ⊥ ρ = ⟨⟨a_{1}, …, a_{r+3}⟩⟩.

Since φ|_{L} ∈ I^{r+2}(W (L)) and ⟨⟨a_{1}, …, a_{r+3}⟩⟩ ∈ I^{r+3}(W (L)), we have that ρ ∈ I^{r+2}(W (L)). But, the dim(ρ) = 2^{r+3} − 6 · 2^{r} = 2^{r+1} < 2^{r+2} - contradiction.

So, we have an exact sequence:

0 → coker(η_{2}^{r+3}) → coker(η^{r+3}).

Lemma 2.4 .

Let X be arbitrary quadric, then coker(η_{ℚ_{(p)}/ℤ_{(p)}}^{m} : H_{et.}^{m}(k, ℚ_{(p)}/ℤ_{(p)}) → H_{nr.}^{m}(k(X), ℚ_{(p)}/ℤ_{(p)})) = 0 for any p ≠ 2 and any m.

Proof

Let Spec(L) ⊂ X will be arbitrary point on our quadric. Since X_{φ}|_{L} is isotropic, it is birationally equivalent to projective space, and hence, coker(η_{A}^{m} : H_{et.}^{m}(L, A) → H_{nr.}^{m}(L(X), A)) = 0 for any group of coefficients A (see [12], 12.10, 7.3). Since ℚ_{(p)}/ℤ_{(p)} is uniquely 2-divisible, by transfer arguments, we have that coker(η_{ℚ_{(p)}/ℤ_{(p)}}^{m} : H_{et.}^{m}(k, ℚ_{(p)}/ℤ_{(p)}) → H_{nr.}^{m}(k(X), ℚ_{(p)}/ℤ_{(p)})) = 0.

The above Lemma shows: coker(η^{r+3}) = coker(η_{ℚ_{(2)}/ℤ_{(2)}}^{r+3} : H_{et.}^{r+3}(k, ℚ_{(2)}/ℤ_{(2)}) → H_{nr.}^{r+3}(k(X_{φ}), ℚ_{(2)}/ℤ_{(2)})).

In this light, we will denote η_{ℚ_{(2)}/ℤ_{(2)}}^{r+3} simply as η^{r+3}.

In [4] B.Kahn interpreted coker(η^{∗}) in terms of motivic cohomology

- see [4], Theorem A.1: there is an exact sequence:

0 → coker(η^{r+3}) → H_{B}^{r+5}(X_{X_{φ}}, ℤ_{(2)}(r + 2)),

where X_{X_{φ}} is a standard simplicial scheme, corresponding to the morphism X_{φ} → Spec(k) (see, for example, [14], Definition 2.3.1), and H_{B}^{r+5}(X_{X_{φ}}, ℤ_{(2)}(r + 2)) = Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ_{(2)}(r + 2)[r + 5]) - the Hom-group in the triangulated category of mixed motives of V.Voevodsky, where ℤ by definition is the motive of a point (M(Spec(k))).

Remark Notice the discrepancy in the notations for η^{m} in [4] in comparison to [5] (where it coinsides with our notation). η^{m} in [4] corresponds to η^{m+1} in [5].

This shows, that we have an exact sequence:

0 → coker(η_{2}^{r+3}) → Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ_{(2)}(r + 2)[r + 5]).

Since we have a nontrivial element γ in coker(η_{2}^{r+3}), it’s image γ′ ∈ Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ_{(2)}(r + 2)[r + 5]) will also be nontrivial.

Using the fact that Hom_{DM_{eff.}^{−}}(M(Spec(L)), ℤ(i)[j]) = 0 for j > i, where L is a field (see [17], Corollary 2.3), and the fact that M(X_{X_{φ}})|_{L} = M(Spec(L)) for any point Spec(L) ⊂ X_{φ} on a quadric (see [17], Lemma 3.8, or [14], Theorem 2.3.4), by usual transfer arguments we have that the groups Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ_{(2)}(i)[j]) have exponent 2 for j > i. That means, that we have an embedding: Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ_{(2)}(r+ 2)[r+5]) → Hom_{DM_{eff.}^{−}}(M(X_{X_{φ}}), ℤ/2(r+2)[r+5]). Let γ′′ be the image of γ′ under this map. We have: γ′′ ≠ 0.

From this point we will drop subscript DM_{eff.}^{−} from Hom, and M(−).

Together with X_{X_{φ}} we can consider X̃_{X_{φ}} := Cone[−1](pr : X_{X_{φ}} → ℤ), wher pr is induced by the natural projection X_{X_{φ}} → Spec(k).

Now, we can use Voevodsky’s cohomological operations.

We have cohomological operations Q_{i} of bidegree (2^{i} − 1)[2^{i+1} − 1], s.t. Q_{i} is a differential (it’s square iz zero), and for quadric X of dimension greater or equal 2^{m} − 1, Q_{m} acts without cohomology on Hom(X̃_{X}, ℤ/2(∗)[∗′]) (see [17], Theorem 3.17, Theorem 3.25 and Lemma 4.11 (notice that in the proof of the later no specific of Pfister case is used))

In particular, in our case, Q_{1}, …, Q_{r+2} will act without cohomology on Hom(X̃_{X_{φ}}, ℤ/2(∗)[∗′]).

Since Hom(ℤ, ℤ/2(i)[j]) = 0 for j > i (see [17], Corollary 2.3), we can identify Hom(X̃_{X_{φ}}, ℤ/2(i)[j]) with Hom(X_{X_{φ}}, ℤ/2(i)[j]) if j > i. Below we will deal only with such cohomology (j > i), so, we will use X_{X_{φ}} instead of X̃_{X_{φ}}.

Let us consider Q_{r−1} ∘ Q_{r−2} ∘ ⋯ ∘ Q_{1}(γ′′).

Lemma 2.5 .

Q_{r−1} ∘ Q_{r−2} ∘ ⋯ ∘ Q_{1}(γ′′) ≠ 0.

Proof

Suppose that Q_{r−1} ∘ Q_{r−2} ∘ ⋯ ∘ Q_{1}(γ′′) = 0. Then there exists such 0 ⩽ i ⩽ r − 2, that Q_{i} ∘ … Q_{1}(γ′′) ≠ 0, but Q_{i+1} ∘ … Q_{1}(γ′′) = 0.

Since Q_{j} has bidegree (2^{j} − 1)[2^{j+1} − 1] (see [17], Theorem 3.17), we have: y := Q_{i} ∘ … Q_{1}(γ′′) ∈ Hom(X_{X_{φ}}, ℤ/2(r − i + 2^{i+1})[r − i + 2^{i+2} + 1]). Since Q_{i+1}(y) = 0 and Q_{i+1} acts without cohomology on Hom(X_{X_{φ}}, ℤ/2(∗)[∗′]), we have that there exists z ∈ Hom(X_{X_{φ}}, ℤ/2(r− i + 1)[r − i + 2]), s.t. y = Q_{i+1}(z).

By [16], Lemma 6.4, [17], Theorem 4.1 (see also [4], Theorem A.1), we have an identification of Hom(X_{X_{φ}}, ℤ/2(r − i + 1)[r − i + 2] with the Ker(K_{r−i+2}^{M}(k)/2 → K_{r−i+2}^{M}(k(X_{φ}))/2).

Since dim(X_{φ}) = 6·2^{r} > 2^{r−i+2} we have by “Kahn-Rost-Sujatha Conjecture” (see [11], Statement 1 from Section 3.2) that Ker(K_{r−i+2}^{M}(k)/2 → K_{r−i+2}^{M}(k(X_{φ}))/2) = 0. We get a contradiction. So, Q_{r−1} ∘ Q_{r−2} ∘ ⋯ ∘ Q_{1}(γ′′) ≠ 0.

So, we get a nontrivial element u := Q_{r−1} ∘ Q_{r−2} ∘ ⋯ ∘ Q_{1}(γ′′) ∈ Hom(X_{X_{φ}}, ℤ/2(2^{r} + 1)[2^{r+1} + 2]).

Since γ′′ came from cohomology group with ℤ_{(2)} coefficients, and Q_{i} = [β, q_{i}] for some operation q_{i} (β here is a Bockstein morphism) (see [17], Theorem 3.17(3)), we have that there exists v ∈ Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) s.t. \overline{v} = u.

The main conclusion is:

Lemma 2.6 .

The group Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) is nontrivial.

Now, we need to relate the motivic cohomology of X_{X_{φ}} and those of X_{φ}. Let X_{φ}^{j} be (as usually) the variety of j-dimensional projective spaces on X_{φ}. By [14], Theorem 4.1, the motive M(X_{φ}) of X_{φ} decomposes as: M(X_{φ}) = ⊕_{i=0,...,2^{r}−1}F_{α}(X_{ρ})(i)[2i], where α := {x_{1}, …, x_{r}} ∈ K_{r}^{M}(k)/2, and F_{α}(X_{ρ}) is so-called higher form of 4-dimensional quadric X_{ρ}, and F_{α}(X_{ρ}) is an extension (in DM_{eff.}^{−}) of M(X_{X_{φ}}), M(X_{X_{φ}^{2^{r}}})(2^{r})[2^{r+1}], M(X_{X_{φ}^{2^{r+1}}})(2·2^{r})[2·2^{r+1}], M(X_{X_{φ}^{2^{r+1}}})(3·2^{r} −1)[3·2^{r+1}−2], M(X_{X_{φ}^{2^{r}}})(4· 2^{r} − 1)[4 · 2^{r+1} − 2] and M(X_{X_{φ}})(5 · 2^{r} − 1)[5 · 2^{r+1} − 2].

More precisely, there exists the following diagram in DM_{eff.}^{−}(k) (we drop M(−) from the notations):

\begin{aligned}& X_{X_{φ}} & X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1}] & & & X_{X_{φ}}(5 · 2^{r} − 1)[5 · 2^{r+1} − 2] \\ ↗ ⋆ & ↓[1] & ↗ ⋆ & ↓[1] & … & ↗ ⋆ & ↓[1] \\ F_{α}(X_{ρ}) & ⟵ & Y_{1} & ⟵ & Y_{2} & Y_{5} & ⟵ & 0 \end{aligned} ,

(see [14], Lemma 3.23, Theorem 3.1, and use octachedron axiom).

In particular, we have an exact sequenses:

Hom(F_{α}(X_{ρ}), ℤ(2^{r} + 1)[2^{r+1} + 2]) ← Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) ←

← Hom(Y_{1}, ℤ(2^{r} + 1)[2^{r+1} + 1]).

and

Hom(Y_{2}, ℤ(2^{r} + 1)[2^{r+1} + 1]) ← Hom(Y_{1}, ℤ(2^{r} + 1)[2^{r+1} + 1]) ←

← Hom(X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1}], ℤ(2^{r} + 1)[2^{r+1} + 1]) ← Hom(Y_{2}, ℤ(2^{r} + 1)[2^{r+1}]).

Notice, that Y_{2} is an extension of some X_{P} (j)[2j], where P are smooth projective varieties and j ⩾ 2^{r+1} > 2^{r} + 1. By [17], Corollary 2.2(1), we have: Hom(Y_{2}, ℤ(2^{r} + 1)[∗]) = 0 and, consequently, Hom(Y_{1}, ℤ(2^{r} +1)[2^{r+1}+1]) = Hom(X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1}], ℤ(2^{r} +1)[2^{r+1}+1]), and the later group is equal to: Hom(X_{X_{φ}^{2^{r}}} , ℤ(1)[1]).

So, we have an exact sequence:

Hom(F_{α}(X_{ρ}), ℤ(2^{r} + 1)[2^{r+1} + 2]) ← Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) ← ← Hom(X_{X_{φ}^{2^{r}}} , ℤ(1)[1]),

where the last map is induced by the map f : X_{X_{φ}} → X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1}+ 1], which is the composition: X_{X_{φ}} → Y_{1}[1] → X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1} + 1] in the diagram above.

By [13], Corollary 3.2.1, Hom(X_{X_{φ}^{2^{r}}} , ℤ(1)[1]) = H^{0}(X_{X_{φ}^{2^{r}}} , O^{∗}) = = H^{0}(Spec(k), O^{∗}) = Hom(ℤ, ℤ(1)[1]), and this isomorphism is induced by projection pr : X_{X_{φ}^{2^{r}}} → ℤ (as usually, I’m dropping M(−)).

So, we get an exact sequence:

Hom(F_{α}(X_{ρ}), ℤ(2^{r} + 1)[2^{r+1} + 2]) ← Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) ← ← Hom(ℤ, ℤ(1)[1]),

where the last map is induced by the composition

g : X_{X_{φ}} →^{f} X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1} + 1] ⟶^{pr(2^{r})[2^{r+1}+1]} ℤ(2^{r} + 1)[2^{r+1} + 1].

Lemma 2.7 .

g = 0

Proof By the very construction of the diagram above - see [14], the proof of Theorem 3.1, the composition Y_{1}[1]→X_{X_{φ}^{2^{r}}} (2^{r})[2^{r+1}+1]→ℤ(2^{r})[2^{r+1}+1]

equals the composition: Y_{1}[1]→F_{α}(X_{ρ})[1]→X_{φ}[1] ⟶^{α_{2^{r}} [1]} ℤ(2^{r})[2^{r+1}+ 1], where α_{2^{r}} ∈ Hom(X_{φ}, ℤ(2^{r})[2^{r+1}]) = CH^{2^{r}}(X_{φ}) is given by the class of a plane section of codimension 2^{r}. Hence, g = 0.

Now, we can finish the proof of the Main Theorem .

From Lemma 2.7 it follows that the map: Hom(X_{X_{φ}}, ℤ(2^{r} +1)[2^{r+1} + 2]) ← Hom(ℤ, ℤ(1)[1]) is trivial. And we have an exact sequence:

Hom(F_{α}(X_{ρ}), ℤ(2^{r} +1)[2^{r+1} +2]) ← Hom(X_{X_{φ}}, ℤ(2^{r} +1)[2^{r+1} +2]) ← 0

Since F_{α}(X_{ρ}) is a direct summand in the motive of X_{φ}, we have (by above) an exact sequence:

Hom(X_{φ}, ℤ(2^{r} + 1)[2^{r+1} + 2]) ← Hom(X_{X_{φ}}, ℤ(2^{r} + 1)[2^{r+1} + 2]) ← 0

By Lemma 2.6 , the group Hom(X_{X_{φ}}, ℤ(2^{r}+1)[2^{r+1}+2]) is nontrivial, on the other hand it has exponent 2 (since 2^{r+1} + 2 > 2^{r} + 1). That means that the group Hom(X_{φ}, ℤ(2^{r} + 1)[2^{r+1} + 2]) contains a torsion element of exponent 2.

By [15], Theorem 4.2.9, we have: Hom(X_{φ}, ℤ(2^{r} + 1)[2^{r+1} + 2]) = CH^{2^{r}+1}(X_{φ}). The Main Theorem is proven.

References

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